so that the problem in redundance of linear dependent equation inside the resul- tant matrix is ...... [45] J. B. Rosser and N. Papamichael. A power series solution ...
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CITY UNIVERSITY OF HONG KONG
Numerical Solution of Partial Differential Equations with the Tau-Collocation Method
Submitted to Department of Mathematics in Partial Fulfillment of the Requirements for the Degree of Master of Philosophy
by
Sam Chi Ngai
September 2004
Abstract
The focus on this thesis is to present our research results in further developing the idea of the Tau-Collocation Method proposed by Liu in 1988. The Tau-Collocation Method, based on the Tau Method, is a numerical method for approximating the solution of ordinary differential equations (ODEs). In this thesis, we define the format of perturbation term for the Tau-Collocation Method so that the idea of the Tau-Collocation Method becomes a complete method. Also, we successfully simulate the recursive formulation of the Tau Method by the Tau-Collocation Method with same perturbation term in both cases. Moreover, we extended the idea and formulation of the Tau-Collocation Method to approximate the solution of 2-dimensional and 3-dimensional linear partial differential equations (PDEs). The format of perturbation term for the Tau-Collocation Method for PDEs is extended from the case for ODEs. It restricts the selection and number of collocation points so that the problem in redundance of linear dependent equation inside the resultant matrix is overcame. Besides, we discuss the approximation to the solution of nonlinear ODEs and PDEs by the Tau-Collocation Method incorporated with iterative schemes in this thesis. And, we proposed a new technique, the Tau-Collocation Method incorporated with the Adomian’s polynomials, to tackle the nonlinear ODEs and PDEs. No linearization process is required when this new technique is applied and this new technique can be applied to the nonlinear ODEs and PDEs with highly nonlinearity. It is a blank new idea in the field of the spectral methods, included the Tau Method. Also, we proposed the Segmented Tau-Collocation Method for the solution of ODEs and PDEs defined on a large domain or experienced with a sharp change over the domain. Moreover, the Tau-Collocation Method with the singularity subtraction technique is proposed in this thesis for tackling the PDEs with boundary singularities. Two problems in the field of linear elastic fracture mechanics are selected to test the effectiveness of the Tau-Collocation Method incorporated with the singularity subtraction technique.
Acknowledgment
I would like to thank City University of Hong Kong for providing me a very good platform to study and research. And, I would like to express my gratitude to my supervisor, Dr. K.M. Liu, for his valuable advices and supports. Thanks Dr Liu to extend my knowledge in mathematics and give me a valuable experience in research training. Finally, I express my thanks to professors and colleagues for their comments, supports and encouragements.
Table of Contents
Chapter 1 Introduction
1
1.1
About This Thesis
1
1.2
History of the Tau Method
3
Chapter 2 The Tau Method
6
2.1
Basic Idea of the Tau Method
6
2.2
Formulation of the Tau Method for the Numerical Solution of
8
Ordinary Differential Equations 2.2.1
2.2.2
Recursive Formulation of the Tau Method
9
2.2.1.1
Undefined Canonical Polynomials
16
2.2.1.2
Multiple Canonical Polynomials
20
Operational Approach to the Tau Method for Ordi-
21
nary Differential Equations
Chapter 3 The Tau-Collocation Method 3.1
Tau-Collocation Method for the Numerical Solution of Ordi-
29 29
nary Differential Equations 3.1.1
Basic Idea of the Tau-Collocation Method for Ordi-
30
nary Differential Equations 3.1.2
Formulation of the Tau-Collocation Method for Ordi-
31
nary Differential Equations 3.1.3
Tau-Collocation Method with Different Format of Per-
42
turbation Term 3.1.4
Error Estimation of the Tau-Collocation Method for Ordinary Differential Equations
47
3.2
Tau-Collocation Method for the Numerical Solution of Partial
49
Differential Equations 3.2.1
Basic Idea of the Tau-Collocation Method for 2-
51
dimensional Partial Differential Equations 3.2.2
Formulation of the Tau-Collocation Method for 2-
55
dimensional Partial Differential Equations 3.2.3
Error Estimation of the Tau-Collocation Method for
69
2-dimensional Partial Differential Equations 3.2.4
Basic Idea of the Tau-Collocation Method for 3-
72
dimensional Partial Differential Equations 3.2.5
Formulation of the Tau-Collocation Method for 3-
77
dimensional Partial Differential Equations
Chapter 4 Advanced Topics in the Tau-Collocation Method 4.1
Numerical Solution of Nonlinear Ordinary and Partial Differ-
88 88
ential Equations 4.1.1
Tau-Collocation Method Incorporated with Linear It-
89
erative Scheme 4.1.1.1
Nonlinear Ordinary Differential Equations
89
4.1.1.2
Nonlinear 2-dimensional Partial Differential
91
Equations 4.1.2
Tau-Collocation Method Incorporated with Quadrate
93
Iterative Scheme 4.1.2.1
Nonlinear Ordinary Differential Equations
94
4.1.2.2
Nonlinear 2-dimensional Partial Differential
95
Equations 4.1.3
Tau-Collocation Method Incorporated with Adomain
97
Polynomial 4.1.3.1
Nonlinear Ordinary Differential Equations
98
4.1.3.2
Nonlinear 2-dimensional Partial Differential
101
Equations
4.2
Segmentation Techniques
103
4.2.1
Step by Step Tau-Collocation Method
105
4.2.2
Segmented Tau-Collocation Method
107
4.2.2.1
110
Segmented Tau-Collocation Method for Ordinary Differential Equtaions
4.2.2.2
Segmented Tau-Collocation Method for 2-
122
dimensional Partial Differential Equtaions
Chapter 5 Application to Linear Elastic Fracture Mechanics
122
5.1
Brief Introduction in Linear Elastic Fracture Mechanics
122
5.2
Singularity Subtraction Technique
123
5.3
Two Mathematical Model in Linear Elastic Fracture Mechan-
124
ics 5.3.1
Motz’ Problem
124
5.3.1.1
126
Subtracted Off the Boundary Singularities by the Singularity Subtraction Technique
5.3.1.2
Formulation
by
the
Segmented
Tau-
127
Collocation Method 5.3.1.3 5.3.2
Numerical Examples
129
Biharmonic Equation for Mode I Crack Problem
131
5.3.2.1
134
Subtracted Off the Boundary Singularities by the Singularity Subtraction Technique
5.3.2.2
Formulation
by
the
Segmented
Tau-
136
Collocation Method 5.3.2.3
Numerical Examples
Chapter 6 Conclusions and Discussions
136
139
6.1
Conclusions
139
6.2
Further Developments
141
References
144
1
Chapter 1
1.1
Introduction
About This Thesis
Most of the phenomena that arise in mathematical physics and engineering fields, such as fluid dynamic, quantum mechanics, fracture mechanics and electricity, can be described by ordinary differential equations (ODEs) and partial differential equations (PDEs). For examples, heat flow and wave propagation phenomena. A complex engineering problem is first formed as an accurate physical model and this physical model is then transferred to mathematical model which is typically described in ODEs and PDEs. Hence, it is important to find out the solution of the mathematical model in order to resolve the engineering problems. Since it is too complicated to have an exact solution of the mathematical model for engineering problems, the solution is usually obtained by numerical methods, such as the finite difference method, the finite element method, the boundary element method and the spectral methods, that give an accuracy approximate solution to the mathematical model and implement upon digital computers. Therefore, researching in the field of numerical method is to develop an efficient, accurate, simple and widely applicable mathematical method and enhance an existing method for servicing the engineering purposes. The focus on the field of numerical methods is to discuss the mathematical ideas, algorithms and formulations of different kinds of ODEs and PDEs rather than a specific engineering problem. In this thesis, discussions will be focus on the field of numerical method. And, the discussions will be narrowed down to the Tau Method and our recently developed methods, the Tau-Collocation Method, which is based on the Tau Method. The Tau Method and the Tau-Collocation Method are a kind of numerical methods for approximating the solution of linear and nonlinear ODEs and PDEs defined on a finite domain with initial, boundary and mixed conditions. The contents of this thesis are divided into five chapters. Chapter 1 in this thesis is introduction. The
2
history of the Tau Method will be presented in this chapter. In Chapter 2, the detail ideas and two usual formulations of the Tau Method for ODEs, the recursive formulation of the Tau Method and the operational approach to the Tau Method, will be discussed. This chapter is the pre-knowledge of the discussion in the TauCollocation Method. In Chapter 3, the detail ideas and formulations of the TauCollocation Method for linear ODEs and PDEs will be discussed. The basic ideas of the Tau-Collocation Method are as same as the basic ideas of the Tau Method except the new format of the perturbation term which will be completely presented in this chapter. For the formulation of the Tau-Collocation Method, it aims to convert the given differential problems into a system of linear algebraic equations by collocating the differential equations at the zeros of the orthogonal polynomials that appeared in the perturbation term. These matters for the cases of both ODEs and PDEs are our research results and will be discussed deeply in this chapter. In Chapter 4, the Tau-Collocation Method incorporates with other methods and techniques will be discussed. The Tau-Collocation Method incorporated with the linear iterative scheme, quadratic iterative scheme and Adomian’s polynomials will be introduced in this chapter. Such combinations of methods are used in approximating the solution of nonlinear ODEs and PDEs. The idea of the Tau-Collocation Method incorporated with Adomian’s polynomials is a blank new idea in the field of the spectral methods, included the Tau Method, and is our latest research result. The Tau-Collocation Method incorporated with the segmentation technique is also introduced in this chapter for the solution of the differential equation defined on a large domain or the solution of given differential problem experienced with a sharp change over the domain. Such combined methods are named as the step by step Tau-Collocation Method and the segmented Tau-Collocation Method, where the details of the segmented Tau-Collocation Method are first proposed in this thesis. In Chapter 5, the applications of the Tau-Collocation Method are discussed. Two linear elastic fracture mechanic problems solved by the segmented Tau-Collocation Method incorporated with the singularity subtraction technique are discussed in this chapter. The examples show the effectiveness of the Tau-Collocation Method.
3
In Chapter 6, we give a conclusion of our research work. Some ideas for further development on the Tau-Collocation Method will be discussed in this chapter.
1.2
History of the Tau Method
The Tau Method was first proposed by L´anczos [18] in 1938. In his paper, the Tau Method was designed to find out an approximate polynomial solution of a given ODE. The idea of the Tau Method is to approximate the solution of given a ODE by substituting an n th degree polynomial solution, with n + 1 unknown coefficients into the perturbed problem, and adding a perturbation term to the right-hand side of the ODE. Then, the perturbed problem is transferred to a balanced system of linear algebraic equations. Finally, this system of linear algebraic equations is solved and all of the unknown coefficients as well as the approximate solution can be obtained. Since then, many researchers, included L´anczos, Ortiz, Samara, Liu and El-Daou, put great effort into further development of the Tau Method in both theoretical and application level. The theoretical development of the Tau Method provides its basic formulations and improves the computational performances of its formulation. With these basic formulations, the idea of the Tau Method can then be implemented. In 1956, L´anczos [19] proposed a recursive formulation of the Tau Method and Ortiz [31, 32] gave more detail on the formulation process in later time. This formulation mainly depends on the sequence of canonical polynomials {Qn (x)}. From 1981 to 1983, Ortiz and Samara [39, 40] introduced an operational approach to the Tau Method. The main idea of the operational approach is to transform a differential problem into a system of linear algebraic equations which has simpler implementation in computer programming. In 1994, El-Daou and Ortiz [8] proposed a recursive formulation of collocation in terms of canonical polynomials technique. This technique only requires O(N ) arithmetic operations. In 1997, Bunchaft [7] enhanced the applicability of the recursive formulation of the Tau Method to handle all kinds of
4
linear ordinary differential operator. The recent theoretical development of the Tau Method is done by Ghoreishi and Hosseini [13] in 2004. They proposed a new preconditioner for the iterative solver of the Tau Method. The method shows a superior performance. Before 1984, the Tau Method could not directly apply to approximate PDEs. Nevertheless, PDEs could be approximated by incorporating the Tau Method with the Method of Line. This hybrid method was named as the Tau-Line Method and was introduced by Liu and Ortiz [24] and El Misiery and Ortiz [10]. It was not until 1984, Ortiz and Samara [41] extended the operational approach to the Tau Method so as to approximate the solution of 2-dimensional partial differential equations directly. From 1998 to 2003, Jos´e Matos, Maria Jo˜ao Rodrigues and Paulo B. Vasconcelos [27, 44] extended the operational approach to handle n-dimensional partial differential equations. The application development of the Tau Method enhances its applicability in approximating different types of ordinary and partial differential equations. Many author successfully solved various differential models in real life applications by using the Tau Method only or incorporating the Tau Method with other methods or techniques. For examples, Aliabadi and Ortiz [4] tried to apply Tau Method repeatedly for solving the moving and free boundary value problem in 1998. Liu and Ortiz [24, 25] tackled the eigenvalue problems for ordinary and partial differential equations directly by the Tau Method since 1983. Also, Freilich and Ortiz [12] handled system of differential equations directly by the Tau Method in 1982. Ortiz, Pham Ngoc Dinh and Pun [34, 36] introduced the Tau Method incorporated with the Newton’s iterative method for solving nonlinear differential equations since 1985. Furthermore, Ortiz [33] proposed the step by step Tau Method in 1975, Onumanyi and Ortiz [30] proposed the segmented Tau Method in 1984 and Ortiz and Pun [37] proposed the Tau-Elements Method in 1986. The ideas of these three methods are based on the Tau Method incorporated with the segmentation technique. And, these methods are used to approximate the solution of the differential equation defined on a large domain or the solution of the differential problems experienced with a sharp change over the domain. The step by step Tau Method can be implemented in
5
parallel programming which is introduced by Escalante [11]. In 1998, Pan and Liu [43] applied the Tau Method incorporated with the singularity subtraction technique to solve the partial differential problems with boundary singularities and tackle the linear elastic fracture mechanics problems. The application of the Tau Method illustrates the effectiveness and widely applicability of the Tau Method and also shows that the Tau Method is a powerful tool for finding out numerical solutions in a variety of differential equations. Beside the theoretical and application development of the Tau Method, Ortiz, El-Daou and A. Pham Ngoc Dinh [9, 35] were working on the structural relations between the Tau Method and other numerical methods, such as the finite element method and the Galerkin method. This approach of studies is fresh in the field of numerical analysis and shows the philosophical feature of the Tau Method, which is the main different between the frameworks of spectral methods. Thanks to the authors putting great effort on researching in the Tau Method and making the Tau Method to become a very powerful numerical method. However, when the given ODEs or PDEs contain variable coefficients or non-polynomial functions, a pre-approximation process is required to convert the non-polynomial terms into polynomials by introducing artificial differential problems which is solved by Tau Method. In order to simplify the computational procedure of the Tau Method, Liu [21] proposed another version of the Tau Method, the Tau-Collocation Method, in 1988. The idea of Tau-Collocation Method was successfully applied into differential eigenvalue problems by Liu [22]. Nevertheless, the perturbation term of the Tau-Collocation Method had not yet defined well. The development of the TauCollocation Method is continued in our research and the results are discussed in this thesis.
6
Chapter 2
The Tau Method
The Tau Method is a kind of numerical method for approximating the solution of differential equations with given supplementary conditions, included initial conditions, boundary conditions and mixed conditions. It is completely different with the other numerical methods, such as the finite difference method and the finite element method, but can be described as the spectral method with same test functions and trial functions, where the trial functions does not satisfy the supplementary conditions. The solution of given differential problem is approximated by a polynomial defined on the global domain when the Tau Method is applied. The approximate polynomial solution is very accuracy with convergence of infinite order. Also, the formulation of the Tau Method is very simple compared with other numerical methods. No integration process, pre-defining for approximate solutions and discretization are required when the Tau Method is applied. In this chapter, the basic idea of the Tau Method will be discussed. Also, the recursive formulation and operational approach to the Tau Method for the numerical solution of linear ordinary equations will be discussed in this chapter. This background information of the Tau Method is the pre-knowledge of the Tau-Collocation Method.
2.1
Basic Idea of the Tau Method
The Tau Method was first conceived by L´anczos [18] in a paper on the Journal of Mathematical Physics. He gave an idea to find out the numerical solution of linear differential equation with rational coefficients in a finite interval by substituting an n th degree polynomial into the given problem and adding a perturbation term on the right hand side of the differential equation. The added perturbation term must satisfy two requirements. The first requirement is to balance the system of linear algebraic equations when an n th degree polynomial with n+1 unknown coefficients is
7
substituted into the differential equation under consideration . Then, the system of linear algebraic equations becomes solvable. Secondly, the perturbation term must be sufficiently small. The idea of perturbation term is the philosophical feature of the Tau Method and is the major difference between the Tau Method and the spectral method. Consider a linear differential equation Dy(x) = f (x) ,
x ∈ [a, b]
,
(2.1)
subject to the supplementary conditions (lr , y) = σr ,
r = 1(1)ν
,
(2.2)
where D :=
αi ν X X
pij x j
i=0 j=0
di dx i
(2.3)
is the class of linear differential operators of order ν with polynomial coefficients. And, lr is a linear point evaluation functional acting on the differentiable function y(x) and its derivatives. Therefore, (lr , y) = σr , r = 1(1)ν, here stands for the initial, boundary or mixed conditions of problem (2.1)-(2.2). The basic idea of the Tau Method is to approximate the exact solution of problem (2.1)-(2.2) by substituting an n th degree polynomial yn (x) =
n X
ai x i
,
(2.4)
i=0
which is defined as Tau approximant, into problem (2.1)-(2.2). Then, an overdetermined system with n+ν +1 linear algebraic equations and n+1 unknown coefficients ai , i = 0(1)n, is formed. In order to find out an approximate polynomial solution yn (x), a perturbation term Hn (x) is added to the right hand side of equation (2.1). So problem (2.1)-(2.2) becomes Dyn (x) = f (x) + Hn (x) ,
x ∈ [a, b]
,
(2.5)
subject to the supplementary conditions (lr , yn ) = σr ,
r = 1(1)ν
,
(2.6)
8
which is defined as the Tau problem(or called the perturbed problem). In order to minimize the value of the perturbation term, L´anczos suggested that the perturbation term should be in Chebyshev polynomial basis because the Chebyshev polynomial, Tn (x), is the best algebraic approximation by a polynomial of degree n and it oscillates an uniform magnitude in x ∈ [−1, 1] and the perturbation term, |Hn (x) − 0|, in range [a, b] does so. Therefore, the perturbation term is chosen in Hn (x) =
ν−1 X
∗ τn−i T n−i (x)
(2.7)
i=0
where τn−i , i = 0(1)ν − 1, are free parameters which balance the overdetermined sys∗ tem of linear algebraic equations and Tn−i (x), i = 0(1)ν − 1, is the shifted Chebyshev
polynomial defined on [a, b]. In fact, the perturbation term can be used in other type of polynomials other than the Chebyshev polynomials under different situations. For examples, the Legendre polynomial basis is always used for solving the singular perturbation problems, the Laguerre polynomial basis is used for solving the differential equation defined on [0, ∞) and the Hermite polynomial basis is used for solving the differential equation defined on (−∞, ∞). Finally, the solution of problem (2.1)-(2.2) is approximated by the approximate polynomial solution (2.4) with Tau degree n. The approximate polynomial solution (2.4) here is the exact polynomial solution of the Tau problem (2.5)-(2.6).
2.2
Formulation of the Tau Method for the Numerical Solution of Ordinary Differential Equations
Once the Tau problem (2.5)-(2.6) is defined, the next step is to formulate the Tau problem and find out its exact solution. This exact solution, which is called the Tau approximant, is the approximate solution of the given ordinary differential problem (2.1)-(2.2). There are two ordinary formulations of the Tau Method, they
9
are the recursive formulation of the Tau Method and the operational approach to the Tau Method which will be discussed in the following sections.
2.2.1 Recursive Formulation of the Tau Method The first conceived formulation of the Tau Method is the recursive formulation which was proposed by L´anczos [19] in 1956 and was further developed by Ortiz [31, 32]. The central idea of recursive formulation is the mapping in accordance with linear differential operator D, i.e. equation (2.3), between the set of independent variables and the unique sequence of canonical polynomials {Qn (x)}, n ∈ N = {0, 1, 2, · · ·}, associated with D. Note that there exists a unique sequences of {Qn (x)} for any linear differential operator D. The canonical polynomials are defined by the functional relationship DQn (x) = x n ,
n ∈ N = {0, 1, 2, · · ·} .
(2.8)
With the functional relationship, the Tau approximant yn (x), defined on equation (2.4), can be written in term of canonical polynomials starting from the Tau problem (2.5) with perturbation term (2.7), which is Dyn (x) = f (x) +
ν−1 X
∗ τn−i T n−i (x)
i=0
⇒
Dyn (x) =
δ X
fk x k +
ν−1 X
τn−i
i=0
k=0
n−i X
(n−i)
cj
xj
j=0
then yn (x) =
δ X k=0
fk Qk (x) +
ν−1 X i=0
τn−i
n−i X
(n−i)
cj
Qj (x)
(2.9)
j=0
(n−i)
where fk , k = 0(1)δ, is the coefficient of x k of f (x) and c j
, j = 0(1)n − i, is the
coefficient of x j of the (n − i) th Chebyshev polynomial. Note that if function f (x) is not a polynomial, it will first be approximated by a polynomial which can be generated from an artificial initial value problem with exact solution of the function f (x). This artificial problem is solved by Tau Method and then a polynomial
10
approximate solution of the function f (x) can be obtained. In equation (2.9), it is clear that the Tau approximant yn (x) depends only on a sequence of canonical polynomials {Qn (x)}. If the sequence of canonical polynomials {Qn (x)} is found, all of the unknown parameters τn−i , i = 0(1)ν − 1, can be determined by the supplementary conditions and the Tau approximant yn (x), i.e. the approximate solution of the given problem (2.1)-(2.2), can also be found. In this stage, it is focused on finding out the sequence of canonical polynomials {Qn (x)} instead of the Tau approximant yn (x). In fact, the sequence of canonical polynomials {Qn (x)} can be generated recursively by Ortiz’s recursive formulation which was proposed by Ortiz [31] in 1969. Ortiz introduced the generating polynomial gn (x) of order n associated with D, which is in the form gn (x) := Dx
n
=
γ X
(n)
bi xi
i=0
(n)
where b i
is the coefficient of x i . His idea was to generate the recurrence rela-
tion of the sequence of canonical polynomials {Qn (x)} by inverting the generating polynomials gn (x). For degree γ ≥ n and by equation (2.8), we have Dx
n
=
γ b (n) γ x
+ D{
γ−1 X
(n)
b i Qi (x)} .
i=0
Therefore, x
γ
=
1 (n) bγ
n
D{x −
γ−1 X
(n)
b i Qi (x)}
i=0
and by equation (2.8) again, we have Qγ (x) =
1 (n) bγ
"
xn −
γ−1 X i=0
(n)
b i Qi (x)
#
.
(2.10)
It is obvious that the recurrence relation of the sequence of canonical polynomials {Qn (x)} is successively formed and can be generated without a given initial value. The following examples show how the sequence of canonical polynomials {Qn (x)} is generated.
11
Example 2.1 Consider D :=
d3 +1 . dx 3
We have Dx
n
d3 = + 1 xn 3 dx = n(n − 1)(n − 2)x n−3 + x n
= D [n(n − 1)(n − 2)Qn−3 (x) + Qn (x)]
,
then, we have Qn (x) = x n − n(n − 1)(n − 2)Qn−3 (x) ,
n ∈ N = {0, 1, 2, · · ·} .
Therefore, by the above recurrence relation, we can obtain Q0 (x) = 1 , Q1 (x) = x , Q2 (x) = x 2
,
Q3 (x) = x 3 − 6 , Q4 (x) = x 4 − 24x , .. .
Example 2.2 Consider D := x
d2 d + 2 dx dx
.
We have Dx
n
d2 d := x 2+ xn dx dx n−1 = n(n − 1)x + nx n−1 = D [n(n − 1)Qn−1 (x) + nQn−1 (x)] = D n 2 Qn−1 (x) ,
12
then, we have (n + 1) 2 Qn (x) = x n+1 ,
n ∈ N = {0, 1, 2, · · ·} .
Therefore, by the above recurrence relation, we can obtain Q0 (x) = x , 1 2 Q1 (x) = x , 4 1 3 Q2 (x) = x , 9 1 4 Q3 (x) = x , 16 1 5 Q4 (x) = x , 25 .. . As the sequence of canonical polynomials {Qn (x)} can be found, the approximate polynomial solution of given problem (2.1)-(2.2) can also be obtained. The following example is a demonstration of the recursive formulation of the Tau Method.
Example 2.3 Consider D :=
d +1 dx
and Dy(x) :=
d y(x) + y(x) = 0 , dx
x ∈ [0, 1]
,
(2.11)
with initial condition y(0) = 1 .
(2.12)
The exact solution of this problem is y(x) = e −x
.
The following steps are the main procedure of the recursive formulation of the Tau Method.
13
1. Setup the Tau problem: The Tau problem of problem (2.11)-(2.12) becomes d yn (x) + yn (x) = Hn (x) , dx
x ∈ [0, 1]
,
(2.13)
with initial condition yn (0) = 1
(2.14)
and we choose the perturbation term Hn (x) = τ T n∗ (x). 2. Generate the recurrence relation of the sequence of canonical polynomials {Q n (x)}: The recursive relation can be generated from d n Dx = + 1 xn dx and by equation (2.10), we have Qn (x) = x n − nQn−1 (x) . Therefore, Q0 (x) = 1 , Q1 (x) = x − 1 , Q2 (x) = x 2 − 2x + 2 Q3 (x) = x 3 − 3x 2 + 6x − 6 . .. .
3. Find out the Tau approximant yn (x) in term the unknown parameters τ : If we take the Tau degree n = 2, by equation (2.13), we have Dy2 (x) = τ (1 − 8x + 8x 2 ) ⇒
y2 (x) = D −1 {τ (1 − 8x + 8x 2 )}
14
and by the recurrence relation of the sequence of canonical polynomials {Qn (x)}, we have y2 (x) = τ (Q0 (x) − 8Q1 (x) + 8Q2 (x)) .
(2.15)
Substituting the sequence of canonical polynomials {Qn (x)} into equation (2.15), we have y2 (x) = τ (8x 2 − 24x + 25) .
(2.16)
4. Determine all value of the unknown parameters τ and then obtain the Tau approximant yn (x): The unknown parameter τ can be determined by substituting the initial condition (2.14) into the equation (2.16). Finally, the Tau approximant becomes y2 (x) =
8 2 24 x − x+1 . 25 25
In Table 1 we give the maximum absolute error E := max|y(x) − yn (x)| for Example 2.3 obtained from the Tau Method (Tau), the Galerkin method (GM) and truncated Taylor series (TSS) with degrees n = 2(1)6 of the approximate polynomial solutions. It shows that the Tau Method and the Galerkin method give similar results and give better convergence than the truncated Taylor series solution. And, in Figure 1 we give the graph of approximate polynomial solutions for Example 2.3 obtained from the Tau Method, the Galerkin method and truncated Taylor series with degree n = 2. The results are compared with the exact solution.
Since the sequence of canonical polynomials {Qn (x)} depends only on the differential operator D, it can be generated once for the same problem with different degrees in the same interval. However, there are some drawbacks. The generation of the sequence of canonical polynomials {Qn (x)} is hard to implement in computer program. Furthermore, some sequence of canonical polynomials Qi ∈ {Qn (x)}, for
15
0.9
0.8
0.7 y 0.6
0.5
0.4
0
0.2
0.4
0.6
0.8
1
x
Exact solution Tau Method Galerkin Method Truncated Taylor series
Figure 1: The graph of approximate polynomial solutions for Example 2.3 obtained from the Tau Method, the Galerkin method and truncated Taylor series with degree n = 2 of approximate polynomial solutions. The results are compared with the exact solution.
16
Table 1: The maximum absolute error E := max|y(x) − yn (x)| for Example 2.3 obtained from the Tau Method (Tau), the Galerkin method (GM) and truncated Taylor series (TTS) with degrees n = 2(1)6 of the approximate polynomial solutions. n
E (Tau)
E (GM)
E (TTS)
2
1.253 × 10 −2
1.062 × 10 −2
1.321 × 10 −1
3
6.792 × 10 −4
8.340 × 10 −4
3.455 × 10 −2
4
2.794 × 10 −5
4.901 × 10 −5
7.121 × 10 −3
5
9.685 × 10 −7
2.304 × 10 −6
1.213 × 10 −3
6
3.777 × 10 −8
9.070 × 10 −8
1.761 × 10 −4
0 6 i 6 n, may not define and may exist more than one polynomial which satisfy DQi = x i , for Qi ∈ {Qn (x)} and 0 6 i 6 n. The details will be discussed in the following sections. In addition, Froes Bunchaft [7] extended the recursive formulation to handle the mentioned difficulties directly.
2.2.1.1 Undefined Canonical Polynomials When the Ortiz’s recursive formulation in Section 2.2.1 has used in finding out the sequence of canonical polynomials {Qn (x)}, there may exist one or more Qi ∈ {Qn (x)}, for 0 6 i 6 n, which are not defined. These Qi were named as undefined canonical polynomials. Before discussing the undefined canonical polynomials, the height h of the linear differential operator D is introduced. Recall equation (2.3), the linear differential operator D is defined as D :=
αi ν X X i=0 j=0
pij x j
di dx i
.
The height h of the linear differential operator D is defined as h := max(αi − i) . i
(2.17)
17
h reflects the number of undefined canonical polynomials when h > 0. For h < 0, the number of undefined canonical polynomials is zero. The following example shows a sequence of canonical polynomials {Qn (x)} contained undefined canonical polynomials.
Example 2.4 Consider D :=
d x2 1 d2 + x + ( + ) , dx 2 dx 4 2
with h = max(−2, 0, 2) = 2 . We have 1 1 n+ Qn (x) = x n − n(n − 1)Qn−2 (x) − Qn+2 (x) , 2 4
n ∈ N = {0, 1, 2, · · ·} .
By the above recurrence relation, we can obtain Q0 (x) = undef ined , Q1 (x) = undef ined , Q2 (x) = 4 − 2Q0 (x) , Q3 (x) = 4x − 6Q1 (x) , Q4 (x) = 4x 2 + 12Q0 (x) − 40 , Q5 (x) = 4x 3 + 60Q1 (x) − 56x , .. . It is clear that Q0 (x) and Q1 (x) are undefined for n = 0, 1, 2, · · · and Qi ∈ {Qn (x)}, 2 6 i 6 n, contain the undefined canonical polynomials Q0 (x) and Q1 (x).
18
In order to overcome the existence of undefined canonical polynomials , Ortiz [32] introduced a more workable definition of Qn (x). The enhanced definition is DQn (x) = x n + Rn (x) where Rn (x) is a polynomial generated by {x k }, for k ∈ S. S is the set containing the index of k such that Qk dissatisfy DQk = x k . When the undefined canonical polynomials exist during computation, an extra term h−1 X
∗ τn−(ν+i) T n−(ν+i) (x)
(2.18)
i=0
will be added to the perturbation term Hn (x). The extra unknown parameters τn−(ν+i) , i = 0(1)h − 1, are determined by equations which are generated from setting the coefficients of Qk , for k ∈ S, to equal to zero. The following example demonstrates how to handle the undefined canonical polynomials.
Example 2.5 Consider a differential equation with linear differential operation D in Example 2.4, 2 d x2 1 d + x + ( + ) y(x) = 0 , x ∈ [0, 1] dx 2 dx 4 2 with boundary conditions y(0) = 1 , y(1) = 0 . The exact solution of this problem is y(x) = e −
x2 4
(1 − x) .
If we take the Tau degree n = 3, the perturbation term will become H3 (x) = τ3 T3∗ (x) + τ2 T2∗ (x) + τ1 T1∗ (x) + τ0 T0∗ (x) .
(2.19)
The term τ1 T1∗ (x) + τ0 T0∗ (x) in equation (2.19) is added because there are two undefined canonical polynomials. After substituting the series solution y3 (x) to the
19
Tau problem with simplification, we get y3 (x) = (−174τ3 − 8τ2 + 2τ1 )Q1 (x) +(95τ3 − 15τ2 − τ1 + τ0 )Q0 (x) +128τ3 x − 192τ3 + 32τ2 Then, the unknown parameters τi , i = 0(1)3, can be determined by the boundary conditions and setting the coefficient of Q0 (x) and Q1 (x) equal to zero. Finally, we get y3 (x) = 1 − x . Note that the degree of the Tau approximant yn (x) is different form the Tau degree n. In Table 2 we give the maximum absolute error E := max|y(x) − yn (x)| and the Tau approximant yn (x) for Example 2.5 obtained from the recursive formulation of the Tau Method with Tau degrees n = 3, 6, 9.
Table 2: The maximum absolute error E := max|y(x) − yn (x)| and the Tau approximant yn (x) with 10 digits accuracy for Example 2.5 obtained from the recursive formulation of the Tau Method with Tau degrees n = 3, 6, 9. n yn (x)
E
3
3.5088 × 10 −2
6
1−x
1 − 1.001435964x − 0.2579343890x 2
+0.3003585950x 3 − 0.04098824194x 4
9
1 − 0.9999998313x − 0.2499940799x 2
+0.2498619725x 3 + 0.03202127146x 4
−0.03308398668x 5 − 0.0005733563390x 6
+0.001768010360x 7
3.8106 × 10 −4 8.8297 × 10 −8
20
2.2.1.2 Multiple Canonical Polynomials Besides the existence of undefined canonical polynomials, there is another difficulty in applying Ortiz’s recursive formulation. The sequence of canonical polynomials {Qn (x)} may exist more than one polynomial which satisfy DQi = x i , for Qi ∈ {Qn (x)} and 0 6 i 6 n. This phenomenon is named as multiple canonical
¯ m (x) are defined and they are the multiple canonical polynomials. If Qm (x) and Q ¯ m (x) polynomials of order m associated with linear differential operator D, Qm (x)−Q ¯ m (x)} = 0. The following example shows must be a polynomial and D{Qm (x) − Q a sequence of canonical polynomials {Qn (x)} contained multiple canonical polynomials.
Example 2.6 Consider D := x
d d2 − (x + 1) +1 . dx 2 dx
We have (1 − n)Qn (x) = x k + n(−n + 2)Qn−1 (x) ,
n ∈ N = {0, 1, 2, · · ·} .
By the above recurrence relation, we can obtain Q0 (x) = 1 or
−x ,
Q1 (x) = undef ined , Q2 (x) = −x 2 , 1 Q3 (x) = − x 3 − 2 1 4 Q4 (x) = − x − 3 .. .
3 2 x , 2 4 3 x − 4x 2 3
,
¯ 1 (x) = −x and we have It is obvious that Q1 (x) = 1 or −x. Let Q1 = 1 and Q
¯ 1 (x)} = 0. D{Q1 (x) − Q
21
In order to overcome the existence of multiple canonical polynomials , an extra term w−1 X i=0
¯ m (x)} , τ¯i D{Qm (x) − Q
(2.20)
where m is the index of multiple canonical polynomials and w is the total number of multiple canonical polynomials with different index m, is added to the perturbation term Hn (x). The extra unknown parameters τ¯i , i = 0(1)w − 1, is determined by the equations which are generated from setting the coefficient of the undefined canonical polynomials Qk , for k ∈ S, equal to zero. When there are multiple canonical polynomials, undefined canonical polynomials must exist. But these undefined canonical polynomials are not necessarily handled by the method in Section 2.2.1.1. In addi¯ (x) are defined and they are the multiple canonical ¯ (x) and Q tion, if Q (x), Q m
m
m
polynomials of order m associated with linear differential operator D, Pan [42] suggested that two extra terms of index m should be added to the perturbation term ¯ (x) pair up. ¯ m (x) and Q no matter how Qm (x), Q m 2.2.2 Operational
Approach
to
the
Tau
Method
for
Ordinary
Differential Equations The operational approach to the Tau Method is another way to formulate the Tau Method. This formulation is first proposed by Ortiz and Samara [39, 40]. The main idea of the operational approach is to transform a differential problem into a system of linear equations which has simpler implementation in computer programming. The operational approach is starting form two related infinite matrices and their relations between the effects of differentiation and multiplication with respect to x of yn (x). Let
O
0
1 η=
O
0 2
0 ..
.
..
.
and µ =
0
O
1 0 1 0
O
1 ..
.
..
.
(2.21)
22
be two infinite matrices. Recall the n th degree polynomial yn (x) in equation (2.4), it can be rewritten in matrix form yn (x) =
n X
ai x i = a n x ,
(2.22)
i=0
where an = (a0 , a1 , a2 , · · · , an , 0, 0, 0, · · ·) and x = (1, x 1 , x 2 , · · ·)0 are infinite row vector and infinite column vector respectively. The effects of differentiation with respect to x of yn (x) is di yn (x) = an η i x . dx i
(2.23)
And, the effects of multiplication with x of yn (x) is x j yn (x) = an µ j x .
(2.24)
Then, the effects of differentiation with respect to x and multiplication with x and a constant pij of yn (x) is pij x j
where γij =
0
0
di yn (x) = an γij x , dx i
(2.25)
is a infinite matrix with first i zero rows and first j zero
0 β columns. The matrix β inside γij is a infinite diagonal matrix and 0, for r 6= t the (r, t) element of β = p (i+r−1)! , for r = t ij (r−1)!
.
Then, the differential operator D in equation (2.3) operated on yn (x) becomes Dyn (x) :=
αi ν X X i=0 j=0
where Π :=
Pν
i=0
P αi
j=0
pij x j
di yn (x) = an Π x , dx i
(2.26)
γij is a banded matrix with h non-zero elements above the
main diagonal and −d non-zero elements below the main diagonal. h here is the height of linear differential operator D defined in equation (2.17) and d := mini (di − i), where di is the smallest index j of the linear differential operator D in equation (2.3) with non-zero coefficient pij , is defined as the depth of
23
linear differential operator D. By equation (2.26), a differential problem can be transformed into a system of linear algebraic equations. There are two different between the operational approach and the traditional Tau Method. Firstly, the n th degree polynomial yn (x) should be changed from x basis to the Chebyshev basis. In fact, the polynomial basis can be changed to other type of orthogonal polynomial other than Chebyshev polynomial under different situations, such as the Legendre polynomial, the Laguerre polynomial or the Hermite polynomial. Secondly, the perturbation term Hn (x) in the Tau problem is implicit when the operational approach is applied. It is not required to determine the the perturbation term Hn (x) as well as the number of free parameters τi before the formulation process. Nevertheless, the perturbation term Hn (x) can be obtained by Hn (x) = Dyn (x) − f (x) after the Tau approximant has been obtained. In general, let yn (x) = an x = αn v
,
(2.27)
where αn = (α0 , α1 , α2 , · · · , αn , 0, 0, 0, · · ·) and v = Cx is orthogonal polynomial basis with coefficient matrix C. The coefficient matrix C is different when different orthogonal polynomial basis has been used. The following example shows the coefficient matrix C of the Chebyshev basis, the shifted Chebyshev basis and the Legendre basis.
Example 2.7 When the polynomial basis is the Chebyshev basis, the coefficient matrix 1 1 0 1 0 1 1 2 0 12 −1 0 2 −1 C= and C = 0 34 0 14 0 −3 0 4 3 4 1 1 0 −8 0 8 8 8 0 8 0 .. .. .. .. . . . .
O
O
.
24
When the polynomial basis is the shifted Chebyshev basis defined on [0, 1], the coefficient matrix
1
0 −1 1 2 −1 1 −4 4 0 2 C = 0 −3 −1 0 4 6 −12 8 1 1 −8 0 −8 0 8 24 −32 .. .. .. . . . 1 −1 2 1 −8 8 . C = −1 18 −48 32 1 −32 160 −256 128 .. .. . .
O
1
O 16 ..
.
O
⇒
And, when the polynomial basis is the Legendre basis, the coefficient matrix 1 1 0 1 0 1 1 1 3 2 −2 3 0 3 0 −1 2 and C = C= 5 3 2 0 − 32 0 0 0 2 5 4 3 1 15 35 4 8 8 5 0 7 0 0 −4 0 8 35 .. . .. .. . . . . .
O
O
.
By equation (2.27), we have an x = α n v ⇒ an C −1 v = αn v
.
Therefore, we have the relation between an and αn as an = αn C
.
(2.28)
25
Then, by equations (2.26)-(2.28), the differential operator D in equation (2.3) operated on yn (x) becomes Dyn (x) :=
αi ν X X i=0 j=0
pij x j
di yn (x) = an Π x dx i = αn C Π C −1 v bv = αn Π
,
(2.29)
b = C Π C −1 . As same as the idea of the Tau Method, the approximate where Π
polynomial solution yn (x) in equation (2.27) is substituted into ordinary differential problem (2.1)-(2.2). Note that yn (x) has infinite many terms here, thus the perturbation term is implicit. Then, we have Dyn (x) = f (x) ,
x ∈ [a, b]
,
(2.30)
subject to the supplementary conditions (lr , yn ) = σr ,
r = 1(1)ν
.
(2.31)
By equation (2.29), equation (2.30) becomes b v = fn C −1 v , αn Π
v ∈ [a, b]
,
(2.32)
where fn = (f0 , f1 , f2 , · · · , fn , 0, 0, · · · , 0) is the coefficient vector of the function f (x). Note that if function f (x) is not a polynomial, it will first be approximated by a polynomial which can be generated from an artificial initial value problem with exact solution of the function f (x). This artificial problem is solved by Tau Method and then a polynomial approximate solution of the function f (x) can be obtained. And, the supplementary conditions (2.31) becomes αn B = σ
,
(2.33)
where σ = (σ1 , σ1 , · · · , σν ) and B = (bij ) where bij = (lj , vi−1 ) for i = 1, 2, 3, · · · and j = 1(1)ν. By combining equation (2.32)-(2.33), we have αn G = s ,
(2.34)
26
b and s = (σ|fn C −1 ). In order to obtain a finite polynomial where G = (B|Π) solution, it is required to truncate the infinite matrix G to a (n + 1) × (n + 1) matrix
Gn , where n + 1 > ν + h, and truncate the infinite row vector s to a finite row vector sn with n + 1 elements. Gn is defined by the first n + 1 rows and n + 1 columns of the infinite matrix G and sn is first n + 1 element of infinite row vector s. Note that the cut off parts of the matrix G to Gn and s to sn are the perturbation term. Therefore, we have αn G n = s n
,
(2.35)
By taking transpose into both side of equation (2.35) and solve the transposed equation by usual method, all of the unknown coefficients αi , i = 0(1)n, can be obtained. By the relation between an and αn in equation (2.28), all of the unknown coefficients ai , i = 0(1)n, can also be obtained and so as the Tau approximant yn (x) in x basis. The following example is a demonstration of operational approach of the Tau Method.
Example 2.8 Refer to the ordinary differential problem in Example 2.3. The following steps are the main procedure of the operational approach to the Tau Method. b and fn C −1 by using shifted Chebyshev basis: 1. Find out the matrix Π b = C (η + I) C −1 Π 1 O 0 O 1 O 1 1 −1 1 0 2 2 1 + = 3 1 −4 4 8 2 0 1 .. .. .. .. .. . . . . . O O 1 O 2 1 = 0 8 1 .. .. . .
1 2 1 2
.. .
1 8
O ..
.
27
and
fn C −1
= (0, 0, 0, · · ·)
O
1 1 2
1 2
3 8
1 2
.. .
1 8
..
.
= (0, 0, 0, · · ·) .
2. Find out the matrices B and σ:
1
−1 and σ = (1) . B= 1 .. . 3. Find out the merged matrices G and s and the truncated matrices Gn and sn :
1 1 0 0 −1 2 1 0 · · · G= 1 0 8 1 .. .
If we take Tau degree n = 2, the 1 G2 = −1 1
and s = (1, 0, 0, 0, · · ·) .
truncated matrices G2 and s2 becomes 1 0 2 1 and s2 = (1, 0, 0) . 0 8
4. Determine all value of the unknown coefficients αi , i = 0(1)n: By taking transpose into both matrices G2 and s2 and solving it by the Gauss elimination method, we have α2 = (0.64, −0.32, 0.04, 0, 0, · · ·) .
28
5. Obtain the Tau approximant yn (x) in x basis: By equation (2.28), the Tau approximant yn (x) in x basis becomes y2 (x) = 0.32x 2 − 0.96x + 1 . Note that the Tau approximant yn (x) obtained from the recursive formulation of the Tau Method in Example 2.3 and operational approach to the Tau Method in this example are the same.
The idea of the Tau Method can be extended to approximate the solution of n-dimensional PDEs. The operational approach for 2-dimensional PDEs is first proposed by Ortiz and Samara [41] and the generalized version in n-dimensinoal PDEs is given by Jos´e Matos, Maria Jo˜ao Rodrigues and Paulo B. Vasconcelos [27, 44] in recent.
29
Chapter 3
The Tau-Collocaiton Method
The Tau-Collocation Method is based on the Tau Method. It is also used in finding the numerical solution of ordinary and partial differential equations in a domain with given supplementary conditions, included initial conditions, boundary conditions and mixed conditions. The Tau-Collocation Method provides a new formulation to the Tau Method in a simple way and enhances the applicability to approximate ordinary and partial differential equations with variable coefficients directly. Moreover, the Tau-Collocation Method gives a new format of the perturbation term of the Tau Method. The perturbation term becomes explicit and changeable when the Tau-Collocation Method is applied and it is selected for simplifying the formulation process rather than minimizing the error of the approximation. The Tau-Collocation Method for the numerical solution of ODEs and PDEs will be discussed in this chapter. These materials are our current research results and some of the results are first purposed in this thesis.
3.1
Tau-Collocation Method for the Numerical Solution of Ordinary differential Equations
The idea of the Tau-Collocation Method for ODEs is first given by Liu [21] in 1986. The method is applied successfully to the numerical solution of eigenvalue problems by Liu [22]. However, the theory, especially the format of the perturbation term, of the Tau-Collocation Method for ODEs has not yet developed well. In our research, we [26] give the detail theory and formulation of the Tau-Collocation Method for ODEs so that the suggested method becomes completed and becomes a powerful tool for approximating solution of ODEs.
30
3.1.1 Basic
Idea
of
the
Tau-Collocation
Method
for
Ordinary
Differential Equatoins Let Dv :=
Pν
di i=0 qi (x) dx i
∈ D, the class of linear differential operators of order P νr di be point ν with variable coefficients qi (x) and let Dvr |x=xr := i=0 qir (xr ) dx i x=xr
evaluation functional acting on y(x). The Tau-Collocation Method for ODEs is
designed to find out the polynomial approximate solution of a linear ODE with variable coefficients Dv y(x) = f (x) ,
x ∈ [a, b]
,
(3.1)
subject to the supplementary conditions Dvr |x=xr y(x) = σr (xr ) ,
r = 1(1)ν
.
(3.2)
As same as the idea of the Tau Method, the exact solution of the problem (3.1)-(3.2) is approximated by substituting an n th degree polynomial yn (x), in equation (2.4), into the problem itself. Then, the associated Tau problem of problem (3.1)-(3.2) is formed by adding a perturbation term Hn (x) to the right hand side of problem (3.1) and is defined as Dv yn (x) = f (x) + Hn (x) ,
x ∈ [a, b]
,
(3.3)
r = 1(1)ν
.
(3.4)
subject to the supplementary conditions Dvr |x=xr yn (x) = σr (xr ) ,
The perturbation term Hn (x) is selected differently from the Tau Method when the Tau-Collocation Method for ODEs is applied. Let τ = (τ0 , τ1 , · · · , τϕ−1 ) be the ϕ free parameters. The perturbation term Hn (x) is chosen as [a,b]
Hn (x) = gn (x; τ ) V n−ν+1 (x) ,
(3.5)
where gn (x; τ ) is a function of x with ϕ free parameters τj , j = 0(1)ϕ − 1, and [a,b]
V n−ν+1 (x), for n − ν + 1 > 0, is an orthogonal polynomial with degree n − ν + 1
31
defined on [a, b]. Usually, the orthogonal polynomial is chosen in the shifted Chebyshev or Legendre polynomial. Note that the free parameters τj , j = 0(1)ϕ − 1, inside function gn (x; τ ) balance the overdetermined system of linear algebraic equations. The format of perturbation term Hn (x) is selected for producing exactly [a,b]
n − ν + 1 zeros of the orthogonal polynomial V n−ν+1 (x). In the case shifted Chebyshev polynomial is used, the zeros can be generated by
b−a 2
cos (2(n−ν+1)−1)π + 2(n−ν+1)
a+b . 2
It is not necessary to consider the detail of the function gn (x; τ ), i.e. the number of free parameter ϕ, throughout the computation because the zeros of the orthogonal [a,b]
polynomial V n−ν+1 (x) are used for collocation during the formulation process of the Tau-Collocation Method for ODEs. In order to find out the approximate polynomial solution yn (x) from the Tau problem (3.3)-(3.4) with perturbation term (3.5), only n + 1 unknown coefficients ai , for i = 0(1)n, is required to found out while the free parameters τj , for j = 0(1)ϕ − 1, can be ignored. With ν linear algebraic equations are given from conditions (3.4), n − ν + 1 extra linear algebraic equations is required for determining the n + 1 unknown coefficients ai , for i = 0(1)n. These extra equations is given by collocating [a,b]
n − ν + 1 zeros of the orthogonal polynomial V n−ν+1 (x) into equation (3.3). Hence, [a,b]
the degree of the V n−ν+1 (x) is set as n−ν +1. Then, a system of n+1 linear algebraic equations with n + 1 unknown coefficients ai , i = 0(1)n, are formed. Finally, the system of linear algebraic equations is solved by the usual method and the Tau approximant yn (x) to the solution of problem (3.1)-(3.2) can be obtained.
3.1.2 Formulation
of
the
Tau-Collocation
Method
for
Ordinary
Differential Equations The formulation of the Tau-Collocation Method aims to convert the Tau problem (3.3)-(3.4) to a system of linear algebraic equations. It is starting form the following theorem.
32
Theorem 3.1 P Let yn (x) = nk=0 ak xk = an xn , where ak , for k = 0(1)n, are the unknown coeffi0
cients, an = (a0 , a1 , a2 , · · · , an ) and xn = (x 0 , x 1 , x 2 , · · · , x n ) . The i th derivative of yn (x) with respect to x can be written as di yn (x) = an Πi (x) , dx i where Πi (x) is a column vector and the r th Proof
0, for r = 0(1)i − 1 element of Πi (x) = r! x (r−i) , for r = i(1)n (r−i)!
.
It is proved directly by induction.
By the Theorem 3.1, equation (3.3) with perturbation term (3.5) can be converted to a matrix and it becomes Dv yn (x) =
ν X
[a,b]
qi (x) an Πi (x) = f (x) + gn (x; τ ) V n−ν+1 (x) ,
i=0
where ξn (x) =
⇒
x ∈ [a, b]
[a,b]
an ξn (x) = f (x) + gn (x; τ ) V n−ν+1 (x)
Pν
i=0 qi (x) Πi (x).
(3.6)
Let xk , k = 0(1)n − v, be the n − ν + 1 collocation
points for equation (3.6). The collocation points are selected as the zeros of the [a,b]
orthogonal polynomial V n−ν+1 (x), which is selected as shifted Chebyshev polynomial in usual. By collocating the collocation points into equation (3.6), we have an Γ = Fn−ν
,
(3.7)
where Γ = (ξn (x0 )|ξn (x1 )| · · · |ξn (xn−ν )) is an (n + 1) × (n − ν + 1) matrix and Fn−ν = (f (x0 ), f (x1 ), · · · , f (xn−ν )). Pr Let Φnr (x) = νi=0 qir (x) Πi (x), for r = 1(1)ν. Similarly, supplementary conditions (3.4) of the Tau problem can be written as a matrix form and it becomes an B = σ
.
(3.8)
33
where B
=
Φn1 (x1 )|Φn2 (x2 )| · · · |Φnν (xν )
σ = (σ1 , σ2 , · · · , σν ).
is a (n + 1) × ν matrix and
Combining equation (3.7)-(3.8), we have a system of linear algebraic equations an G = s n
(3.9)
where G = (B|Γ) is an (n + 1) × (n + 1) matrix and sn = (σ|Fn−ν ) is a row vector with n + 1 components. By taking transpose in the both sides of equation (3.9), we have 0
G an = s n
.
(3.10)
Then, by solving the system of linear algebraic equations (3.10) with the usual method, all unknown coefficients ai , for i = 0(1)n, can be obtained and so as the Tau approximant yn (x) to the solution of problem (3.1)-(3.2). The following example demonstrates the simply formulation process of the Tau-Collocation Method.
Example 3.1 Refer to the ordinary differential problem in Example 2.5. The following steps are the main procedure of the Tau-Collocation Method. 1. Setup the Tau problem: If we take Tau degree n = 3 and use the shifted Chebyshev basis in the perturbation term Hn (x), we have the Tau problem d2 d x2 1 y (x) + x y (x) + ( + )y3 (x) = gn (x; τ ) T 2∗ (x) , 3 3 dx 2 dx 4 2 with boundary conditions y3 (0) = 1 , y3 (1) = 0 .
2. Formulate the matrices Γ and Fn−ν :
x ∈ [0, 1]
,
34
By equation (3.6), we have (a0
a1
a2
x2 4
1 2
+ 3 x + x + 1x 4 2 a3 ) 2 + 2x 2 + x 4 + 1 x 2 4 5 2 6x + 3x 3 + x4 + 12 x 3
= gn (x; τ ) T 2∗ (x) . (3.11)
Then, we collocate the zeros of T 2∗ (x), i.e. ± 2√1 2 + 12 , into (3.11) and we have
(a0
a1
a2
0.6821383476 0.5053616524 1.435794890 0.2204551101 = (0, 0) a3 ) 3.954080987 0.8896892381 7.411092011 2.053731513
.
(3.12)
3. Formulate the matrices B and σ:
From the boundary conditions, we have (a0
a1
a2
1 1 0 1 = (1, 0) a3 ) 0 1 0 1
.
(3.13)
4. Formulate the system of linear algebraic equations and solve it to obtain the Tau approximant: Combining equations (3.12)-(3.13) and taking transpose in both side of the combined equation, we have the system of linear algebraic equations 1 0 0 0 1 1 1 1 0.6821383476 1.435794890 3.954080987 7.411092011 0.5053616524 0.2204551101 0.8896892381 2.053731513
By solving the equation (3.14), we can obtain the Tau approximant
1
0 0 0
y3 (x) = 1 − 0.9883874828x − 0.2380798838x 2 + 0.2264673666x 3
. (3.14)
35
with 10 digits accuracy. In Table 3 we give the maximum absolute error E := max|y(x) − yn (x)| for Example 3.1 obtained from the Tau-Collocation Method (TC) with degrees n = 1(1)8 of the approximate polynomial solutions. The results are compared with the results obtained from the recursive formulation of the Tau Method (Tau). Note that in the case of the same Tau degree is used, the degree of the Tau approximant generated from the recursive formulation of the Tau Method may be different from the one generated from the Tau-Collocation Method. The results show that the accuracy of the Tau-Collocation Method is similar to the Tau Method. Also, the graph of error curves of the approximate solutions for Example 3.1 obtained from the TauCollocation Method with degrees n = 7 and 8 are shown in Figure 2. It shows a remarkable equioscillatory behavior.
Table 3: The maximum absolute error E := max|y(x) − yn (x)| for Example 3.1 obtained from the recursive formulation of the Tau Method (Tau) and the TauCollocation Method (TC) with degrees n = 1(1)8 of approximate polynomial solutions. n
E (Tau)
E (TC)
n
E (Tau)
E (TC)
1
3.5088 × 10 −2
3.5088 × 10 −2
5
3.1631 × 10 −5
3.0562 × 10 −5
2
1.2640 × 10 −2
1.0865 × 10 −2
6
1.6337 × 10 −6
1.6506 × 10 −6
3
4.9440 × 10 −3
4.8886 × 10 −3
7
8.8297 × 10 −8
9.5120 × 10 −8
4
3.8106 × 10 −4
3.6397 × 10 −4
8
4.2741 × 10 −9
4.3420 × 10 −9
36
4E-9 8E-8
2E-9
4E-8
E
E
0 0
0.2
0.4
0.6
0.8
0 0
1
0.2
0.4
0.6
0.8
1
x
x -4E-8
-2E-9
-8E-8
-4E-9
n=7
n=8
Figure 2: The error curves of the approximate solutions for Example 3.1 obtained from the Tau-Collocation Method with degrees n = 7 and 8.
Example 3.2 Consider the linear differential equation discussed in Khalifa et al. [17] d4 y(x) + 4y(x) = 0 , dx 4
[−1, 1]
,
with supplementary conditions y(−1) = e 2
,
y(1) = 1 , 2
d y(−1) = 2e −2 tan(1) , dx 2 d2 y(1) = −2 tan(1) . dx 2 The exact solution of this problem is y(x) = sec(1)e 1−x cos x . The Tau-Collocation Method can be applied directly to this problem directly. In Table 4 we give the maximum absolute error E := max|y(x) − yn (x)| for
37
Example 3.2 obtained from the Tau-Collocation Method (TC) with degrees n = 4(2)18 of the approximate solutions and compared with the results of the Chebyshev expansion method (CEM) [17].
Table 4: The maximum absolute error E := max|y(x) − yn (x)| for Example 3.2 obtained from the Tau-Collocation Method (TC) and the Chebyshev expansion method (CEM) with degrees n = 4(2)18 of approximate polynomial solutions. n
E (TC)
E (CEM)
n
E (TC)
E (CEM)
4
1.8357 × 10 −2
−
12 9.0601 × 10 −11
3.57 × 10 −7
6
5.1610 × 10 −3
−
14 1.7822 × 10 −13
1.25 × 10 −9
8
1.6647 × 10 −5
4.22 × 10 −3
16 2.6947 × 10 −16
6.29 × 10 −12
10 3.9606 × 10 −8
2.91 × 10 −5
18 −
4.84 × 10 −14
Although it is not necessary to know the detail of gn (x; τ ) of perturbation term Hn (x) in equation (3.5) during the computation process, the next two examples show the detail of function gn (x; τ ) with different Tau degrees n. These two examples give a better understanding about the function gn (x; τ ) as well as the Tau-Collocation Method.
Example 3.3 Consider the linear differential equation (ex + 1)
d2 y(x) − y(x) = 0 , dx 2
[−1, 1]
with boundary conditions y(−1) = 1 + e , y(1) = 1 + e−1
.
,
38
The exact solution of this problem is y(x) = 1 + e−x
.
By Tau-Collocation Method , we have the following results with 20 digits accuracy: n = 3 y3 (x) = 1.9200481044281764545 − 0.99546478888337456066x +0.62303253038706732401x2 − 0.17973640476042689622x3 max|y(x) − y3 (x)| = 7.9952 × 10 −2 H3 (x) = 1.2460650607741346480ex − 1.0784184285625613773xex −0.6739830436540418065 − 0.08295363967918681664x −0.62303253038706732401x2 + 0.17973640476042689622x3 g3 (x; τ0 , τ1 ) = (2x2 − 1)−1 (2τ1 ex + 6τ0 xex + 2τ1 − 1.9200481044281764545 +(6τ0 + 0.99546478888337456066)x − τ1 x2 − τ0 x3 ) τ0 = −0.17973640476042689622 τ1 = 0.62303253038706732400
n = 4 y4 (x) = 1.9989881584206103941 − 0.98705120467064981184x +0.49974703960515259853x2 − 0.18814998897315164504x3 +0.044345436789480785815x4 max|y(x) − y4 (x)| = 4.9804 × 10 −3 H4 (x) = 0.99949407921030519706ex − 1.1288999338389098702xex +0.53214524147376942978x2ex − 0.99949407921030519704 −0.14184872916826005836x + 0.03239820186861683125x 2 +0.18814998897315164504x3 − 0.044345436789480785815x4 g4 (x; τ0 , τ1 ) = (4x3 − 3x)−1 (0.99949407921030519706ex + 6τ1 xex
39
+12τ0 x2 ex − 0.99949407921030519704 +(6τ1 + 0.98705120467064981184)x +(12τ0 − 0.49974703960515259853)x2 − τ1 x3 − τ0 x4 ) τ0 = 0.044345436789480785815 τ1 = −0.18814998897315164503
n = 5 y5 (x) = 2.0004900563382728630 − 0.99995268495730302324x +0.49736180471485234626x2 − 0.16649169257093018427x3 +0.045228773762118569175x4 − 0.0087568161155682493660x5 max|y(x) − y5 (x)| = 4.9006 × 10 −4 H5 (x) = 0.99472360942970469252ex − 0.99895015542558110562xex +0.54274528514542283010x2ex − 0.17513632231136498732x3ex −1.0057664469085681705 + 0.00100252953172191762x +0.04538348043057048384x2 − 0.00864462974043480305x3 −0.045228773762118569175x4 + 0.0087568161155682493660x5 g5 (x; τ0 , τ1 ) = (8x4 − 8x2 + 1)−1 (0.99472360942970469252ex −0.99895015542558110562xex + 12τ1 x2 ex + 20τ0 x3 ex −1.0057664469085681705 + 0.00100252953172191762x +(12τ1 − 0.49736180471485234626)x2 +(20τ0 + 0.16649169257093018427)x3 − τ1 x4 − τ0 x5 ) τ0 = −0.0087568161155682493660 τ1 = 0.045228773762118569175
40
n = 6 y6 (x) = 2.0000024949979804913 − 1.0001135267242646326x +0.50000062374949512283x2 − 0.16622171323941069190x3 +0.041629869396268541895x4 − 0.0088659536801261324115x5 +0.0014476466714996224315x6 max|y(x) − y6 (x)| = 2.5589 × 10 −5 H6 (x) = 1.0000012474989902457ex − 0.99733027943646415140xex +0.49955843275522250274x2ex − 0.17731907360252264823x3ex +0.043429400144988672945x4ex − 1.0000012474989902456 +0.00278324728780048120x − 0.00044219099427262009x 2 −0.01109736036311195633x3 + 0.001799530748720131050x4 +0.0088659536801261324115x5 − 0.0014476466714996224315x6 g6 (x; τ0 , τ1 ) = (16x5 − 20x3 + 5x)−1 (1.0000012474989902457ex −.99733027943646415140xex + .49955843275522250274x2ex +20τ1 x3 ex + 30τ0 x4 ex − 1.0000012474989902456 +0.00278324728780048120x − 0.00044219099427262011x 2 +(20τ1 + 0.16622171323941069190)x3 +(30τ0 − 0.041629869396268541895)x4 − τ1 x5 − τ0 x6 ) τ0 = 0.0014476466714996224315 τ1 = −0.0088659536801261324115
Example 3.4 Consider the linear differential equation cos2 x
d2 1 d 3 y(x) + sin(2x) y(x) + y(x) = 0 , 2 dx 2 dx 4
[0, 1]
,
41
with initial conditions y(0) = 1 , d y(0) = 1 . dx The exact solution of this problem is y(x) = 2
p p x x cos(x)sin( ) + cos(x)cos( ) . 2 2
By Tau-Collocation Method , we have the following results with 20 digits accuracy: n = 3 y3 (x) = 1 + x − 0.37997822745369237044x2 −0.27333703475632542749x3 max|y(x) − y3 (x)| = 3.1906 × 10 −3 H3 (x) = −0.75995645490738474088cos2(x) −1.6400222085379525649xcos2(x) + 0.5sin(2x) −0.37997822745369237044xsin(2x) −0.41000555213448814124x2sin(2x) + 0.75 + 0.75x −0.28498367059026927783x2 − 0.20500277606724407062x3 g3 (x; τ0 , τ1 ) = (8x2 − 8x + 1)−1 (2τ1 cos2 (x) + 6τ0 xcos2 (x) +0.5sin(2x) + τ1 xsin(2x) + 1.5τ0 x2 sin(2x) + 0.75 + 0.75x +0.75τ1 x2 + 0.75τ0 x3 ) τ0 = −0.27333703475632542748 τ1 = −0.37997822745369237044
n = 4 y4 (x) = 1 + x − 0.38268020291190776350x2 −0.24756299728670879988x3 − 0.019559125677695730085x4
42
max|y(x) − y4 (x)| = 1.5813 × 10 −3 H4 (x) = −0.76536040582381552700cos2(x) −1.4853779837202527993xcos2(x) −0.23470950813234876102x2cos2 (x) + 0.5sin(2x) −0.38268020291190776350xsin(2x) −0.37134449593006319982x2sin(2x) −0.039118251355391460170x3sin(2x) + 0.75 + 0.75x −0.28701015218393082262x2 − 0.18567224796503159991x3 −0.014669344258271797564x4 g4 (x; τ0 , τ1 ) = (32x3 − 48x2 + 18x − 1)−1 (−0.76536040582381552700cos2(x) +6τ1 xcos2 (x) + 12τ0 x2 cos2 (x) + 0.5sin(2x) −0.38268020291190776350xsin(2x) + 1.5τ1 x2 sin(2x) +2τ0 x3 sin(2x) + 0.75 + 0.75x − 0.28701015218393082262x2 +0.75τ1 x3 + 0.75τ0 x4 ) τ0 = −0.019559125677695730085 τ1 = −0.24756299728670879988
3.1.3 Tau-Collocation Method with Different Format of Perturbation Term The idea of perturbation term is the central idea and philosophical feature of the Tau Method. Ortiz, El-Daou and Pham Ngoc Dinh worked on the structural relations between the Tau Method and other numerical methods [9, 35]. They showed that the Tau Method with different format of the perturbation term can be used in simulating other numerical methods, such as the finite element method and the Galerkin method. In our research, we try to use the Tau-Collocation Method to
43
simulate the Tau Method by using the perturbation term which is defined as same as the recursive formulation of the Tau Method. In the case of the linear differential operator with polynomial coefficients, the approximate solution obtained from the Tau-Collocation Method may not be as accuracy as the result obtained from the recursive formulation of Tau Method. However, with the same perturbation term as the recursive formulation of the Tau Method, the Tau-Collocation Method becomes backward applicable to the recursive formulation of the Tau Method. The same Tau approximant, i.e. the same result, can be obtained from both methods. The formulation of the Tau-Collocation Method is slightly modified when the format of the perturbation term is chosen different form the suggested perturbation term in Section 3.1.1. In this section, it is restricted to discuss the linear differential equation with polynomial coefficients with the height h > 0. This restriction is limited by the applicability of the recursive formulation of Tau Method. Taking the Tau problem (2.5)-(2.6) into consideration, the perturbation term Hn (x) in equation (2.5) is chosen in the form which follows the rule of recursive formulation instead of the form in equation (3.5) and it is Hn (x) =
ν−1 X
∗ τn−i T n−i (x)
(3.15)
i=0
where τn−i , i = 0(1)ν − 1, is a free parameter which balances the overdetermined ∗ system of linear equations and Tn−i (x), i = 0(1)ν − 1, is shifted Chebyshev polyno-
mial. Moreover, there is some adjustments in the perturbation term Hn (x) due to the undefined canonical polynomials and multiple canonical polynomials. Like the recursive formulation, if there exist undefined canonical polynomials associated with the linear differential operator D in equation (2.3), it is required to add the extra term (2.18) to the perturbation term (3.15). Nevertheless, the tiresome of finding out the recursive relation of the canonical polynomials is avoided. Since the extra term barely depends on the height of linear differential operator D, it is only required to determine the height h by equation (2.17) instead of finding out the undefined canonical polynomials by generating the recursive relation. Also, it is not required to handle the multiple canonical polynomials associated with the linear differen-
44
tial operator D of the Tau problem (2.5)-(2.6) when the Tau-Collocation Method is applied, because the value of the extra term generated from multiple canonical polynomials always equals to zero. Consequently, the perturbation term can be formed without generating the recursive relation of the canonical polynomials. The formulation here is slightly different from the formulation of the Tau-Collocation Method while perturbation term in equation (3.5) is used. By Theorem 3.1, the Tau problem (2.5) can be converted to a matrix and it becomes an ξn (x) = f (x) + = f (x) +
ν−1 X
∗ τn−i T n−i (x)
i=0 ν+h−1 X
+
h−1 X
∗ τn−ν−i T n−ν−i (x)
i=0
∗ τn−i T n−i (x)
i=0
= f (x) + τn ςn (x) where τn = (τn , τn−1 , τn−2 , · · · , τn−(ν+h−1) ) and ∗ ∗ ∗ ςn (x) = (T n∗ (x), T n−1 (x), T n−2 (x), · · · , T n−(ν+h−1) (x)) t . Then, we have
an ξn (x) − τn ςn (x) = f (x) and let An+ν+h ζn (x) = f (x) where An+ν+h =
an |τn
and ζn (x) =
ξn |−ςn .
(3.16) Since there are totally
(n + 1) + (ν + h) unknowns, included unknown coefficients ai , i = 0(1)n, and free parameters τj , j = n(−1)n − (ν + h − 1), to be determined subject to ν supplementary conditions from equation (2.6), n + 1 + h equations are required to find out all of the unknowns from equation (3.16). The n + 1 + h equations can be generated by collocating uniform mesh points xk , where xk = a + k∆x for k = 0(1)n + h and ∆x =
b−a , n+h
into equation (3.16). Note that the Tau problem (2.5)-(2.6) is in the
range [a, b]. Therefore, we have An+ν+h Ψ = Fn+h
(3.17)
where Ψ = (ζn (x0 ), ζn (x1 ), · · · , ζn (xn+h )) is an (n + 1 + ν + h) × (n + 1 + h) matrix and Fn+h = (f (x0 ), f (x1 ), · · · , f (xn+h )).
45
By equation (3.8), the supplementary conditions (2.6) can be written as a matrix and it becomes
⇒
An+ν+h
an B = σ B = σ O
(3.18)
where B is an (n + 1) × ν matrix and the size of the zero matrix O is (ν + h) × ν. Combining the equations (3.17) and (3.18), we have a system of linear algebraic equations An+ν+h P = qn where P =
B |Ψ O
(3.19)
is an (n + 1 + ν + h) × (n + 1 + ν + h) matrix and qn = (σ|Fn+h )
is a row vector with (n + 1 + ν + h) components. Taking transpose in both sides of equation (3.19), we have 0
P An+ν+h = qn
(3.20)
By solving the system of equations (3.20) with the usual method, all unknown coefficients ai , i = 0(1)n, and free parameters τj , j = n(−1)n − (ν + h − 1), can be determined. Then, the Tau approximant yn (x) to the solution of problem (3.1)-(3.2) can also be obtained. It is much simpler to obtain the same Tau approximant by Tau-Collocation Method, especially for the case that the undefined canonical polynomials or multiple canonical polynomials exists. In table 6 we give eight sample problems with the same Tau approximant’s yn (x) obtained by recursive formulation of the Tau Method and Tau-Collocation Method. Some of the sample equations come from the Automation of the Tau Method [38]. Note that the sample equation 3, 6, 7 and 8 contain undefined canonical polynomials and the sample equation 4 contains multiple canonical polynomials. And, the following example is a demonstration of the Tau-Collocation Method with perturbation term which is defined as same as the Tau Method.
Example 3.5 Refer to the ordinary differential problem in Example 2.4. Firstly, it should be find
46
out the value of the height h. According to equation (2.17), we know that h = 2 in this problem. If we take Tau degree n = 3, the perturbation term of this problem is H3 (x) = τ3 T 3∗ (x) + τ2 T 2∗ (x) + τ1 T 1∗ (x) + τ0 T 0∗ (x) . By equation (3.16), we have
(a0
a1
a2
a3
τ3
τ2
τ1
τ0 )
x2 4
+
1 2
x+ + 4 2 + 2x 2 + x4 + 12 x 2 5 x 1 3 3 6x + 3x + 4 + 2 x = 0 (3.21) 32x 3 − 48x 2 + 18x − 1 2 8x − 8x + 1 2x − 1 1 . x3 4
1 x 2
Then, we use a set of uniform mesh points { 0, 0.2, 0.4, 0.6, 0.8, 1} to collocate into equation (3.21) and append the collocated matrix to the boundary conditions 1 1 0 1 0 1 0 1 = (1, 0) . (a0 a1 a2 a3 τ3 τ2 τ1 τ0 ) 0 0 0 0 0 0 0 0
After solving the whole system of linear algebraic equations, we get the Tau approx-
imant y3 (x) = 1 − x . It is obvious that the result given by the Tau-Collocation Method is as same as the recursive formulation of the Tau Method. In Table 5 we give the maximum absolute error E := max|y(x) − yn (x)| and the Tau approximant yn (x) for Example 3.5
47
obtained from the Tau-Collocation Method with Tau degrees n = 3, 6, 9. All of the results in Table 5 are as same as the recursive formulation, see Table 2.
Table 5: The maximum absolute error E := max|y(x) − yn (x)| and the Tau approximant yn (x) with 10 digits accuracy for Example 3.5 obtained from the TauCollocation Method with Tau degrees n = 3, 6, 9. n yn (x)
E
3
3.5088 × 10 −2
6
1−x
1 − 1.001435964x − 0.2579343890x 2
+0.3003585950x 3 − 0.04098824194x 4
9
1 − 0.9999998313x − 0.2499940799x 2
+0.2498619725x 3 + 0.03202127146x 4
3.8106 × 10 −4 8.8297 × 10 −8
−0.03308398668x 5 − 0.0005733563390x 6
+0.001768010360x 7
3.1.4 Error Estimation of the Tau-Collocation Method for Ordinary Differential Equations In practical, the Tau-Collocation Method for ODEs is applied to find out the approximate solution of a given ODE without knowing its exact or analytical solutions. Therefore, error estimation is necessary for the Tau-Collocation Method for ODEs. Define an error function en (x) := y(x) − yn (x)
(3.22)
for problem (3.1)-(3.2) approximated by the Tau-Collocation Method, where yn (x) here is the Tau approximant. By subtracting problem (3.1)-(3.2) and Tau problem (3.3)-(3.4), an ordinary differential problem for error function (3.22) is formed and
48
Table 6: Eight sample problems with same Tau approximant’s yn (x) obtained by the recursive formulation of the Tau Method and the Tau-Collocation Method. Sample
Differential equation
Interval
Supplementary
Analytical solution
conditions 1.
y 00 (x) + y(x) = 0
[0, 1]
y(0) = 1
cos(x)
y 0 (0) = 1
2.
y 00 (x) + y(x) = 0
[0, 1]
y(0) = 1
sin(x)
y(1) = sin(1)
3.
xy 00 (x) + 2y 0 (x)
[0, 1]
−4xy = −2x 4.
xy 00 (x) − (x + 1)y 0 (x)
x 2 y 00 (x) + xy 0 (x)
[0, 1]
(x 2 − 1)y 0 (x)
5sinh(2x) xsinh(2)
1 + x + ex
y 0 (0) = 2 y(1) − y 0 (1) = 1
[0.5, 1]
+y(x) = 0
6.
0.5 +
y 0 (0) = 0
+y(x) = 0
5.
y(1) = 5.5
y(0.5) = 1
cos(ln(2x))
y 0 (0.5) = 0 1
[0.5, 1]
y(0.5) = 1
5x − (3|x 2 − 1|) 2
[0.5, 1]
y(1) = 1
1 ((x 2 3(x 2 +1)
−xy(x) = −5 7.
(x 2 + 1)y 0 (x)
1
+xy(x) = x(x2 + 1)
8.
x 6 y 00 (x) −(4 − 6x 2 )y(x) = 0
+ 1) 2
+(2(x 2 + 1)) 2 )
[0.5, 1]
y(0.5) = y(1) =
1 e
1 e4
1
e−x2
49
it is Dv en (x) = −Hn (x) ,
x ∈ [a, b]
,
(3.23)
r = 1(1)ν
,
(3.24)
subject to the supplementary conditions Dvr |x=xr en (x) = 0 ,
where the symbols follow the symbols defined on Section 3.1.1. Note that Hn (x) in equation (3.23) can be obtained by substituting back the determined Tau approximant yn (x) into equation (3.3). Then, the Tau-Collocation Method with Tau degree greater than n is applied to problem (3.23)-(3.24) and the approximate solution to error function (3.22), which is the error estimation of problem (3.1)-(3.2) approximated by the Tau-Collocation Method for ODEs, can be obtained.
3.2
Tau-Collocation Method for the Numerical Solution of Partial Differential Equations
In our research, we extended the Tau-Collocation Method to find out the numerical solution of 2-dimensional and 3-dimensinoal PDEs on a finite domain with given supplementary conditions, included initial conditions, boundary conditions and mixed conditions. The Tau-Collocation Method for 2-dimensinoal PDEs is first purposed by Sam, the author of this thesis, and Liu [47] and the Tau-Collocation Method for 3-dimensional PDEs is first presented in this thesis. As same as the Tau-Collocation Method for ODEs, Tau-Collocation Method for PDEs is a simple and direct method for the solution of different types of PDEs. Before further discussing the Tau-Collocation Method for PDEs, two definitions and one theorem of the Kr¨onecker Products given in the Kr¨onecker Products and Matrix Calculus with Applications [14], are recalled first. The Kr¨onecker Products is used through out the formulation of the Tau-Collocation Method for PDEs. Also, a (k + 1) × (k + 1) matrix is defined first. This matrix is used in the algebraic
50
operations on polynomial.
Definition 3.1 For any n × m matrix X = (xij ), for i = 0(1)n − 1 and j = 0(1)m − 1, the vec operator is defined as vec(X) = [x00
x10
···
xn−1 0
x01
x11
···
xn−1 m−1 ]
0
.
Note that the vec operator take no effects on any n × 1 matrix (column vector), i.e. A = vec(A), if A is a n × 1 matrix. Definition 3.2 For any n × m matrix X = (xij ), for i = 0(1)n − 1 and j = 0(1)m − 1, and any r × s matrix Z, the Kr¨onecker Products X ⊗ Z is defined x00 Z x01 Z x02 Z x10 Z x11 Z x12 Z X ⊗Z = .. . xn−1 0 Z xn−1 1 Z xn−1 2 Z
as ···
x0 m−1 Z
· · · x1 m−1 Z · · · xn−1 m−1 Z
,
where X ⊗ Z is a nr × ms matrix. Let α be any scalar and let A, B and C be any
matrices. The following are the properties and rules for the Kr¨onecker Products, where the proofs are given by the Kr¨onecker Products and Matrix Calculus with Applications [14]. 1. Production with a constant: A ⊗ (αB) = α(A ⊗ B) . 2. Distributive property: (A + B) ⊗ C = A ⊗ C + B ⊗ C
.
3. Distributive property: A ⊗ (B + C) = A ⊗ B + A ⊗ C
.
4. Associative property: A ⊗ (B ⊗ C) = (A ⊗ B) ⊗ C 5. Transposition: (A ⊗ B)0 = A0 ⊗ B 0
.
6. Mixed product rule: (A ⊗ B) (C ⊗ D) = AC ⊗ BD 7. Inversion: (A ⊗ B) −1 = A −1 ⊗ B −1
.
.
.
51
Theorem 3.2 For any matrices X, Y, Z and B with size n × m, m × r, r × s and n × s respectively and any constant k, the equation kXY Z = B implies 0
(Z ⊗ k X) vec(Y ) = vec(B) . Proof The proof follows Kr¨onecker Products and Matrix Calculus with Applications [14].
The next is to define a (k + 1) × (k + 1) matrix. The Matrix follows the operational approach to the Tau Method [41] but it is a finite matrix in here.
Definition 3.3 Let a (k + 1) × (k + 1) matrix
ηk
=
0
O
1 0
2 0 .. .. . .
O
k
0
,
where k is constant and ηk = (ηij ), for ηij = jδi,j+1 , i = 0(1)k, j = 0(1)k and δi,j is Kr¨onecker delta.
3.2.1 Basic Idea of the Tau-Collocation Method for 2-dimensinoal Partial Differential Equations The Tau-Collocation Method for 2-dimensional PDEs follows the idea of the Tau Method and is an extension of the Tau-Collocation Method for ODEs. It is used in finding the polynomial approximate solution to the solution of 2-dimensional PDEs with variable coefficients defined on a finite rectangular domain directly. This method is applicable to the PDEs with initial, boundary and mixed supplementary conditions.
52
Let L :=
P νx
rx =0
P νy
∂ rx +ry ry =0 qrx ry (x, y) ∂x rx ∂y ry
∈ L , the class of linear partial dif-
ferential operators in two variables x and y of order νx and νy respectively with P P rx +ry variable coefficients qrx ry (x, y). Let Dyp := κrxx=0 κryy=0 θrx ry (x, y) ∂x∂ rx ∂y ry be a linear partial differential operators of order κx in x and order κy in y with variable P ωx P ωy ∂ rx +ry coefficients θrx ry (x, y). Also, let Dxq := rx =0 ry =0 ξrx ry (x, y) ∂x rx ∂y ry be a lin-
ear partial differential operators of order ωx in x and order ωy in y with variable coefficients ξrx ry (x, y). Consider a linear partial differential equation with variable coefficients on a rectangular domain [ax , bx ] × [ay , by ], where ax , bx , ay and by ∈ R, Lu(x, y) = f (x, y) ,
(x, y) ∈ [ax , bx ] × [ay , by ]
,
(3.25)
subject to the supplementary conditions Dyp |x=xp u(x, y) = σyp (y) ,
p = 1(1)Nx
,
(3.26)
Dxq |y=yq u(x, y) = σxq (x) ,
q = 1(1)Ny
,
(3.27)
where Nx and Ny are positive constants, depending on νx and νy . By the basic idea of the Tau Method and the Tau-Collocation Method for ODEs, a polynomial with nxth degree in x and nyth degree in y unx ny (x, y) =
ny nx X X
a ix iy x ix y iy
(3.28)
ix =0 iy =0
is substituted into problem (3.25)-(3.27). An overdetermined system of linear algebraic equations with (nx + 1)(ny + 1) unknown coefficients aix iy , where ix = 0(1)nx and iy = 0(1)ny , are formed. Let τx = (τx0 , τx1 , · · · , τx ϕx −1 ) and τy = (τy0 , τy1 , · · · , τy ϕy −1 ) be the ϕx and ϕy free parameters respectively. A perturbation term Hnx ny (x, y) with unknown parameters τx and τy is added to the right-hand side of equation (3.25) so as to construct a balanced system of linear algebraic equations for determining the approximate polynomial solution unx ny (x, y). Then, problem (3.25)-(3.27) becomes Lunx ny (x, y) = f (x, y) + Hnx ny (x, y) ,
(x, y) ∈ [ax , bx ] × [ay , by ]
, (3.29)
53
subject to the supplementary conditions Dyp |x=xp unx ny (x, y) = σyp (y) ,
p = 1(1)Nx
,
(3.30)
Dxq |y=yq unx ny (x, y) = σxq (x) ,
q = 1(1)Ny
,
(3.31)
which is defined as the associated Tau problem of problem (3.25)-(3.27). As the idea of Tau-Collocation Method for ODEs, the format of perturbation term in equation (3.29) is chosen as [a ,b ]
[a ,b ]
y x Hnx ny (x, y) = gnx ny (x, y; τx , τy ) V x,nxx −N (x) V y,nyy −N (y) x +1 y +1
(3.32)
[a ,b ]
[a ,b ]
y x where V x,nxx −N (x) and V y,nyy −N (y) are orthogonal polynomials of degree x +1 y +1
nx − Nx + 1 defined on [ax , bx ] and degree ny − Ny + 1 defined on [ay , by ] respectively. The orthogonal polynomials are chosen as the shifted Chebyshev or Legendre polynomials in usual. Note that the degree of the orthogonal polynomials should not be negative, hence the selection of nx and ny should be satisfy nx > Nx − 1 and ny > Ny − 1 respectively. In equation (3.32), gnx ny (x, y; τx , τy ) is a function of x, y, ϕx free parameters τx and ϕy free parameters τy . The free parameters τx and τy balance the overdetermined system of linear algebraic equations. Besides balancing the overdetermined system of linear algebraic equations, the another purpose for choosing the perturbation term Hnx ny (x, y) in the form of equation (3.32) is to produce exactly nx − Nx + 1 and ny − Ny + 1 collocation points [a ,b ]
[a ,b ]
y x which are the zeros of orthogonal polynomial V x,nxx −N (x) and V y,nyy −N (y) rex +1 y +1
[a ,b ]
x (x) spectively. Let xi , for i = 0(1)nx − Nx , be the nx − Nx + 1 zeros of V x,nxx −N x +1
[a ,b ]
y and let yj , for j = 0(1)ny − Ny , be the ny − Ny + 1 zeros of V y,nyy −N (y). By coly +1
locating (xi , yj ), i = 0(1)nx − Nx and j = 0(1)ny − Ny , zeros into problem (3.29), (nx − Nx + 1)(ny − Ny + 1) collocation equations are generated. These collocation equations are used in finding out the unknown coefficients aij , i = 0(1)nx and j = 0(1)ny while all of the free parameters τx and τy can be ignored. Note that it is not necessary to consider the detail of gnx ny (x, y; τx , τy ), i.e. the number of free parameters ϕx and ϕy , throughout the computation because the zeros of orthogonal [a ,b ]
[a ,b ]
y x polynomials V x,nxx −N (x) and V y,nyy −N (y) are used for collocation. x +1 y +1
54
In order to find out the solution unx ny (x, y) of the Tau problem (3.29)-(3.31) with perturbation term (3.32), (nx − Nx + 1)(ny − Ny + 1) collocation equations is not sufficient to do so and (nx + 1)(ny + 1) − (nx − Nx + 1)(ny − Ny + 1) more linear algebraic equations are required. The rest of the linear algebraic equations are gen[a ,b ]
x erated from conditions (3.30)-(3.31). Let V x,nxx +1 (x) be an orthogonal polynomial
[a ,b ]
y of degree nx + 1 defined on [ax , bx ] and V y,nyy +1 (y) be an orthogonal polynomial of
[a ,b ]
[a ,b ]
y x degree ny + 1 defined on [ay , by ]. Usually, V x,nxx +1 (x) and V y,nyy +1 (y) are selected as
[a ,b ]
[a ,b ]
y x same basis as the orthogonal polynomials V x,nxx −N (x) and V y,nyy −N (y) in the y +1 x +1
perturbation term respectively, i.e. the shifted Chebyshev or Legendre polynomials. [a ,b ]
y By collocating ny +1 zeros of V y,nyy +1 (y) into Nx conditions (3.30) and nx +1 zeros of
[a ,b ]
x V x,nxx +1 (x) into Ny conditions (3.31), Nx (ny + 1) + Ny (nx + 1) linear algebraic equa-
tions are generated. However, there are Nx Ny redundant linear dependent equations appear in the generated Nx (ny + 1) + Ny (nx + 1) linear algebraic equations. After several row echelon form operations of the system of linear algebraic equations have been done, such Nx Ny redundant linear dependent equations descript the same unknown coefficients which also be descript by other linear algebraic equations. These overlapped properties happen at the corner points between two nearby conditions in the rectangular domain. Every touching between two conditions gives one redundant linear dependent equations. It is obvious that there are Nx Ny touching in a rectangular domain so that there are Nx Ny redundant linear dependent equations, see Figure 3. Eventually, Nx (ny + 1) + Ny (nx + 1) − Nx Ny linear algebraic equations, which are linearly independent equations, are generated from the conditions (3.30)-(3.31). After the collocation process for the Tau problem (3.29)-(3.31) has been done, (nx − Nx + 1)(ny − Ny + 1) + Nx (ny + 1) + Ny (nx + 1) − Nx Ny = n x ny − n x N y + n x − n y N y + N x N y − N x + n y − N y + 1 +ny Nx + Nx + nx Ny + Ny − Nx Ny = (nx + 1)(ny + 1) linear algebraic equations are generated. Therefore, (nx + 1)(ny + 1) unknown co-
55
y
x Touching between two nearby conditions Figure 3: The geometrical presentation of the Nx Ny redundant linear dependent equations.
efficients aix iy , ix = 0(1)nx and iy = 0(1)ny , can be obtained by solving the whole system of linear algebraic equations by usual method. Then, the Tau approximant unx ny (x, y) to the solution of problem (3.25)-(3.27) can also be obtained.
3.2.2 Formulation of the Tau-Collocation Method for 2-dimensional Partial Differential Equations As same as the idea of the Tau-Collocation Method for ODEs, the formulation of the Tau-Collocation Method for 2-dimensional PDEs aims to convert the Tau problem (3.29)-(3.31) to a system of linear algebraic equations. The formulation is starting form the following theorem.
Lemma 3.1 Let yn (x) =
Pn
i=0
ai xi = an xn = xn an be a polynomial in x of degree n, where 0
xn = (x 0 , x 1 , x 2 , · · · , x n ) and an = (a0 , a1 , a2 , · · · , an ) . Then, the effects of differentiation with respect to x of yn (x) is 0 dr yn (x) = an η nr xn = xn (η nr ) an r dx
.
56
Proof The proof follows directly by induction. Note that an η nr xn is a 1 × 1 matrix and it’s transpose equals to itself.
Lemma 3.2 For any matrixes X, A and B with size n × m, 1 × n and 1 × m respectively and any constant k, then (B ⊗ k A) vec(X) = k A X B
.
Proof Since k A X B is a 1×1 matrix, by Theorem 3.2, we have k A X B = vec k A X B =
(B ⊗ k A) vec(X). Lemma 3.3
Let a bivariate polynomial of degree nx in x and degree ny in y unx ny (x, y) =
ny nx X X
ix =0 iy =0
aix iy x ix y iy = (yny ⊗ xnx ) vec(Anx ny ) ,
where Anx ny = (aix iy ), for ix = 0(1)nx and iy = 0(1)ny , xnx = (x 0 , x 1 , x 2 , · · · , x nx ) and yny = (y 0 , y 1 , y 2 , · · · , y ny ). Then, the effect of combined differentiation with respect to x and y of unx ny (x, y) is ∂ rx +ry 0 0 unx ny (x, y) = (yny (η nryy ) ⊗ xnx (η nrxx ) ) vec(Anx ny ) . r r x y ∂x ∂y Proof By Lemma 3.2, ∂ rx +ry ∂ rx +ry u (x, y) = xn A n n y n n n ∂x rx ∂y ry x y ∂x rx ∂y ry x x y y
.
Therefore, by Lemma 3.1, ∂ rx +ry d ry rx 0 x A y = x (η ) A y n n n n nx nx ny ny nx ∂x rx ∂y ry x x y y dy ry 0 = xnx (η nrxx ) Anx ny η nryy yny .
57
Apply Lemma 3.2 again, 0
0
0
0
0
xnx (η nrxx ) Anx ny η nryy yny = ((η nryy yny ) ⊗ xnx (η nrxx ) ) vec(Anx ny ) = (yny (η nryy ) ⊗ xnx (η nrxx ) ) vec(Anx ny ) .
Theorem 3.3 Let a bivariate polynomial of degree nx in x and degree ny in y unx ny (x, y) =
ny nx X X
ix =0 iy =0
aix iy x ix y iy = (yny ⊗ xnx ) vec(Anx ny ) ,
where Anx ny = (aix iy ), for ix = 0(1)nx and iy = 0(1)ny , xnx = (x 0 , x 1 , x 2 , · · · , x nx ) and yny = (y 0 , y 1 , y 2 , · · · , y ny ). Also, let g(x, y) be any function with variables x and y. Then g(x, y)
∂ rx +ry 0 0 unx ny (x, y) = (yny (η nryy ) ⊗ g(x, y) xnx (η nrxx ) ) vec(Anx ny ) . r r x y ∂x ∂y
Proof The proof follows the proprieties and rules in Theorem 3.2.
The formulation of the Tau-Collocation Method for 2-dimensional PDEs is divided conceptually in two parts.
They are the formulation of the
linear PDE and the formulation of the conditions of the given problem.
The
following steps show the formulation of the linear PDE (3.29). Let P νx P νy 0 r 0 Πnx ny (x, y) = rx =0 ry =0 (yny (η nyy ) ⊗ qrx ry (x, y) xnx (η nrxx ) ) vec(Anx ny ). By Theo-
rem 3.3, equation (3.29) with perturbation term (3.32) becomes
Πnx ny (x, y) vec(Anx ny ) = f (x, y) + Hnx ny (x, y) , [a ,b ]
(3.33)
[a ,b ]
y x where Hnx ny (x, y) = gnx ny (x, y; τx , τy ) V x,nxx −N (x) V y,nyy −N (y). The next is to x +1 y +1
collocate (nx − Nx + 1)(ny − Ny + 1) zero points generated from the perturbation term Hn (x) into equation (3.33). Let xix , ix = 0(1)nx − Nx , and yiy , iy = 0(1)ny − Ny , [a ,b ]
x be the nx − Nx + 1 zeros of orthogonal polynomial V x,nxx −N (x) and ny − Ny + 1 x +1
58
[a ,b ]
y zeros of orthogonal polynomial V y,nyy −N (y) respectively. By collocating (xix , yiy ), y +1
ix = 0(1)nx − Nx and iy = 0(1)ny − Ny , zeros into equation (3.33), we have Γ vec(Anx ny ) = F where
Γ =
Πnx ny (x0 , y0 ) Πnx ny (x0 , y1 ) .. . Πnx ny (x0 , yny −Ny ) Πnx ny (x1 , y0 ) .. . Πnx ny (xnx −Nx , yny −Ny )
(3.34)
,
is a (nx − Nx + 1)(ny − Ny + 1) × (nx + 1)(ny + 1) matrix and f (x0 , y0 ) f (x0 , y1 ) .. . F = f (x0 , yny −Ny ) , f (x1 , y0 ) .. . f (xnx −Nx , yny −Ny )
is a column vector with (nx − Nx + 1)(ny − Ny + 1) elements. The equation (3.29) now is successfully converted to a set of linear algebraic equations. The following steps show the formulation of Nx conditions (3.30) of the Tau probP P r 0 lem (3.29)-(3.31). Let Πyp nx ny (x, y) = κrxx=0 κryy=0 (yny (η nyy ) ⊗ θrx ry (x, y) xnx (η nrxx ).
By Theorem 3.3, conditions (3.30) become
Πyp nx ny (xp , y) vec(Anxny ) = σyp (y) ,
p = 1(1)Nx
.
(3.35) [a ,b ]
y (y). By Let yiy , iy = 0(1)ny , be the ny + 1 zeros of orthogonal polynomial V y,nyy +1
collocating these ny + 1 zeros into equation (3.35), we have Γy vec(Anx ny ) = Fy
(3.36)
59
where
Γy
=
Πy1 nx ny (x1 , y0 ) Πy1 nx ny (x1 , y1 ) .. . Πy1 nx ny (x1 , yny ) Πy2 nx ny (x2 , y0 ) .. . ΠyNx nx ny (xNx , yny )
is a Nx (ny + 1) × (nx + 1)(ny + 1) matrix and σ (y ) y1 0 σy2 (y1 ) .. . Fy = σy1 (yny ) σy2 (y0 ) .. . σyNx (yny )
,
,
is a column vector with Nx (ny + 1) elements.
The following steps show the formulation of conditions (3.31) of the Tau problem P P 0 r 0 (3.29)-(3.32). Let Πxq nx ny (x, y) = ωrxx=0 ωryy=0 (yny (η nyy ) ⊗ ξrx ry (x, y) xnx (η nrxx ) ). By Theorem 3.3, condition (3.31) become
Πxq nx ny (x, yq ) vec(Anx ny ) = σxq (x) ,
q = 1(1)Ny
.
(3.37) [a ,b ]
x Let xi , for i = 0(1)nx , be the nx + 1 zeros of orthogonal polynomial V x,nxx +1 (y). By
collocating the nx + 1 zeros into equation (3.37), we have Γx vec(Anx ny ) = Fx
(3.38)
60
where
Γx
Πx1 nx ny (x0 , y1 ) Πx1 nx ny (x1 , y1 ) .. . = Πx1 nx ny (xnx , y1 ) Πx2 nx ny (x0 , y2 ) .. . ΠxNy nx ny (xnx , yNy )
is a Ny (nx + 1) × (nx + 1)(ny + 1) matrix and σ (x ) x1 0 σx1 (x1 ) .. . Fx = σx1 (xnx ) σx2 (x0 ) .. . σxNy (xnx ) is a column vector with Ny (nx + 1) elements.
,
,
By merging equation (3.34), (3.36) and (3.38), we have a resulting system of linear algebraic equations
0
Γ F Γy vec(Anm ) = Fy Γx Fx
,
(3.39)
where (Γ, Γy , Γx ) is a ((nx + 1)(ny + 1) + Nx Ny ) × (nx + 1)(ny + 1) matrix and 0
(F , Fy , Fx ) is a column vector with ((nx + 1)(ny + 1) + Nx Ny ) elements. Since there are Nx Ny redundant linear dependent equations inside system (3.39), the rank of system (3.39) is (nx + 1)(ny + 1). Therefore, all of the unknown coefficients aix iy , ix = 0(1)nx and iy = 0(1)ny , can be obtained by solving the system of linear algebraic equations (3.39) by usual method without finding out free parameters τx and τy .
61
And, the Tau approximant unx ny (x, y) to the solution of problem (3.25)-(3.27) can be obtained. The following examples show the effectiveness of the Tau-Collocation Method applied into different types of PDEs. And, the first example is a demonstration of Tau-Collocation Method.
Example 3.6 Consider a second order PDE ∂2 u(x, y) = 4xy + e x , ∂x∂y
(x, y) ∈ [0, 1] × [0, 1]
,
(3.40)
with supplementary conditions ∂ u(0, y) = y , ∂y u(x, 0) = 2 ,
y ∈ [0, 1]
,
(3.41)
x ∈ [0, 1]
.
(3.42)
The exact solution of this problem is u(x, y) = x 2 y 2 + ye x +
y2 −y+2 . 2
The following steps are the main procedure of the Tau-Collocation Method. 1. Setup the Tau problem: If we take Tau degree nx = ny = 2 and use the shifted Chebyshev basis in the perturbation term Hnx ny (x, y), the Tau problem of problem (3.40)-(3.42) becomes ∂2 u2 2 (x, y) = 4xy + e x + g2 2 (x, y; τx , τy ) T 2∗ (x) T 2∗ (y) , ∂x∂y
(3.43)
where (x, y) ∈ [0, 1] × [0, 1], with supplementary conditions ∂ u2 2 (0, y) = y , ∂y u2 2 (x, 0) = 2 ,
y ∈ [0, 1]
,
(3.44)
x ∈ [0, 1]
.
(3.45)
2. Formulate the matrices Γ and F : Since
0 Π2 2 (x, y) = ( 1 y 1 y 2 ) 1 0 = 0 0 0 0 1 2x
0 0
0
0
0 0 0 1 2 0 0 ⊗ ( 1 x x ) 1 0 0 vec(A2 2 ) 2 0 0 2 0 (3.46) 0 2y 4xy vec(A2 2 ) .
62
By collocation the zeros of T 2∗ (x) and T 2∗ (y), i.e. ( 21 + ( 12 −
√ 2 1 , 4 2
+
√ 2 ) 4
and ( 21 −
of equation (3.43), 0 0 0 0 Γ = 0 0 0 0
√ 2 1 , 4 2
−
√ 2 ), 4
√ 2 1 , 4 2
+
√ 2 ), 4
( 21 +
√ 2 1 , 4 2
−
√ 2 ), 4
into equation (3.46) and the right-hand side
we have 0 0 1 1.7071067812 0 1.7071067812 2.9142135624 0 0 1 1.7071067812 0 0.2928932188 0.5 0 0 1 0.2928932188 0 1.7071067812 0.5 0 0 1 0.2928932188 0 0.2928932188 0.0857864377
and
5.2621888818 2.8479753194 F = 1.6577131182 1.2434995558
3. Formulate the matrices Γy and Fy : Since
0 Πy1 2 2 (0, y) = ( 1 y 1 y 2 ) 1 0 = 0 0 0 1 0 0
0 0
0
0 0 ⊗( 1 0 0 2 0 2y 0 0 vec(A2 2 )
By collocation the zeros of T 3∗ (x), i.e.
√ 3 4
+ 21 ,
1 2
and −
√ 3 4
1 0 0 ) 0 1 0 vec(A2 2 ) 0 0 1 .
Fy
0.9330127019 = 0.5 0.0669872981
(3.47)
+ 21 , into equation (3.47)
and the right-hand side of equation (3.44), we have 0 0 0 1 0 0 1.8660254038 0 0 Γy = 0 0 0 1 0 0 1 0 0 0 0 0 1 0 0 0.1339745962 0 0 and
0
63
4. Formulate the matrices Γx and Fx : Since
1 Πx1 2 2 (x, 0) = ( 1 0 0 ) 0 0 = 0 x x2 0 0
0 0
0
0
1 0 0 1 2 1 0 ⊗ ( 1 x x ) 0 1 0 vec(A2 2 ) 0 1 0 0 1 (3.48) 0 0 0 0 vec(A2 2 ) .
By collocation the zeros of T 3∗ (x), i.e.
√ 3 4
+ 21 ,
1 2
and −
√ 3 4
+ 21 , into equation (3.48)
and the right-hand side of equation (3.45), we have 1 0.9330127019 0.8705127019 0 0 0 0 0 0 Γx = 1 0.5 0.25 0 0 0 0 0 0 1 0.0669872981 0.0044872981 0 0 0 0 0 0 and
2
Fx = 2 2
5. Formulate the system of linear algebraic equations and solve it to obtain the Tau approximant: By combining matrices Γ, Γy and Γx and also column vectors F , Fy and Fx , we have 0 0 5.2621888818 0 0 1 1.7071067812 0 1.7071067812 2.9142135624
0
0
0
0
1
1.7071067812
0
0.2928932188
0.5
2.8479753194
0
0
0
0
1
0.2928932188
0
1.7071067812
0.5
1.6577131182
0
0
0
0
1
0.2928932188
0
0.2928932188
0.0857864377
1.2434995558
0
0
0
1
0
0
1.8660254038
0
0
0.9330127019
0
0
0
1
0
0
1
0
0
0.5
0
0
0
1
0
0
0.1339745962
0
0
0.0669872981
1
0.9330127019
0.8705127019
0
0
0
0
0
0
2
1
0.5
0.25
0
0
0
0
0
0
2
1
0.0669872981
0.0044872981
0
0
0
0
0
0
2
By solving the above matrix by the Gauss Elimination Method, we can obtain 1 0 0 0 0 0 0 0 0 2
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
0
0
0
0
0.9112017449
0
0
0
0
0
1
0
0
0
0.8416424739
0
0
0
0
0
0
1
0
0
0.5
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
0
0
1
1
0
0
0
0
0
0
0
0
0
0
.
.
(3.49)
64
Note that the last row of matrix (3.49) is a zero row which is the redundant linear dependent equation generated from the two nearby conditions (3.44) and (3.45) during the collocation process. From matrix (3.49), we can obtain the Tau approximant u2 2 (x, y) = 2 + 0.9112017449xy + 0.8416424739x 2y + 0.5y 2 + x 2 y 2 with 10 digits accuracy. The Tau-Collocation Method can be applied directly to this problem without any pre-approximation has been done for the right-hand side functions of both equation and conditions. In Table 7 we give the maximum absolute error E := max|u(x, y) − unx ny (x, y)| for Example 3.6 obtained from the TauCollocation Method with Tau degrees nx = ny = 2(1)8.
Table 7: The maximum absolute error E := max|u(x, y) − unx ny (x, y)| for Example 3.6 obtained from the Tau-Collocation Method with Tau degrees nx = ny = 2(1)8 of approximate polynomial solutions. nx , ny
E
n x , ny
E
2, 2
4.2961 × 10 −2
6, 6
1.0999 × 10 −7
3, 3
2.0633 × 10 −3
7, 7
3.3212 × 10 −9
4, 4
7.9550 × 10 −5
8, 8
8.6603 × 10 −11
5, 5
2.9222 × 10 −6
Example 3.7 Consider a fourth order PDE with variable coefficients x ∂4 ∂2 u(x, t) + −1 u(x, t) = 0 , ∂t 2 sin x ∂x 4 with initial conditions
u(x, 0) = x − sin x ,
(x, t) ∈ [0, π] × [0, 1]
,
65
∂ u(x, 0) = sin x − x , ∂t
x ∈ [0, π]
,
and boundary conditions u(0, t) = 0 , 2
∂ u(0, t) = 0 , ∂x 2 u(π, t) = πe −t , ∂2 u(π, t) = 0 , t ∈ [0, 1] ∂x 2
,
The exact solution of this problem is u(x, t) = (x − sin x)e −t
,
see WazWaz [52]. The Tau-Collocation Method can be applied directly to this problem without any pre-approximation has been done for the variable coefficient. Note that number of conditions Nx = 4 and Nt = 2 in this problem, hence there are Nx Nt = 8 redundant linear dependent equations inside the system of linear algebraic equations. In Table 8 we give the maximum absolute error E := max|u(x, t) − unx nt (x, t)| for Example 3.7 obtained from the Tau-Collocation Method with Tau degrees nx = nt = 7(1)14. The graphs of the error surfaces of our approximations with Tau degrees nx = nt = 7 and 8 are shown in Figure 4.
Example 3.8 Consider the inhomogeneous second order PDE in operational approach to the Tau Method [41] (1 + x + x 2 )
∂2 1 ∂2 ∂2 u(x, y) − (x + y) u(x, y) + y(2 − y) u(x, y) = 2y 2 ∂x 2 2 ∂x∂y ∂y 2
,
where (x, y) ∈ [−1, 1] × [−1, 1], with either of the two sets of boundary conditions:
∂ u(x, −1) ∂x
∂ u(x, 1) ∂y
= 2x , = 2x 2 ,
u(±1, y) = y 2 ,
x ∈ [−1, 1] y ∈ [−1, 1]
, .
u(x, ±1) = x 2 , or u(±1, y) = y 2 ,
x ∈ [−1, 1]
,
y ∈ [−1, 1]
.
66
Table 8: The maximum absolute error E := max|u(x, t) − unx nt (x, t)| for Example 3.7 obtained from the Tau-Collocation Method with Tau degrees nx = nt = 7(1)14 of approximate polynomial solutions. nx , nt
E
n x , nt
E
7, 7
1.0502 × 10 −2
11, 11 4.5385 × 10 −8
8, 8
2.0163 × 10 −4
12, 12 1.8714 × 10 −9
9, 9
9.9633 × 10 −5
13, 13 4.3112 × 10 −10
10, 10 1.0383 × 10 −6
14, 14 1.7078 × 10 −12
0.0002
0.01
0.00015 0.005
0.0001 5E-5
0
0 -5E-5
-0.005 -1E-4 -0.01 1 0.8 2.5 3 0.6 0.4 1.5 x2 y 1 0.2 0 0 0.5
nx = n t = 7
-0.00015 1
2.5 3 0.8 0.6 1.5 x2 y 0.4 0.2 1 0 0 0.5
n x = nt = 8
Figure 4: The error curves of the approximate solutions for Example 3.7 obtained from the Tau-Collocation Method with Tau degrees nx = nt = 7 and 8.
67
The exact solution of this problem is u(x, y) = x 2 y 2
.
The exact solution unx ny (x, y) = x 2 y 2 can be obtained from the Tau-Collocation Method when Tau degrees nx , ny > 2.
Example 3.9 Consider the second order PDE in Ke¸san [16] ∂ 2 u(x, t) ∂ 2 u(x, t) = +6, ∂t 2 ∂x 2
(x, t) ∈ [−1, 1] × [0, 1]
,
with initial conditions u(x, 0) = x 2 ∂ u(x, 0) = 4x , ∂t
, x ∈ [−1, 1]
,
and boundary conditions u(−1, t) = (−1 + 2t) 2 u(1, t) = (1 + 2t) 2 ,
, t ∈ [0, 1]
.
The exact solution of this problem is u(x, t) = (x + 2t) 2
.
The exact solution unx nt (x, t) = x 2 + 4xt + 4t 2 can be obtained from the TauCollocation Method when Tau degrees nx , nt > 2.
Example 3.10 This example is the Klein-Gordon equation ∂2 ∂2 u(x, t) − u(x, t) + u(x, t) = 2 sin x , ∂t 2 ∂x 2
(x, t) ∈ [0, 1] × [0, 1]
with initial conditions u(x, 0) = sin x , ∂ u(x, 0) = 1 , ∂t
x ∈ [0, 1]
,
,
68
and boundary conditions u(0, t) = sin t , u(1, t) = sin 1 + sin t ,
t ∈ [0, 1]
.
The exact solution of this problem is u(x, t) = sin x + sin t , see WazWaz [52]. This problem is solved by using the Tau-Collocation Method with Tau degrees nx = nt = 4(1)10. In Table 9 we give the maximum absolute error E := max|u(x, t) − unx nt (x, t)| for Example 3.10 obtained from the Tau-Collocation Method with Tau degrees nx = ny = 4(1)10. Very accuracy results can be obtained.
Table 9: The maximum absolute error E := max|u(x, t) − unx nt (x, t)| for Example 3.10 obtained from the Tau-Collocation Method with Tau degrees nx = nt = 4(1)10. nx , nt
E
n x , nt
E
4, 4
2.4734 × 10 −2
8, 8
3.6479 × 10 −5
5, 5
4.1113 × 10 −3
9, 9
3.3698 × 10 −6
6, 6
1.2470 × 10 −3
10, 10 7.0092 × 10 −7
7, 7
1.4787 × 10 −4
Example 3.11 This example is the Saint-Venant’s torsion problem for a prismatic bar, which is discussed by the operational approach to the Tau Method [41] and the new implementation of the Tau Method for PDE’s [27] ∂2 ∂2 u(x, y) + u(x, y) = −2 , ∂x 2 ∂y 2
(x, y) ∈ [−1, 1] × [−1, 1]
,
with boundary conditions u(±1, y) = u(x, ±1) = 0 ,
x ∈ [−1, 1] and y ∈ [−1, 1]
.
69
The exact solution of this problem is 32 u(x, y) = 3 π
∞ X
k=1,3,5,···
(−1) k3
k−1 2
cosh kyπ 2 1− cosh kπ 2
!
cos
kxπ 2
.
This problem is solved by using the Tau-Collocation Method (TC) with Tau degrees nx = ny = 4, 6, 8 and 16 and compared with the results of the operational approach to the Tau Method (OA) [41] and new implementation of the Tau Method for PDE’s with Chebyshev basis (NI-T) and canonical basis (NI-C) [27]. In Table 10 we give the maximum absolute error E := max|u(x, y) − unx ny (x, y)| for Example 3.11 obtained from the mentioned methods with Tau degrees nx = ny = 4, 6, 8 and 16. Similar results can be obtained form these three versions of the Tau Method.
Table 10: The maximum absolute error E := max|u(x, y) − unx ny (x, y)| for Example 3.11 with Tau degrees n = m = 4, 6, 8 and 16, obtained from the Tau-Collocation Method for PDE (TC), the operational approach to the Tau Method (OA) and the new implementation of the Tau Method for PDE’s with Chebyshev basis (NI-T) and canonical basis (NI-C). nx , ny
E (TC)
E (OA)
E (NI-T)
E (NI-C)
4, 4
1.494 × 10 −2
1.6 × 10 −2
1.8 × 10 −2
1.1 × 10 −2
6, 6
3.742 × 10 −3
2.5 × 10 −3
−
−
8, 8
1.289 × 10 −3
4.0 × 10 −4
1.1 × 10 −3
1.8 × 10 −3
16, 16
4.237 × 10 −5
−
5.3 × 10 −5
2.4 × 10 −3
3.2.3 Error Estimation of the Tau-Collocation Method for 2-dimensional Partial Differential Equations In practical, the Tau-Collocation Method for 2-dimensional PDEs is applied to find out the approximate solution of a given 2-dimensional PDE without knowing
70
its exact or analytical solutions. Therefore, error estimation is necessary for the Tau-Collocation Method for 2-dimensional PDEs. As same as the case of ODEs, define an error function enx ny (x, y) := u(x, y) − unx ny (x, y)
(3.50)
for problem (3.25)-(3.27) approximated by the Tau-Collocation Method, where unx ny (x, y) here is the Tau approximant. By subtracting problem (3.25)-(3.27) and Tau problem (3.29)-(3.31), a 2-dimensional partial differential problem for error function (3.50) is formed and it is Lenx ny (x, y) = −Hnx ny (x, y) ,
(x, y) ∈ [ax , bx ] × [ay , by ]
,
(3.51)
subject to the supplementary conditions Dyp |x=xp enx ny (x, y) = 0 ,
p = 1(1)Nx
,
(3.52)
Dxq |y=yq enx ny (x, y) = 0 ,
q = 1(1)Ny
,
(3.53)
where the symbols follow the symbols defined on Section 3.2.1. Then, the TauCollocation Method with Tau degree greater than nx and ny in x and y respectively is applied to problem (3.51)-(3.53) and the approximate solution to error function (3.50), which is the error estimation of problem (3.25)-(3.27) approximated by the Tau-Collocation Method, can be obtained. The following examples show the effectiveness of the error estimation of the Tau-Collocation Method applied into different types of PDEs.
Example 3.12 Refer to the second order partial different problem discussed in Example 3.6. The error estimation of the Tau-Collocation Method discussed in this section is applied to this example.
In Table 11 we give the maximum absolute value
Ee := max |enx ny nex ney (x, y)|, where enx ny nex ney (x, y) is the approximate solution of the error function with degree nex in x and ney in y, for Example 3.12 obtained from
71
the Tau-Collocation Method with Tau degrees nx = ny = 2(1)8 and correspondingly nex = ney = 3(1)9 and compared with the results of maximum absolute error E := max|u(x, y) − unx ny (x, y)| for Example 3.12 obtained from the Tau-Collocation Method with Tau degrees nx = ny = 2(1)8. The results of the error estimation are closed to the corresponding maximum absolute errors of the problem very much.
Table 11: The maximum absolute value Ee := max |enx ny nex ney (x, y)| and maximum absolute error E := max|u(x, y) − unx ny (x, y)| for Example 3.12 obtained from the Tau-Collocation Method with Tau degrees nx = ny = 2(1)8 and correspondingly nex = ney = 3(1)9. nx , ny
ne x , ne y
Ee
E
2, 2
3, 3
4.2777 × 10 −2
4.2961 × 10 −2
3, 3
4, 4
2.0800 × 10 −3
2.0633 × 10 −3
4, 4
5, 5
8.0832 × 10 −5
7.9550 × 10 −5
5, 5
6, 6
2.9002 × 10 −6
2.9222 × 10 −6
6, 6
7, 7
1.0969 × 10 −7
1.0999 × 10 −7
7, 7
8, 8
3.3300 × 10 −9
3.3212 × 10 −9
8, 8
9, 9
8.7212 × 10 −11
8.6603 × 10 −11
Example 3.13 Refer to the different problem discussed in Example 3.7. The error estimation of the Tau-Collocation Method discussed in this section is applied to this example. In Table 12 we give the maximum absolute value Ee := max |enx ny nex net (x, t)|, where enx ny nex net (x, t) is the approximate solution of the error function with degree nex in x and net in t, for Example 3.13 obtained from the Tau-Collocation Method with Tau degrees nx = nt = 7(1)14 and correspondingly nex = net = 8(1)15 and compared with the results of maximum absolute error E := max|u(x, t) − unx nt (x, t)|
72
for Example 3.13 obtained from the Tau-Collocation Method with Tau degrees nx = nt = 7(1)14. The results of the error estimation are closed to the corresponding maximum absolute errors of the problem. The graphs of the approximate solution of the error function for Example 3.13 obtained from the Tau-Collocation Method with Tau degrees nx = nt = 7 with nex = net = 8 and nx = nt = 8 with nex = net = 9 are shown in Figure 5. The graphics show remarkable equioscillatory behavior. Table 12: The maximum absolute value Ee := max |enx ny nex net (x, t)| and maximum absolute error E := max|u(x, t) − unx nt (x, t)| for Example 3.13 obtained from the Tau-Collocation Method with Tau degrees nx = nt = 7(1)14 and correspondingly nex = net = 8(1)15. nx , nt
ne x , ne t
Ee
E
7, 7
8, 8
1.4822 × 10 −2
1.0502 × 10 −2
8, 8
9, 9
2.7372 × 10 −4
2.0163 × 10 −4
9, 9
10, 10
1.3223 × 10 −4
9.9633 × 10 −5
10, 10 11, 11
1.0242 × 10 −6
1.0383 × 10 −6
11, 11 12, 12
4.4385 × 10 −8
4.5385 × 10 −8
12, 12 13, 13
2.6186 × 10 −9
1.8714 × 10 −9
13, 13 14, 14
8.5306 × 10 −10
4.3112 × 10 −10
13, 13 14, 14
1.2608 × 10 −12
1.7078 × 10 −12
3.2.4 Basic Idea of the Tau-Collocation Method for 3-dimensinoal Partial Differential Equations The Tau-Collocation Method can be extended to find out the polynomial approximate solution of 3-dimensional PDEs with variable coefficients on a finite cubical domain directly. This method is applicable to the PDEs with initial, boundary and
73
0.01
0.0002
0.005
0.0001
0
0
-0.005 -0.0001 -0.01 -0.0002 -0.015 1
0.8
0.6 0.4 t 0.2
3 2 2.5 1 1.5 x 0.5 0 0
1
nx = nt = 7 with nex = net = 8
0.8 0.6 t 0.4 0.2
2.5 3 1.5 x2 1 0 0 0.5
nx = nt = 8 with nex = net = 9
Figure 5: The graphs of the approximate solution of the error function for Example 3.13 obtained from the Tau-Collocation Method with Tau degrees nx = nt = 7 with nex = net = 8 and nx = nt = 8 with nex = net = 9.
mixed supplementary conditions. This topic is also our current research result and is first proposed in this thesis. Let S = {x, y, z} be the basis of x, y and z. And, let L3 :=
νy νx X νz X X
qrx ry rz (x, y, z)
rx =0 ry =0 rz =0
∂ rx +ry +rz ∂x rx ∂y ry ∂z rz
,
where L3 ∈ L , the class of linear partial differential operators in three variables s of order νs , for s ∈ S, with variable coefficients qrx ry rz (x, y, z). Let Dyzp :=
κy κx X κz X X
θrx ry rz (x, y, z)
rx =0 ry =0 rz =0
∂ rx +ry +rz ∂x rx ∂y ry ∂z rz
be a linear partial differential operators of order κs in s, for s ∈ S, with variable coefficients θrx ry rz (x, y, z). Also, let Dxzq :=
ωy ωx X ωz X X
rx =0 ry =0 rz =0
ξrx ry rz (x, y, z)
∂ rx +ry +rz ∂x rx ∂y ry ∂z rz
74
be a linear partial differential operators of order ωs in s, for s ∈ S, with variable coefficients ξrx ry rz (x, y, z). And, let Dxyr :=
ζy ζx ζz X X X
χrx ry rz (x, y, z)
rx =0 ry =0 rz =0
∂ rx +ry +rz ∂x rx ∂y ry ∂z rz
be a linear partial differential operators of order ζs in s, for s ∈ S, with variable coefficients χrx ry rz (x, y, z). Consider a linear partial differential equation with variable coefficients on a cubical domain [ax , bx ] × [ay , by ] × [az , bz ], where as , bs ∈ R, for s ∈ S, L3 u(x, y, z) = f (x, y, z) ,
(x, y, z) ∈ [ax , bx ] × [ay , by ] × [az , bz ]
(3.54)
subject to the supplementary conditions Dyzp |x=xp u(x, y, z) = σyzp (y, z) ,
p = 1(1)Nx
,
(3.55)
Dxzq |y=yq u(x, y, z) = σxzq (x, z) ,
q = 1(1)Ny
,
(3.56)
Dxyr |z=zr u(x, y, z) = σxyr (x, y) ,
r = 1(1)Nz
,
(3.57)
where Ns are positive constants, depending on νs , for s ∈ S. As same as the idea of Tau-Collocation Method for ODEs and 2-dimensional PDEs, a polynomial with n th s degree in s, for s ∈ S, unx ny nz (x, y, z) =
ny nz nx X X X
a ix iy iz x ix y iy z iz
(3.58)
ix =0 iy =0 iz =0
is substituted into problem (3.54)-(3.57). An overdetermined system of linear algebraic equations with (nx + 1)(ny + 1)(nz + 1) unknown coefficients aix iy iz , where is = 0(1)ns for s ∈ S, are formed. A perturbation term Hnx ny nz (x, y, z) with ϕx , ϕy and ϕz free parameters τx , τy and τz respectively, where τx = (τx0 , τx1 , · · · , τx ϕx −1 ), τy = (τy0 , τy1 , · · · , τy ϕy −1 ) and τz = (τz0 , τz1 , · · · , τz ϕz −1 ), are added to the right-hand side of equation (3.54) so as to construct a balanced system of linear algebraic equations for determining the approximate polynomial solution unx ny nz (x, y, z). Then, problem (3.54)-(3.57) becomes L3 unx ny nz (x, y, z) = f (x, y, z) + Hnx ny nz (x, y, z) ,
(3.59)
75
where (x, y, z) ∈ [ax , bx ] × [ay , by ] × [az , bz ], subject to the supplementary conditions Dyzp |x=xp unx ny nz (x, y, z) = σyzp (y, z) ,
p = 1(1)Nx
,
(3.60)
Dxzq |y=yq unx ny nz (x, y, z) = σxzq (x, z) ,
q = 1(1)Ny
,
(3.61)
Dxyr |z=zr unx ny nz (x, y, z) = σxyr (x, y) ,
r = 1(1)Nz
,
(3.62)
which is defined as the associated Tau problem of problem (3.54)-(3.56). The format of perturbation term in equation (3.59) is followed it’s structure in the TauCollocation Method for ODEs and 2-dimensional PDEs and is defined as [a ,b ]
[a ,b ]
[a ,b ]
y x z Hnx ny nz (x, y, z) = gnx ny nz (x, y, z; τx , τy , τz ) V x,nxx −N (x) V y,nyy −N (y) V z,nzz −N (z) x +1 y +1 z +1
(3.63) [a ,b ]
s where V s,nss +N , ns > Ns − 1, are orthogonal polynomials of degree ns + Ns + 1 s +1
defined on [as , bs ], for s ∈ S. Usually, the orthogonal polynomials are chosen as the shifted Chebyshev or Legendre polynomials.
Inside equation (3.63),
gnx ny nz (x, y, z; τx , τy , τz ) is a function of x, y, z, ϕx free parameters τx , ϕy free parameters τy and ϕz free parameters τz . These free parameters τx , τy and τz balance the overdetermined system of linear algebraic equations. The another purpose for choosing the perturbation term Hnx ny nz (x, y, z) in the form of equation (3.63) is to produce exactly nx − Nx + 1, ny − Ny + 1 and nz − Nz + 1 collocation points which are the zeros of orthogonal polynomial [a ,b ]
[a ,b ]
[a ,b ]
y x z V x,nxx −N (x), V y,nyy −N (y) and V z,nzz −N (z) respectively. Let xix , yiy and ziz , x +1 y +1 z +1
for ix = 0(1)nx − Nx , iy = 0(1)ny − Ny and iz = 0(1)nz − Nz , be the nx − Nx + 1, ny − N y + 1
[a ,bz ] V z,nzz −N (z) z +1
and
n z − Nz + 1
respectively.
iy = 0(1)ny − Ny
and
zeros
of
[a ,b ]
x V x,nxx −N (x), x +1
[a ,b ]
y V y,nyy −N (y) y +1
and
By collocating (xix , yiy , ziz ), for ix = 0(1)nx − Nx , iz = 0(1)nz − Nz ,
zeros
into
problem
(3.59),
(nx − Nx + 1)(ny − Ny + 1)(nz − Nz + 1) collocation equations are generated. These collocation equations are used in finding out the unknown coefficients aix iy iz , ix = 0(1)nx , iy = 0(1)ny and iz = 0(1)nz , while all of the free parameters τx , τy and τz can be ignored. Note that it is not necessary to consider the detail of
76
gnx ny nz (x, y, z; τx , τy , τz ), i.e. the number of free parameters ϕx , ϕy and ϕz , through[a ,b ]
x out the computation because the zeros of orthogonal polynomials V x,nxx −N (x), x +1
[a ,b ]
[a ,b ]
y z V y,nyy −N (y) and V z,nzz −N (z) are used for collocation. y +1 z +1
Since (nx −Nx +1)(ny −Ny +1)(nz −Nz +1) collocation equations is not sufficient to find out the solution unx ny nz (x, y, z) of problem (3.59)-(3.62) with perturbation term (3.63), (nx + 1)(ny + 1)(nz + 1) − (nx − Nx + 1)(ny − Ny + 1)(nz − Nz + 1) more linear algebraic equations are required to do so. The rest of the linear algebraic [a ,b ]
[a ,b ]
y x equations are generated from conditions (3.60)-(3.62). Let V x,nxx +1 (x), V y,nyy +1 (y)
[a ,b ]
z and V z,nzz +1 (z) be the orthogonal polynomial of degree nx + 1, ny + 1 and nz + 1 de-
[a ,b ]
[a ,b ]
y x fined on [ax , bx ], [ay , by ] and [az , bz ] respectively. Usually, V x,nxx +1 (x), V y,nyy +1 (y) and
[a ,b ]
[a ,b ]
z x V z,nzz +1 (z) are selected as same basis as the orthogonal polynomials V x,nxx −N (x), x +1
[a ,b ]
[a ,b ]
y z V y,nyy −N (y) and V z,nzz −N (z) respectively, i.e. the shifted Chebyshev or Legendre y +1 z +1
[a ,b ]
[a ,b ]
y z polynomials. By collocating (ny + 1)(nz + 1) zeros of V y,nyy +1 (y) and V z,nzz +1 (z) into
[a ,b ]
[a ,b ]
x z Nx conditions (3.60), (nx + 1)(nz + 1) zeros of V x,nxx +1 (x) and V z,nzz +1 (z) into Ny
[a ,b ]
[a ,b ]
y x conditions (3.61) and (nx +1)(ny +1) zeros of V x,nxx +1 (x) and V y,nyy +1 (y) into Nz con-
ditions (3.62), Nx (ny + 1)(nz + 1) + Ny (nx + 1)(nz + 1) + Nz (nx + 1)(ny + 1) linear algebraic equations are generated. As same as the Tau-Collocation Method for 2-dimensional PDEs, there are redundant linear dependent equations inside the Nx (ny + 1)(nz + 1) + Ny (nx + 1)(nz + 1) + Nz (nx + 1)(ny + 1) linear algebraic equations which generated from the conditions. After several row echelon form operations of the system of linear algebraic equations have been done, these redundant linear dependent equations descript the same unknown coefficients which also be descript by other linear algebraic equations. These overlapped properties happen at the corner points between two nearby conditions in the rectangular domain. Every touching between two conditions gives one redundant linear dependent equations. For 3-dimensional case, there are Nx Ny touching at x − y plane with (nz + 1) levels, Nx Nz touching at x − z plane with (ny + 1) levels and Ny Nz touching at y − z plane with (nx + 1) levels.
However, there are not totally
Nx Ny (nz + 1) + Nx Nz (ny + 1) + Ny Nz (nx + 1) redundant linear dependent equations because the corners of the cubic domain is counted thrice. The number of
77
overlapped points at the corners touching by both x − y plane, z − x plane and y − z plane should be reduced. The number of such overlapped points at the corners are Nx Ny Nz . So the total number of redundant linear dependent equations is R := Nx Ny (nz + 1) + Nx Nz (ny + 1) + Ny Nz (nx + 1) − Nx Ny Nz , see Figure 6. Finally, Nx (ny + 1)(nz + 1) + Ny (nx + 1)(nz + 1) + Nz (nx + 1)(ny + 1) − R linear algebraic equations, which are linearly independent equations, are generated from the conditions (3.60)-(3.62). After the collocation process for problem (3.59)-(3.62) has been done, (nx − Nx + 1)(ny − Ny + 1)(nz − Nz + 1) +Nx (ny + 1)(nz + 1) + Ny (nx + 1)(nz + 1) + Nz (nx + 1)(ny + 1) − R = (nx + 1)(ny + 1)(nz + 1) linear algebraic equations are generated. Therefore, (nx + 1)(ny + 1)(nz + 1) unknown coefficients aix iy iz , is = 0(1)ns for s ∈ S, can be obtained by solving the whole system of linear algebraic equations by usual method. Then, the Tau approximant unx ny nz (x, y, z) to the solution of problem (3.54)-(3.57) can also be obtained.
3.2.5 Formulation of the Tau-Collocation Method for 3-dimensional Partial Differential Equations The formulation of the Tau-Collocation Method for 3-dimensional PDEs aims to convert the Tau problem (3.59)-(3.61) to a system of linear algebraic equations. The formulation is extended form the Tau-Collocation Method for 2-dimensional PDEs. An additional theorem is discussed first.
Lemma 3.4 Let a polynomial of degree nx , ny and nz in x, y and z respectively unx ny nz (x, y, z) =
ny nz nx X X X
ix =0 iy =0 iz =0
aix iy iz x ix y iy z iz = (znz ⊗ yny ⊗ xnx ) vec(Anx (ny nz ) ) ,
78
z
z
y
x
y
x Touching between two nearby conditions in x-y plane
Touching between two nearby conditions in x-z plane z
z
y
y
x
x Touching between two nearby conditions in y-z plane
Overlapped point at the corners touching by both three planes
Figure 6: The geometrical presentation of the R redundant linear dependent equations in a cubic domain.
79
where
Anx (ny nz )
is
a
a0(00)
a0(10)
···
a0(ny 0)
a0(01)
···
a0(ny nz )
a1(00) a1(10) · · · a1(ny 0) a1(01) · · · a1(ny nz ) = .. . anx (00) anx (10) · · · anx (ny 0) anx (01) · · · anx (ny nz )
(nx + 1) × (ny + 1)(nz + 1)
matrix,
xnx = (x 0 , x 1 , x 2 , · · · , x nx ),
yny = (y 0 , y 1 , y 2 , · · · , y ny ) and znz = (z 0 , z 1 , z 2 , · · · , z nz ). Then, the effect of com-
bined differentiation with respect to x, y and z of unx ny nz (x, y, z) is ∂ rx +ry +rz rz 0 ry 0 rx 0 un n n (x, y, z) = znz (η nz ) ⊗ yny (η ny ) ⊗ xnx (η nx ) vec(Anx (ny nz ) ) . ∂x rx ∂y ry ∂z rz x y z Proof By Lemma 3.2, ∂ rx +ry +rz ∂ rx +ry +rz u (x, y, z) = x A z ⊗ y n n n n n (n n ) nz ny ∂x rx ∂y ry ∂z rz x y z ∂x rx ∂y ry ∂z rz x x y z
.
Therefore, by Lemma 3.1,
∂ rx +ry +rz xn An (n n ) znz ⊗ yny ∂x rx ∂y ry ∂z rz x x y z ∂ ry +rz rx 0 xnx (η nx ) Anx (ny nz ) znz ⊗ yny = ∂y ry ∂z rz h i0 ∂ ry +rz rx 0 = znz ⊗ yny xnx (η nx ) Anx (ny nz ) ∂y ry ∂z rz h i0 ∂ ry +rz rx 0 = znz ⊗ yny vec xnx (η nx ) Anx (ny nz ) , ∂y ry ∂z rz rx 0 where xnx (η nx ) Anx (ny nz ) znz ⊗ yny is a 1 × 1 matrix so that it’s transpose i0 h 0 equals to itself and xnx (η nrxx ) Anx (ny nz ) is a (ny + 1)(nz + 1) × 1 matrix so that vec operation take no effects on it. Then, by Lemma 3.3, we have h i0 ∂ ry +rz rx 0 znz ⊗ yny vec xnx (η nx ) Anx (ny nz ) ∂y ry ∂z rz h i0 rz 0 ry 0 rx 0 = znz (η nz ) ⊗ yny (η ny ) vec xnx (η nx ) Anx (ny nz ) h i0 0 0 0 = znz (η nrzz ) ⊗ yny (η nryy ) xnx (η nrxx ) Anx (ny nz ) 0 0 0 0 = xnx (η nrxx ) Anx (ny nz ) znz (η nrzz ) ⊗ yny (η nryy ) ,
80
where
0 znz (η nrzz )
⊗
r 0 yny (η nyy )
h
0 xnx (η nrxx )
Anx (ny nz )
i0
is a 1 × 1 matrix so that it’s
transpose equals itself. Finally, by Lemma 3.2 again, we have
=
0 xnx (η nrxx )
Anx (ny nz )
0 znz (η nrzz )
0 yny (η nryy )
0
⊗ 0 0 0 znz (η nrzz ) ⊗ yny (η nryy ) ⊗ xnx (η nrxx ) vec(Anx (ny nz ) ) .
Theorem 3.4 Let a polynomial of degree nx , ny and nz in x, y and z respectively unx ny nz (x, y, z) =
ny nz nx X X X
ix =0 iy =0 iz =0
aix iy iz x ix y iy z iz = (znz ⊗ yny ⊗ xnx ) vec(Anx (ny nz ) ) ,
where
Anx (ny nz )
is
a
a a0(10) · · · a0(ny 0) a0(01) · · · a0(ny nz ) 0(00) a1(00) a1(10) · · · a1(ny 0) a1(01) · · · a1(ny nz ) = .. . anx (00) anx (10) · · · anx (ny 0) anx (01) · · · anx (ny nz )
(nx + 1) × (ny + 1)(nz + 1)
matrix,
xnx = (x 0 , x 1 , x 2 , · · · , x nx ),
yny = (y 0 , y 1 , y 2 , · · · , y ny ) and znz = (z 0 , z 1 , z 2 , · · · , z nz ). Also, let g(x, y, z) be any function with variables x, y and z. Then ∂ rx +ry +rz g(x, y, z) rx ry rz unx ny nz (x, y, z) ∂x ∂y ∂z rz 0 ry 0 rx 0 = znz (η nz ) ⊗ yny (η ny ) ⊗ g(x, y, z) xnx (η nx ) vec(Anx (ny nz ) ) . Proof The proof follows the proprieties and rules in Theorem 3.2 ∂ rx +ry +rz g(x, y, z) rx ry rz unx ny nz (x, y, z) ∂x ∂y ∂z 0 0 0 = znz (η nrzz ) ⊗ g(x, y, z) yny (η nryy ) ⊗ xnx (η nrxx ) vec(Anx (ny nz ) ) 0 0 0 = znz (η nrzz ) ⊗ yny (η nryy ) ⊗ g(x, y, z) xnx (η nrxx ) vec(Anx (ny nz ) ) .
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The formulation of the Tau-Collocation Method for 3-dimensional PDEs follows Tau-Collocation Method for 2-dimensional PDEs. Let
=
Πnx ny nz (x, y, z) νy νx X νz X X rz 0 ry 0 rx 0 znz (η nz ) ⊗ yny (η ny ) ⊗ qrx ry rz (x, y, z) xnx (η nx )
,
Πyzp nx ny nz (x, y, z) κy κx X κz X X 0 0 0 znz (η nrzz ) ⊗ yny (η nryy ) ⊗ θrx ry rz (x, y, z) xnx (η nrxx )
,
rx =0 ry =0 ry =0
=
rx =0 ry =0 rz =0
=
Πxzq nx ny nz (x, y, z) ωy ωx X ωz X X rz 0 ry 0 rx 0 znz (η nz ) ⊗ yny (η ny ) ⊗ ξrx ry rz (x, y, z) xnx (η nx ) rx =0 ry =0 rz =0
and Πyzr nx ny nz (x, y, z) ζy ζx ζz X X X rz 0 ry 0 rx 0 = znz (η nz ) ⊗ yny (η ny ) ⊗ χrx ry rz (x, y, z) xnx (η nx )
.
rx =0 ry =0 rz =0
By Theorem 3.4, the Tau problem (3.59)-(3.62) becomes Πnx ny nz (x, y, z) vec(Anx (ny nz ) ) = f (x, y, z) + Hnx ny nz (x, y, z) ,
(3.64)
where (x, y, z) ∈ [ax , bx ] × [ay , by ] × [az , bz ], subject to the supplementary conditions Πyzp nx ny nz (x, y, z) vec(Anx (ny nz ) ) = σyzp (y, z) ,
p = 1(1)Nx
,
(3.65)
Πxzq nx ny nz (x, y, z) vec(Anx (ny nz ) ) = σxzq (x, z) ,
q = 1(1)Ny
,
(3.66)
Πxyr nx ny nz (x, y, z) vec(Anx (ny nz ) ) = σxyr (x, y) ,
r = 1(1)Nz
,
(3.67)
where Ns are positive constants, depending on νs , for s ∈ S. Let xix , yiy and ziz , for ix = 0(1)nx − Nx , iy = 0(1)ny − Ny and iz = 0(1)nz − Nz , be the nx − Nx + 1, ny − Ny + 1 and nz − Nz + 1 zeros of the orthogonal polynomial [a ,b ]
[a ,b ]
[a ,b ]
y x z V x,nxx −N (x), V y,nyy −N (y) and V z,nzz −N (z) respectively. Also, let αix , βiy and x +1 y +1 z +1
82
γiz , for ix = 0(1)nx , iy = 0(1)ny and iz = 0(1)nz , be the nx + 1, ny + 1 and nz + 1 ze[a ,b ]
[a ,b ]
[a ,b ]
y x z ros of the orthogonal polynomial V x,nxx +1 (x), V y,nyy +1 (y) and V z,nzz +1 (z) respectively.
By collocating (nx − Nx + 1)(ny − Ny + 1)(nz − Nz + 1) zero points (xix , yiy , ziz ), for ix = 0(1)nx − Nx , iy = 0(1)ny − Ny and iz = 0(1)nz − Nz , into equation (3.64) and collocating (βiy , γiz ), (αix , γiz ) and (αix , βiy ), for ix = 0(1)nx , iy = 0(1)ny and iz = 0(1)nz , zero points into equations (3.65), (3.66) and (3.67) respectively, we can obtain Γ vec(Anx (ny nz ) ) = F ,
where
(3.68)
Γyz vec(Anx (ny nz ) ) = Fyz ,
p = 1(1)Nx
,
(3.69)
Γxz vec(Anx (ny nz ) ) = Fxz ,
q = 1(1)Ny
,
(3.70)
Γxy vec(Anx (ny nz ) ) = Fxy ,
r = 1(1)Nz
,
(3.71)
Γ =
Πnx ny nz (x0 , y0 , z0 ) Πnx ny nz (x0 , y0 , z1 ) .. . Πnx ny nz (x0 , y0 , znz −Nz ) Πnx ny nz (x0 , y1 , z0 ) .. . Πnx ny nz (xnx −Nx , yny −Ny , znz −Nz )
,
is a (nx − Nx + 1)(ny − Ny + 1)(nz − Nz + 1) × (nx + 1)(ny + 1)(nz + 1) matrix, Πyz1 nx ny nz (x1 , β0 , γ0 ) Πyz1 nx ny nz (x1 , β0 , γ1 ) .. . Γyz = , Πyz1 nx ny nz (x1 , β0 , γnz −Nz ) Πyz2 nx ny nz (x2 , β1 , γ0 ) .. . ΠyzNx nx ny nz (xNx , βny −Ny , γnz −Nz )
83
is a Nx (ny + 1)(nz + 1) × (nx + 1)(ny + 1)(nz + 1) matrix, Πxz1 nx ny nz (α0 , y1 , γ0 ) Πxz1 nx ny nz (α0 , y1 , γ1 ) .. . Γxz = Πxz1 nx ny nz (α0 , y1 , γnz −Nz ) Πxz2 nx ny nz (α1 , y2 , γ0 ) .. . ΠxzNy nx ny nz (αnx −Nx , yNy , γnz −Nz )
is a Ny (nx + 1)(nz + 1) × (nx + 1)(ny + 1)(nz + 1) matrix, Πxy1 nx ny nz (α0 , β0 , z1 ) Πxy1 nx ny nz (α0 , β1 , z1 ) .. . Γxy = Πxy1 nx ny nz (α0 , βny −Ny , z1 ) Πxy2 nx ny nz (α1 , β0 , z2 ) .. . ΠxyNz nx ny nz (αnx −Nx , βny −Ny , γNz )
is a Nz (nx + 1)(ny + 1) × (nx + 1)(ny + 1)(nz + 1) matrix, f (x0 , y0 , z0 ) f (x0 , y0 , z1 ) .. . F = f (x0 , y0 , znz −Nz ) f (x0 , y1 , z0 ) .. . f (xnx −Nx , yny −Ny , znz −Nz )
,
,
,
84
is a column vector with (nx − Nx + 1)(ny − Ny + 1)(nz − Nz + 1) elements, σyz1 (β0 , γ0 ) σyz1 (β0 , γ1 ) .. . Fyz = , σyz1 (β0 , γnz −Nz ) σyz2 (β1 , γ0 ) .. . σyzNx (βny −Ny , γnz −Nz ) is a column vector with Nx (ny + 1)(nz + 1) elements, σxz1 (α0 , γ0 ) σxz1 (α0 , γ1 ) .. . Fxz = σxz1 (α0 , γnz −Nz ) σxz2 (α1 , γ0 ) .. . σxzNy (αnx −Nx , γnz −Nz )
is a column vector with Ny (nx + 1)(nz + 1) elements, and σxy1 (α0 , β0 ) σxy1 (α0 , β1 ) .. . Fxy = σxy1 (α0 , βnz −Nz ) σxy2 (α1 , β0 ) .. . σxyNz (αnx −Nx , βny −Ny )
,
,
is a column vector with Nz (nx + 1)(ny + 1) elements. Recall
the
number
of
redundant
linear
dependent
equations
R := Nx Ny (nz + 1) + Nx Nz (ny + 1) + Ny Nz (nx + 1) − Nx Ny Nz . By merging equation
85
(3.68)-(3.71), we have a resulting system of linear algebraic equations Γ F Γyz Fyz , vec(Anx (ny nz ) ) = Γxz Fxz Γxy Fxy
(3.72)
0
where (Γ, Γyz , Γxz , Γxy ) is a ((nx + 1)(ny + 1)(nz + 1) + R) × (nx + 1)(ny + 1)(nz + 1) 0
matrix and (F , Fyz , Fxz , Fxy ) is a column vector with (nx + 1)(ny + 1)(nz + 1) + R elements. Since there are R redundant linear dependent equations inside system (3.72), the rank of system (3.72) is (nx + 1)(ny + 1)(nz + 1). Therefore, all of the unknown coefficients aix iy iz , is = 0(1)ns for s ∈ S, can be obtained by solving the system of linear algebraic equations (3.72) by usual method without finding out free parameters τs , for s ∈ S. And, the Tau approximant unx ny nz (x, y, z) to the solution of problem (3.54)-(3.57) can be obtained. The following examples show the results of the Tau-Collocation Method applied into different types of 3-dimensional PDEs.
86
Example 3.14 Consider a wave problem ∂2 1 ∂2 ∂2 u(x, y, t) = u(x, y, t) + u(x, y, t) , ∂t 2 2 ∂x 2 ∂y 2
(x, y, t) ∈ [0, π] × [0, π] × [0, 1]
with initial conditions
u(x, y, 0) = 1 , ∂ u(x, y, 0) = sin x cos y , ∂t
(x, y) ∈ [0, π] × [0, π]
,
and boundary conditions u(0, y, t) = 1 , u(π, y, t) = 1 ,
(y, t) ∈ [0, π] × [0, 1]
,
u(x, 0, t) = 1 + sin x sin t , u(x, π, t) = 1 − sin x sin t ,
(x, t) ∈ [0, π] × [0, 1]
.
The exact solution of this problem is u(x, y, t) = 1 + sin x cos y sin t , see WazWaz [52]. The Tau-Collocation Method can be applied directly to this problem without any pre-approximation has been done for the right-hand side function of the conditions. In Table 13 we give the maximum absolute error E := max|u(x, y, t) − unx ny nt (x, y, t)| for Example 3.14 obtained from the TauCollocation Method with Tau degrees nx = ny = nt = 4(1)12.
,
87
Table 13: The maximum absolute error E := max|u(x, y, t) − unx ny nt (x, y, t)| for Example 3.14 obtained from the Tau-Collocation Method with Tau degrees nx = ny = nt = 4(1)12. nx , ny , nt
E
n x , ny , nt
E
4, 4, 4
1.9942 × 10 −2
9, 9, 9
3.2269 × 10 −7
5, 5, 5
3.7365 × 10 −3
10, 10, 10 1.9995 × 10 −8
6, 6, 6
2.9623 × 10 −4
11, 11, 11 2.3405 × 10 −9
7, 7, 7
3.5598 × 10 −5
12, 12, 12 6.2605 × 10 −10
8, 8, 8
2.8766 × 10 −6
Example 3.15 Consider a 2-dimensional heat flow problem ∂ ∂2 ∂2 u(x, y, t) = u(x, y, t) + u(x, y, t) , ∂t ∂x 2 ∂y 2
(x, y, t) ∈ [0, π] × [0, π] × [0, 1]
,
with initial conditions
u(x, y, 0) = sin x cos y ,
(x, y) ∈ [0, π] × [0, π]
,
and boundary conditions u(0, y, t) = u(π, y, t) = 0 , ∂ ∂ u(x, 0, t) = u(x, π, t) = 0 , ∂t ∂t
(x, t) ∈ [0, π] × [0, 1]
,
(y, t) ∈ [0, π] × [0, 1]
.
The exact solution of this problem is u(x, y, t) = e −2t sin x cos y
,
see WazWaz [52]. The Tau-Collocation Method can be applied directly to this problem without any pre-approximation has been done for the right-hand side
88
function of the conditions.
In Table 14 we give the maximum absolute error
E := max|u(x, y, t) − unx ny nt (x, y, t)|
for
Example
3.15
obtained
from
the
Tau-Collocation Method with Tau degrees nx = ny = nt = 4(1)12.
Table 14: The maximum absolute error E := max|u(x, y, t) − unx ny nt (x, y, t)| for Example 3.15 obtained from the Tau-Collocation Method with Tau degrees nx = ny = nt = 4(1)12. nx , ny , nt
E
n x , ny , nt
E
4, 4, 4
1.9789 × 10 −2
9, 9, 9
2.8869 × 10 −7
5, 5, 5
3.4720 × 10 −3
10, 10, 10 1.6267 × 10 −8
6, 6, 6
3.2088 × 10 −4
11, 11, 11 2.0078 × 10 −9
7, 7, 7
4.6109 × 10 −5
12, 12, 12 2.4547 × 10 −10
8, 8, 8
2.7158 × 10 −6
89
Chapter 4
Advanced Topics in the Tau-Collocation Method
The Tau-Collocation Method not only can approximate the solution of linear ordinary and partial differential equations, but can be used in finding the numerical approximate solution of nonlinear ordinary and partial differential equations [26, 47]. The solution of nonlinear differential equations can be handled by the TauCollocation Method with incorporation of iterative schemes or Adomian’s polynomials [5, 48, 51]. Also, the Tau-Collocation Method incorporated with segmentation technique can be used in tackled the differential problem defined on a large domain or the solution of the differential problem experienced with a sharp change over the defined domain. In general, a given differential problem is first handled by the corporate method, then the Tau-Collocation Method acted as the final processor to find out the approximate solution of the treated problem. These advanced topics in the Tau-Collocation Method are our current research result and will be discussed in this chapter.
4.1
Numerical solution of Nonlinear Ordinary and Partial Differential Equations
The solution of nonlinear ordinary and partial differential equations can also be approximated by the Tau-Collocation Method with incorporation of iterative schemes, such as the linear iterative scheme and the quadratic iterative scheme (Newton’s iterative scheme). This idea is followed the idea of the Tau Method with incorporation of such iterative schemes, see [34, 36, 37, 39, 40, 50]. In our research, we also try another approach to approximate the solution of nonlinear differential equations by incorporation of the Tau-Collocation Method and Adomian’s polynomials. No linearization process is required when this approach is applied. As we
90
know, this approach is fresh in the research area of the spectral methods and is first proposed in this thesis.
4.1.1 Tau-Collocation Method Incorporated with Linear Iterative Scheme In our research, we succeed to incorporated the Tau-Collocation Method with the linear iterative scheme to approximate the solution of nonlinear ODEs and 2dimensional PDEs. The process of formulating the linearized differential equation is very simple.
4.1.1.1 Nonlinear Ordinary Differential Equations Let ν be the order of the nonlinear ODE. The linear iterative scheme can be applied to the following nonlinear ODE mi ν X i X X
qijr (x)
i=0 j=0 r=0
r i dj d y(x) y(x) = f (x) , j dx dx i
x ∈ [a, b]
,
(4.1)
j
d where mi , i = 0(1)ν, is the highest power of the term dx j y(x), for j = 0(1)i, qijr (x) is j r i d d the variable coefficient of the term dx y(x), for i = 0(1)ν, r = 0(1)mi and j y(x) dx i
j = 0(1)i, and a and b are constant. Let y(k) (x) be the k th iterative approximation
to y(x) and let yn(k) (x) be the numerical approximate solution to y(k) (x) with degree n. By the linear iterative scheme, equation (4.1) becomes mi ν X i X X i=0 j=0 r=0
qijr (x)
dj y(k−1) (x) dx j
r
di y(k) (x) = f (x) , dx i
k = 1, 2, 3, · · ·
. (4.2)
A nonlinear ODE is first linearizated into a linear ODE. An initial guess y(0) (x), which can be a polynomial satisfied the given supplementary conditions, is substituted into equation (4.2) with k = 1, so that equation (4.2) becomes a linearized ODE. The Tau-Collocation Method is applied to find out the approximate solution of this linearized ODE and then the Tau approximant yn(1) to the solution of the linearized ODE can be obtained. Then, the Tau approximant yn(1) is substituted into
91
y(1) of equation (4.2) with k = 2 for next iteration, and the Tau-Collocation Method is applied again to the linearized ODE. Therefore, the Tau approximant yn(2) can be obtained. The process is repeated by using the result of k th iteration to activate (k + 1) th iteration and a sequence of approximations {yn(k) (x)}, for k = 1, 2, 3, · · · is generated. If the process is convergent, a fixed point will be reached after several iterations by inserting {y(0) (x), yn(1) (x), yn(2) (x), · · ·} into problem (4.2). The iterative process is ended when the maximum absolute value of the difference between two consecutive estimates max|yn(k+1) (x) − yn(k) (x)|, for k = 1, 2, 3, · · ·, is less than a tolerance error ε which is set as ε = 1 × 10 −12 in usual. The following example shows the results obtained from the Tau-Collocation Method with incorporation of the linear iterative scheme.
Example 4.1 Consider a nonlinear ODE with boundary conditions d d2 y(x) + ( y(x)) 3 = 0 , 2 dx dx
[0, 1]
,
with boundary conditions d 3 , y(0) + y(0) = √ dx 2 d 1 y(1) = . dx 2 The exact solution of this problem is 1
y(x) = (2x + 2) 2
,
see Sachdev [46]. By the linear iterative scheme, the given problem can be linearized as d2 y(k) (x) + dx 2
d y(k−1) (x) dx
2
d y(k) (x) = 0 , dx
k = 1, 2, 3, · · ·
.
In Table 15 we give the maximum absolute error E := max|y(x) − yK(n) (x)| and number of iterations K for Example 4.1 obtained from the Tau-Collocation Method with incorporation of the linear iterative scheme (TC-LIS) with degrees n = 4(1)12
92
of the approximate solution and ε = 1 × 10 −12 . The results are compared with the results of the operational approach to the Tau Method with incorporation of the linear iterative scheme (OA-LIS). An initial gauss y(0) (x) = 1 is used in both cases. Similar accuracy of results can be obtained from both methods.
Table 15: The maximum absolute error E := max|y(x) − yK(n) (x)| and number of iterations K for Example 4.1 obtained from the Tau-Collocation Method (TC-LIS) and the operational approach to the Tau Method (OA-LIS) with incorporation of the linear iterative scheme with degrees n = 4(1)12 of the approximate solution and ε = 1 × 10 −12 . n
K
E (TC-LIS)
K
E (OA-LIS)
4
19 4.0948 × 10 −4
19 7.9982 × 10 −4
5
17 1.0382 × 10 −4
18 1.1726 × 10 −4
6
16 3.2375 × 10 −6
16 6.1837 × 10 −6
7
15 1.6006 × 10 −6
16 1.6886 × 10 −6
8
14 4.7776 × 10 −8
15 9.3220 × 10 −8
9
14 2.9611 × 10 −8
14 3.1054 × 10 −8
10
14 8.8082 × 10 −10
14 1.7365 × 10 −9
11
14 6.1007 × 10 −10
14 6.4085 × 10 −10
12
14 1.8753 × 10 −11
14 3.6080 × 10 −11
4.1.1.2 Nonlinear 2-dimensional Partial Differential Equations The Tau-Collocation Method for 2-dimensional PDEs can also incorporate with linear iterative scheme to approximate the solution of nonlinear 2-dimensional PDEs. Let νx and νy be the order in x and y respectively of the nonlinear PDE. The linear
93
iterative scheme can be applied to the following nonlinear PDE νy iy m i x i y νx X ix X X X X
ix =0 iy =0 rx =0 ry =0 r=0
r ∂ ix +iy ∂ rx +ry qix iy rx ry r (x, y) u(x, y) u(x, y) ∂x rx ∂y ry ∂x ix ∂y iy
= f (x, y) ,
(4.3)
where (x, y) ∈ [ax , bx ] × [ay , by ], mix iy , ix = 0(1)νx and iy = 0(1)νy , is the highest power of the term
∂ rx +ry u(x, y), ∂x rx ∂y ry
for rx = 0(1)ix and ry = 0(1)iy , qix iy rx ry r (x, y)
is the variable coefficient and ax ,bx , ay and by are constant. Let u(k) (x, y) be the k th iterative approximation to u(x, y) and let unx ny (k) (x, y) be the numerical approximate solution to u(k) (x, y) with degree nx and ny in x and y receptivity. By the linear iterative scheme, equation (4.3) becomes νy iy m i x i y νx X ix X X X X
ix =0 iy =0 rx =0 ry =0 r=0
= f (x, y) ,
r ∂ rx +ry ∂ ix +iy qix iy rx ry r (x, y) u (x, y) u(k) (x, y) (k−1) ∂x rx ∂y ry ∂x ix ∂y iy
k = 1, 2, 3, · · ·
.
(4.4)
As same as the case for ODEs, an initial guess u(0) (x, y), which can be a polynomial satisfied the given supplementary conditions, is substituted into equation (4.4) with k = 1 so as to start the iteration process. A sequence of approximations {unx ny (k) (x, y)}, for k = 1, 2, 3, · · ·, can be obtained from applying the TauCollocation Method to the linearized equation (4.4) in every stages of k = 1, 2, 3, · · ·. If the process is convergent, a fixed point will be reached after several iterations and the iterative process is ended when the maximum absolute value of the difference between two consecutive estimates max|unx ny (k+1) (x, y) − unx ny (k) (x, y)|, for k = 1, 2, 3, · · ·, is less than a tolerance error ε which is set as ε = 1 × 10 −12 in usual. The following example shows the results obtained from the Tau-Collocation Method with incorporation of the linear iterative scheme.
Example 4.2 Consider a nonlinear PDE ∂ ∂2 u(x, t) − 2 u(x, t) − (u(x, t)) 2 = f (x, t) , ∂t ∂t
(x, t) ∈ [0, 1] × [0, 1]
,
94
where f (x, t) = e t sin πx (1 + π 2 − e t sin πx), with initial conditions u(x, 0) = sin πx ,
x ∈ [0, 1]
,
and boundary conditions u(0, t) = 0 , u(1, t) = 0 ,
t ∈ [0, 1]
.
The exact solution of this problem is u(x, t) = e t sin πx , see Ortiz and Pham Ngoc Dinh [34]. By the linear iterative scheme, the given problem can be linearized as ∂2 ∂ u(k) (x, t) − 2 u(k) (x, t) − u(k−1) u(k) = f (x, y) , ∂t ∂t
k = 1, 2, 3, · · ·
.
The Tau-Collocation Method can be applied directly to this problem without any pre-approximation has been done for the right-hand side functions of both problem and condition. In Table 16 we give the maximum absolute error E := max|u(x, t) − unx nt (K) (x, t)| and number of iterations K for Example 4.2 obtained from the Tau-Collocation Method with incorporation of linear iterative scheme with Tau degrees nx = nt = 4(1)10 and ε = 1 × 10 −9 . An initial gauss u(0) (x, y) = 0 is used in this example.
4.1.2 Tau-Collocation Method Incorporated with Quadratic Iterative Scheme In our research, we succeed to incorporated the Tau-Collocation Method with the quadratic iterative scheme to approximate the solution of nonlinear ODEs and 2-dimensional PDEs. The quadratic iterative scheme gives better convergence then the linear iterative scheme. Also, it can be applied to wider types of differential equations. However, the process of formulating the linearized differential equations
95
Table 16: The maximum absolute error E := max|u(x, t) − unx nt (K) (x, t)| and number of iterations K for Example 4.2 obtained from the Tau-Collocation Method with incorporation of linear iterative scheme with Tau degrees nx = nt = 4(1)10 and ε = 1 × 10 −9 . nx , nt
K
4,4
E
n x , nt
K
E
13 7.3382 × 10 −2
8,8
12 1.2757 × 10 −6
5,5
12 2.6617 × 10 −2
9,9
12 9.5285 × 10 −7
6,6
12 4.2462 × 10 −4
10,10
12 4.1416 × 10 −9
7,7
12 1.4100 × 10 −4
by the quadratic iterative scheme is not as easy as the linear iterative scheme. The computation process of the Tau-Collocation Method with incorporation of the quadratic iterative scheme is as same as the Tau-Collocation Method with incorporation of the linear iterative scheme, only the process of formulating the linearized differential equations is different.
4.1.2.1 Nonlinear Ordinary Differential Equations Let ν be the order of the nonlinear ODE. The quadratic iterative scheme can be applied to the following nonlinear differential equation d dν G x, y(x), y(x), · · · , ν y(x) = f (x) , x ∈ [a, b] dx dx
,
(4.5)
where a and b are constant. Let y(k) (x) be the k th iterative approximation to y(x) and let yn(k) (x) be the numerical approximate solution to y(k) (x) with degree n. d dν Also, let G(k) = G x, y(k) (x), dx y(k) (x), · · · , dx ν y(k) (x) . By the quadratic iterative scheme, equation (4.5) becomes
ν X di di y(k) (x) − y(k−1) (x) dx i dx i ∂ i=0
∂G(k−1) di y (x) dx i (k−1)
= f (x) − Fk−1
,
96
where k = 1, 2, 3, · · ·. Then, the nonlinear ODE is changed into a linearized ODE. The rest of the computation process is exactly as same as the Tau-Collocation Method incorporated with linear iterative scheme. The following example shows the results obtained from the Tau-Collocation Method incorporated with the quadratic iterative scheme.
Example 4.3 Consider a nonlinear ODE with boundary conditions discussed in Example 4.1. By the quadratic iterative scheme, the given problem can be linearized as 2 3 d d d d2 y(k) (x) + 3 y(k−1) (x) y(k) (x) = 2 y(k−1) (x) , k = 1, 2, 3, · · · dx 2 dx dx dx In Table 15 we give the maximum absolute error E := max|y(x) − yn(K) (x)| and number of iterations K for Example 4.3 obtained from the Tau-Collocation Method with incorporation of the quadratic iterative scheme (TC-QIS) with degrees n = 4(1)12 of the approximate solution and ε = 1×10 −12 . The results are compared with the results of the operational approach to the Tau Method with incorporation of the quadratic iterative scheme (OA-QIS). An initial gauss y(0) (x) = 1 is used in both cases. 4.1.2.2 Nonlinear 2-dimensional Partial Differential Equations Let u = u(x, y) and let νx and νy be the order in x and y respectively of the nonlinear PDE. The quadratic iterative scheme can be applied to the following nonlinear PDE
∂ ∂2 ∂ νx ∂ ∂ νx +νy G x, y, u, u, u, · · · , u, u, · · · , u ∂x ∂x 2 ∂x νx ∂y ∂x νx ∂y νy
= f (x, y) . (4.6)
where (x, y) ∈ [ax , bx ] × [ay , by ]. Let u(k) be the k th iterative approximation to u and let unx ny (k) be the numerical approximate solution to u(k) with degree nx and ny in x and y receptivity. Also, let G(k) = G(x, y, u(k) ,
∂ ∂2 ∂ νx ∂ ∂ νx +νy u(k) , u , · · · , u , u , · · · , u(k) ) (k) (k) (k) ∂x ∂x 2 ∂x νx ∂y ∂x νx ∂y νy
.
97
Table 17: The maximum absolute error E := max|y(x) − yK(n) (x)| and number of iterations K for Example 4.3 obtained from the Tau-Collocation Method (TC-QIS) and the operational approach to the Tau Method (OA-QIS) with incorporation of the quadratic iterative scheme with degrees n = 4(1)12 of the approximate solution and ε = 1 × 10 −12 . n
K
E (TC-QIS)
K
E (OA-QIS)
4
6
4.0948 × 10 −4
6
7.9982 × 10 −4
5
6
1.0382 × 10 −4
6
1.1726 × 10 −4
6
6
3.2375 × 10 −6
6
6.1837 × 10 −6
7
6
1.6006 × 10 −6
6
1.6886 × 10 −6
8
6
4.7776 × 10 −8
6
9.3220 × 10 −8
9
6
2.9611 × 10 −8
6
3.1054 × 10 −8
10
6
8.8081 × 10 −10
6
1.7365 × 10 −9
11
6
6.1009 × 10 −10
6
6.4085 × 10 −10
12
6
1.8732 × 10 −11
6
3.6058 × 10 −11
By the Quadratic Iterative Scheme, equation (4.6) becomes νy νx X X ∂ G(k−1) ∂ rx +ry ∂ rx +ry u(k) − u(k−1) = f (x, y) − G(k−1) rx +ry r r r r ∂ x y x y ∂x ∂y ∂x ∂y ∂( ∂x rx ∂y ry u(k−1) ) rx =0 ry =0 where k = 1, 2, 3, · · ·. Then, the given nonlinear PDE is changed into a linearized PDE. The rest of the computation process is exactly as same as the Tau-Collocation Method incorporated with linear iterative scheme. The following example shows the results obtained from the Tau-Collocation Method with incorporation of the quadratic iterative scheme.
Example 4.4 Consider a nonlinear PDE discussed in Example 4.2. By the quadratic iterative
,
98
scheme, the given problem can be linearized as 2 ∂2 ∂ u(k) (x, t) − 2 u(k) (x, t) − u(k−1) (x, t)u(k) (x, t) = f (x, t) − u(k−1) (x, t) ∂t ∂t
,
where k = 1, 2, 3, · · ·. The Tau-Collocation Method can be applied directly to this problem without any pre-approximation has been done for the right-hand side functions of both problem and condition. In Table 18 we give the maximum absolute error max |u(x, t) − unx nt (K) (x, t)| and number of iterations K for Example 4.4 obtained from the Tau-Collocation Method with incorporation of quadratic iterative scheme with Tau degrees nx = nt = 4(1)10 and ε = 1 × 10 −9 . An initial gauss u(0) (x, y) = 0 is used in this problem.
Table 18: The maximum absolute error max |u(x, t) − unx nt (K) (x, t)| and number of iterations K for Example 4.4 obtained from the Tau-Collocation Method with incorporation of quadratic iterative scheme with Tau degrees nx = nt = 4(1)10 and ε = 1 × 10 −9 . nx , nt
K
E
n x , nt
K
E
4,4
4
7.3383 × 10 −2
8,8
4
1.2782 × 10 −6
5,5
4
2.6405 × 10 −2
9,9
4
9.5270 × 10 −7
6,6
4
4.2462 × 10 −4
10,10
4
4.1652 × 10 −9
7,7
4
1.4044 × 10 −4
4.1.3 Tau-Collocation Method Incorporated with Adomian Polynomial In our research, we developed a new approach, other than the Tau-Collocation Method incorporated with iterative scheme, to approximate the solution of nonlinear ordinary and partial differential equations by mixing up the Tau-Collocation Method and Adomian’s polynomials [5, 48, 51]. The Adomian’s polynomials come from the
99
Adomian decomposition method [1, 2, 52]. The role of the Adomian’s polynomials is to convert the nonlinear terms of the differential equation into a set of polynomials. No linearization process is required for the suggested method and it can be used in approximating the solution of nonlinear differential equations with highly nonlinear terms, such as trigonometric and exponential nonlinearity. However, the suggested method is not applicable to the fully nonlinear differential equations. 4.1.3.1 Nonlinear Ordinary Differential Equations The Tau-Collocation Method with incorporation of the Adomian’s polynomials can be applied into a nonlinear differential equation Dv y(x) + F (y(x)) = f (x) ,
x ∈ [a, b]
,
(4.7)
subject to the supplementary conditions Dvr |x=xr y(x) = σr (xr ) ,
r = 1(1)ν
,
(4.8)
where F (y(x)) is the nonlinear term of the given problem and other symbols follow P the symbols in Section 3.1.1. Let y(x) = ∞ i=0 y(i) (x), where y(i) (x), i = 0(1)∞, are the decomposed solutions to the problem (4.7)-(4.8) and let yn(i) (x) be the numerical approximate solution to y(i) (x) with degree n, for i = 0(1)∞. The nonlinear term F (y(x)) in equation (4.7) can be written in term of the Adomian’s polynomials ! i ∞ ∞ X X X 1 di j F λ y(j) (x) (4.9) F (y(x)) = Ai = i i! dλ j=0 i=0 i=0 λ=0 P i i d j where Ai = i!1 dλ , i = 0(1)∞, are the Adomian’s polynomiiF j=0 λ y(j) (x) λ=0
als. Note that Ai depends on y(0) (x), y(1) (x), · · · y(i) (x) only, for i = 0(1)∞. Then, equation (4.7) can be written as
Dv y(0) (x) + y(1) (x) + · · · = f (x) − (A0 + A1 + · · ·) ,
x ∈ [a, b]
.
Then, problem (4.7)-(4.8) can be decomposed into infinity many sub-problems by the principle of superposition and it become Dv y(0) (x) = f (x) Dv y(i+1) (x) = −Ai and D | D | vr x=xp y(0) (x) = σr (xr ) vr x=xp y(i+1) (x) = 0
, (4.10)
100
where x ∈ [a, b], r = 0(1)ν and i = 0(1)∞. Firstly, the Tau-Collocation Method is used in finding out the approximate solution yn(0) (x) to y(0) (x). Then, this approximate solution yn(0) (x) is substituted into Adomian’s polynomials A0 in equation (4.10) with i = 0 and the Tau-Collocation Method is applied again to obtain the approximate solution yn(1) (x) to y(1) (x). The process is repeated by substituting the results of approximate solution yn(i) (x), i = 0(1)k, to Adomian’s polynomials Ak+1 in problem (4.10) so as to activate (k + 1) th differential problem. Note that the differential operators of all sub-problems (4.10) with the Adomian’s polynomials are identical for i = 0(1)∞. Hence, the resultant matrix G in equation (3.9), refer Section 3.1.2, of these sub-problems are identical. Then, the approximate solution yn(i) (x), i = 0(1)∞, can be obtained by multiplying (G0 ) −1 to different column vectors sn in equation (3.10). The process is ended when the maximum absolute value of the max|yn(i) (x)|, for i = 1, 2, 3, · · ·, is less than a tolerance error ε which is set as ε = 1 × 10 −12 in usual. Finally, a sequence of approximations {yn(i) (x)}, i = 0(1)K, where K ∈ N such that max|yn(K) (x)| < ε, is generated. Then, the approximate P solution of problem (4.7)-(4.8) is obtained by K i=0 yn(i) (x) which is the truncated
series of yn(i) (x), i = 0(1)∞. The following example shows the results obtained by the Tau-Collocation Method with incorporation of the Adomian’s polynomials.
Example 4.5 Consider a nonlinear ODE with boundary conditions discussed in Example 4.1. The Adomian’s polynomials of the given problem are as following 3 d A0 = y0 (x) dx 2 d d A1 = 3 y0 (x) y1 (x) dx dx ! 2 d d d d A2 = 3 y0 (x) y1 (x) + y0 (x) y2 (x) dx dx dx dx 3 2 d d d d d d A3 = y1 (x) + 6 y0 (x) y1 (x) y2 (x) + 3 y0 (x) y3 (x) dx dx dx dx dx dx .. .
101
In Table 19 we give the maximum absolute error E := max|y(x) −
PK
i=0
yi(n) (x)| for
Example 4.5 obtained by the Tau-Collocation Method incorporated with the Adomian’s polynomials with Tau degrees n = 4(1)12 and ε = 1 × 10 −12 .
Table 19: The maximum absolute error E := max|y(x) −
PK
i=0
yi(n) (x)| for Exam-
ple 4.5 obtained by the Tau-Collocation Method incorporated with the Adomian’s polynomials with Tau degrees n = 4(1)12 and ε = 1 × 10 −12 . n
K
EK
E
4
37 4.9577 × 10 −13
4.0948 × 10 −4
5
40 7.4812 × 10 −13
1.0382 × 10 −4
6
33 4.8377 × 10 −13
3.2375 × 10 −6
7
37 6.3094 × 10 −13
1.6006 × 10 −6
8
36 5.3337 × 10 −13
4.7776 × 10 −8
9
36 7.7406 × 10 −13
2.9611 × 10 −8
10 36 6.5814 × 10 −13
8.8145 × 10 −10
11 36 6.9758 × 10 −13
6.0874 × 10 −10
12 36 6.7850 × 10 −13
2.0035 × 10 −11
Example 4.6 Consider a nonlinear ODE with trigonometric nonlinearity h πi d2 d y(x) − y(x) sinh y(x) = 0 , x ∈ 0, dx 2 dx 4
with initial conditions
y(0) = 0 , d y(0) = 1 . dx The exact solution of this problem is y(x) = sinh −1 (tan x)
,
,
102
see Sachdev [46]. The Tau-Collocation Method incorporated with the Adomian’s polynomials can be applied to this problem directly, while the Tau-Collocation Method incorporated with iterative schemes can hardly to do so. The Adomian’s polynomials of the given problem are as following d y0 (x) dx d d = − cosh (y0 (x)) y1 (x) y0 (x) − sinh (y0 (x)) y1 (x) dx dx 1 2 d = − sinh (y0 (x)) (y1 (x)) y0 (x) 2 dx d d − cosh (y0 (x)) y2 (x) y0 (x) − cosh (y0 (x)) y1 (x) y1 (x) dx dx d − sinh (y0 (x)) y2 (x) dx 1 d = − cosh (y0 (x)) (y1 (x))3 y0 (x) 6 dx d d y0 (x) y2 (x) − 1/2 sinh (y0 (x)) (y1 (x))2 y1 (x) − sinh (y0 (x)) y1 (x) dx dx d d − cosh (y0 (x)) y3 (x) y0 (x) − cosh (y0 (x)) y2 (x) y1 (x) dx dx d d − cosh (y0 (x)) y1 (x) y2 (x) − sinh (y0 (x)) y3 (x) dx dx .. .
A0 = − sinh (y0 (x)) A1 A2
A3
In Table 20 we give the maximum absolute error E := max|y(x) −
PK
i=0
yi(n) (x)| for
Example 4.6 obtained by the Tau-Collocation Method incorporated with the Adomian’s polynomials with Tau degrees n = 4(1)12 and ε = 1 × 10 −12 . The graph of error curves of the approximate solutions for Example 4.6 obtained by the TauCollocation Method incorporated with the Adomian’s polynomials with Tau degrees n = 7 and 8 and K = 16 and 14 respectively are shown in Figure 7. It shows a remarkable equioscillatory behavior.
4.1.3.2 Nonlinear Partial Differential Equations The idea of the Tau-Collocation Method with incorporation of the Adomian’s
103
Table 20: The maximum absolute error E := max|y(x) −
PK
i=0
yi(n) (x)| for Exam-
ple 4.6 obtained by the Tau-Collocation Method incorporated with the Adomian’s polynomials with Tau degrees n = 4(1)12 and ε = 1 × 10 −12 . n
K
EK
E
4
18 6.5131 × 10 −13
2.0484 × 10 −3
5
17 4.2540 × 10 −13
1.3295 × 10 −4
6
15 9.6960 × 10 −13
2.0507 × 10 −5
7
16 2.1197 × 10 −13
2.7627 × 10 −6
8
14 9.9672 × 10 −13
3.7167 × 10 −7
9
15 4.1978 × 10 −13
5.6027 × 10 −8
10 15 2.0122 × 10 −13
8.5479 × 10 −9
11 15 2.8979 × 10 −13
1.2373 × 10 −9
12 15 2.5950 × 10 −13
1.9831 × 10 −10
polynomials can be extended to handle the nonlinear PDEs in any dimensions without any modifications. In our research, we only tried to apply the suggested method to 2-dimensonal nonlinear PDEs. Consider a 2-dimensonal nonlinear PDE Lu(x, y) + F (u(x, y)) = f (x, y) ,
(x, y) ∈ [ax , bx ] × [ay , by ]
,
(4.11)
subject to the supplementary conditions Dyp |x=xp u(x, y) = σyp (y) ,
p = 1(1)Nx
,
(4.12)
Dxq |y=yq u(x, y) = σxq (x) ,
q = 1(1)Ny
,
(4.13)
where F (u(x, y)) is the nonlinear term of the given problem and other symbols P∞ follow Section 3.2.1. Let u(x, y) = i=0 u(i) (x, y), where u(i) (x, y), i = 0(1)∞, are the decomposed solutions to the problem (4.11)-(4.13) and let unx ny (i) (x, y) be
104
x 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0 3E-7 -5E-7 2E-7
-1E-6
-1E-6 1E-7 -2E-6 0 -2E-6
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
x
n = 7, K = 16
n = 8, K = 14
Figure 7: The error curves of the approximate solutions for Example 4.6 obtained by the Tau-Collocation Method with incorporation of the Adomian’s polynomials with Tau degrees n = 7 and 8 and K = 16 and 14 respectively.
the numerical approximate solution to u(i) (x, y) with degree nx and ny in x and y respectively, for i = 0(1)∞. The nonlinear term F (u(x, y)) in equation (4.11) can be written in term of the Adomian’s polynomials, which is as same as the case of ODE, F (u(x, y)) =
∞ X i=0
where Ai =
1 di F i! dλ i
P
∞ X 1 di Ai = F i! dλ i i=0
i j λ u (x, y) (j) j=0
λ=0
i X j=0
! j λ u(j) (x, y)
(4.14) λ=0
, i = 0(1)∞, are the Adomian’s poly-
nomials. And, problem (4.11)-(4.13) can be decomposed into infinity many subproblems by the principle of superposition and the problem become Lu(i+1) (x, y) = −Ai Lu (x, y) = f (x) (0) Dyp |x=xp u(i+1) (x, y) = 0 Dyp |x=xp u(0) (x, y) = σyp (y) and D | D | xq y=yq u(i+1) (x, y) = 0 xq y=yq u(0) (x, y) = σxq (x)
, (4.15)
where [ax , bx ] × [ay , by ], p = 1(1)Nx , q = 1(1)Ny and i = 0(1)∞. The computation
process of the Tau-Collocation Method with incorporation of the Adomian’s polyno-
105
mials for nonlinear PDEs is exactly as same as the case of ODEs. Finally, the numerP ical approximate solution of problem (4.11)-(4.13) is obtained by K i=0 unx ny (i) (x, y), i = 1(0)K, where K ∈ N such that max|unx ny (K) (x, y)| < ε. The following example
shows the results obtained by the Tau-Collocation Method with incorporation of the Adomian’s polynomials.
Example 4.7 Consider a nonlinear PDE discussed in Example 4.2. The Adomian Polynomials of the given problem are as following 2 d A0 = u0 (x) dx d d A1 = 2 u0 (x) u1 (x) dx dx 2 d d d A2 = u1 (x) + 2 u0 (x) u2 (x) dx dx dx d d d d A3 = 2 u1 (x) u2 (x) + 2 u0 (x) u3 (x) dx dx dx dx .. . where k = 1, 2, 3, · · ·. The Tau-Collocation Method can be applied directly to this problem without any pre-approximation has been done for the right-hand side functions of both problem and condition. In Table 21 we give the maximum absolute error max |u(x, t) − unx nt (K) (x, t)| for Example 4.7 obtained by the TauCollocation Method incorporated with Adomian’s polynomials with Tau degrees nx = nt = 4(1)10 and ε = 1 × 10 −9 .
4.2
Segmentation Techniques
When the Tau Method or the Tau-Collocation Method is solely applied to a differential problem defined on a large domain or the solution of the differential
106
Table 21: The maximum absolute error max |u(x, t) − unx nt (K) (x, t)| for Example 4.7 obtained by the Tau-Collocation Method incorporated with Adomian’s polynomials with Tau degrees nx = nt = 4(1)10 and ε = 1 × 10 −9 . nx , nt
K
4,4
E
n x , nt
K
E
29 7.3486 × 10 −2
8,8
24 1.2758 × 10 −6
5,5
23 2.6617 × 10 −2
9,9
24 9.5557 × 10 −7
6,6
24 4.2417 × 10 −4
10,10
24 2.2801 × 10 −9
7,7
24 1.4155 × 10 −4
problem experienced with a sharp change over the defined domain, the obtained results may not be converged as fast as the problems discussed before. In order to get more accuracy results with lower degrees by the Tau Method in such differential problems, the given problem is first divided into many sub-problems and then the Tau Method is used in finding out all of the approximation solutions of the subproblems. These segmentation techniques are first purposed by Ortiz and Onumanyi [30, 33] for the case of ODEs. And, the techniques is extended to handle the PDEs by Ortiz and Pun [37]. In our research, we success mixed the idea of the segmentation technique with the Tau-Collocation Method for approximate the solution of both ODEs and PDEs which are defined on a long interval or the solution of ODEs and PDEs experienced with a sharp change over the defined domain.
4.2.1 Step by Step Tau-Collocation Method The step by step Tau-Collocation Method follows the idea of the step by step Tau Method [33]. It is first purposed by Liu et al. [26]. The suggested method aims to divide the given ODE, with initial conditions only, into many sub-problems in a shorter interval. And, the Tau-Collocation Method is used in finding out the approximate solutions of the sub-problems one by one. A piecewise polynomial
107
approximations can be obtained finally. Consider ordinary different problem (3.1)-(3.2) in Section 3.1.1 limited to initial conditions. The interval x ∈ [a, b] is first divided into a set of small subintervals. Let P = {a = a1 < b1 = a2 < b2 = a3 < · · · < bm = b} be partition of the interval [a, b] with m subintervals Ji = [ai , bi ], for i = 1(1)m. Let yi (x) be the solution of subproblems defined on [ai , bi ], i = 1(1)m, and let yi,n (x) be the numerical approximate solution to yi (x), for i = 1(1)m. Then, we have the first sub-problem Dv y1 (x) = f (x) ,
x ∈ [a1 , b1 ]
,
(4.16)
with initial conditions Dvr |x=xr y1 (x) = σr (xr ) ,
r = 1(1)ν
.
(4.17)
And, the other sub-problems are defined as Dv yi (x) = f (x) ,
x ∈ [ai , bi ]
,
(4.18)
with additional conditions d r−1 y (x) = i dx r−1 x=ai
d r−1 y (x) i−1 dx r−1 x=bi−1
,
(4.19)
where i = 2(1)m and r = 1(1)ν. Conditions (4.19) are additional conditions for linking up solution yi−1 (x) and yi (x), for i = 2(1)m, and making the system wellposed. The associate Tau problem of the above sub-problems are Dv yi,n (x) = f (x) + Hi,n (x) ,
x ∈ [ai , bi ] and i = 1(1)m
with initial conditions Dvr |x=xr yi,n (x) = σr (xr ) d r−1 d r−1 y (x) = y (x) dx r−1 i,n dx r−1 i−1,n x=bi−1
x=ai
,
for i = 1zs,
,
for i = 2(1)m
(4.20)
,
(4.21)
where Hi,n (x) is the corresponding perturbation term of i th sub-problem, i = 1(1)m
and r = 1(1)ν.
108
The formulation of the Tau-Collocation Method is applied to formulate the Tau problems (4.20)-(4.21) in sequence starting form i = 1. Finally, a piecewise polynomial approximate solution yn,i (x), for i = 1(1)m, to the problem (3.1)-(3.2), limited to initial conditions, can be obtained.
4.2.2 Segmented Tau-Collocation Method In our research, we succeed to incorporate the Tau-Collocation Method with the segmentation technique for approximating the solution of both ordinary and partial differential equations with initial, boundary and mixed conditions defined on a large domain or the solution of the ODEs and PDEs experienced with a sharp change over the domain. The details of the ideas and formulations of the suggested methods are discussed in this section. 4.2.2.1 Segmented Tau-Collocation Method for Ordinary Differential Equations The segmented Tau-Collocation Method for ODEs follows the idea of the Segmented Tau Method [30]. The suggested method aims to divide the given ordinary differential problem with initial, boundary and mixed conditions into many subproblems in a shorter intervals and then the Tau-Collocation Method is applied to find out the approximate solutions of the sub-problems. All the sub-problems are formulated by the Tau-Collocation Method and the resultant matrix of every subproblem is combined into one matrix. Finally, piecewise polynomials approximation can be obtained by solve this combined matrix. Consider ordinary different problem (3.1)-(3.2) in Section 3.1.1. As same as the idea of step by step Tau-Collocation Method, the interval x ∈ [a, b] is first divided into a set of small subintervals. Let P = {a = a1 < b1 = a2 < b2 = a3 < · · · < bm = b} be partition of the interval [a, b] with m subintervals Ji = [ai , bi ], for i = 1(1)m. Let yi (x) be the solution of sub-problems defined on [ai , bi ], i = 1(1)m, and let yi,n (x) be the approximate solution to yi (x), for i = 1(1)m. Then, we have the following
109
sub-problems Dv yi (x) = f (x) ,
x ∈ [ai , bi ] zs,
(4.22)
with supplementary conditions Dvi,r |x=xi,r yi (x) = σr (xi,r ) ,
(4.23)
where Dvi,r is the corresponding differential operator Dvr defined on Section 3.1.1 of i th sub-problem, for r = 1(1)ν and i = 1(1)m, and other symbols follow the symbols defined on Section 3.1.1. In order to link up solution yi−1 (x) and yi (x), for i = 2(1)m, and making the system well-posed, (m − 1) × ν additional conditions r−1 d r−1 d yi−1 (x) − yi (x) = 0 , (4.24) r−1 r−1 dx dx x=bi−1 x=ai
where i = 2(1)m and r = 1(1)ν, are added to the given problem. Note that the
orders of all sub-problems remain ν. Sub-problems (4.22)-(4.24) can be formulated by the Tau-Collocation Method, however, there are some modifications of the formulation process. Refer to Section 3.1.2, equation (4.22) can be written in matrix form ai,n Γi = Fn−ν ,
i = 1(1)m ,
(4.25)
where ai,n and Γi are the corresponding unknown coefficients an and matrix Γ in equation (3.7) respectively of i th sub-problem, i = 1(1)m. Also, conditions (4.23) can be written as ai,n Bi = σi ,
i = 1(1)m ,
(4.26)
where ai,n , Bi and σi are the corresponding matrices an , B and σ in equation (3.8) respectively of i th sub-problem, i = 1(1)m. Let Πj,i (x) be the corresponding matrix Πi (x) of
di y (x), dx i j,n
j = 1(1)m, in Theorem 3.1. Then, additional conditions (4.24)
becomes ai−1,n Πi−1,r−1 (bi−1 ) − ai,n Πi,r−1 (ai ) = 0 ,
r = 1(1)ν
,
(4.27)
110
where ai,n is the corresponding unknown coefficients an of i th sub-problem, i = 2(1)m. Let
Ci =
Πi−1,0 (bi−1 ) Πi−1,1 (bi−1 ) · · · Πi−1,ν−1 (bi−1 ) −Πi,0 (ai )
−Πi,1 (ai )
···
−Πi,ν−1 (ai )
,
where i = 2(1)m. The additional conditions (4.27) can be written as a matrix form
ai−1,n |ai,n Ci = Oν
,
(4.28)
where Oν is a row vector with ν zero elements. In order to find out the unknown coefficients ai,n , i = 1(1)m, of every sub-problem, equations (4.25), (4.26) and (4.28) should be combined into one system of linear algebraic equations. Let =
B∗
B1
C1
O
B2
C2
B3 ..
O
.
..
.
Cm
Bm−1 Bm
Also, let
G
∗
Γ1
=
O
Γ2 Γ3
B∗
..
O
.
. Γm−1 Γm
,
where O is zero matrix. Then, equations (4.25), (4.26) and (4.28) can be formulated as
a1,n |a2,n | · · · |am,n G
∗
=
σ1 |σ2 | · · · |σm |0|0| · · · |0|Fn−ν |Fn−ν | · · · |Fn−ν
. (4.29)
111
Finally, the piecewise polynomial approximate solutions yn,i (x), for i = 1(1)m, to the problem (3.1)-(3.2) can be obtained by solving equation (4.29) with usual method. 4.2.2.2 Segmented Tau-Collocation Method for 2-dimensional Partial Differential Equations The segmented Tau-Collocation Method can be extended to handle the PDEs with initial, boundary and mixed conditions defined on a large domain or the solution of the PDEs experienced a sharp change over the defined domain. The suggested method follows the idea of the segmented Tau-Collocation Method for ODEs and the Tau-Element Method [37] and is first purposed in this thesis. The segmented TauCollocation Method for PDEs aims to divide the given PDE into many sub-problems in a smaller domain and then the Tau-Collocation Method is applied to find out the approximate solutions of the sub-problems. All the sub-problems are formulated by the Tau-Collocation Method and the resultant matrix of every sub-problem is combined into one matrix. Finally, piecewise polynomials approximations can be obtained by solve this combined matrix. Consider partial different problem (3.25)-(3.27) in Section 3.2.1. Both of the interval x ∈ [ax , bx ] and y ∈ [ay , by ] are first divided into a set of small subintervals. Let Px = {ax = ax1 < bx1 = ax2 < bx2 = ax3 < · · · < bxmx = bx } be partition of the interval [ax , bx ] with mx subintervals Jxix = [axix , bxix ], for ix = 1(1)mx , and let Py = {ay = ay1 < by1 = ay2 < by2 = ay3 < · · · < bymy = by } be partition of the in terval [ay , by ] with my subintervals Jyiy = ayiy , byiy , for iy = 1(1)my . Also, let uix iy (x, y) be the solution of sub-problems defined on [aix , bix ]× aiy , biy , ix = 1(1)mx and iy = 1(1)my , and let uix iy ,nx ny (x, y) be the numerical approximate solution to uix iy (x, y), for ix = 1(1)mx and iy = 1(1)my . Then, we have the following subproblems Luix iy (x, y) = f (x, y) ,
(x, y) ∈ [axix , bxix ] × ayiy , byiy
with supplementary conditions Dix ,yp x=xi ,p uix iy (x, y) = σix ,yp (y) , x
p = 1(1)Nx
,
,
(4.30)
(4.31)
112 Diy ,xq y=yi
y ,q
uix iy (x, y) = σiy ,xq (x) ,
q = 1(1)Ny
,
(4.32)
where Dix ,yp and Diy ,xq are the corresponding differential operator Dyp and Dxq respectively defined on Section 3.2.1 of (ix , iy ) th sub-problem, ix = 1(1)mx and iy = 1(1)my , and other symbols follow the symbols defined on Section 3.2.1. The geometrical presentation of the sub-problems shows in Figure 8. In order to link up solution uix −1iy (x, y) and uix iy (x, y), (mx − 1) Nx my additional conditions p−1 ∂ ∂ p−1 u (x, y) − u (x, y) = 0 , i −1i i i x y x y ∂x p−1 ∂x p−1 x=bxix −1 x=axix
(4.33)
where ix = 2(1)mx , iy = 1(1)my and p = 1(1)Nx , are added to the given problem. Also, in order to link up solution uix iy −1 (x, y) and uix iy (x, y), (my − 1) Ny mx additional conditions q−1 ∂ ∂ q−1 u (x, y) − u (x, y) = 0 , i i −1 i i x y x y q−1 ∂y q−1 ∂y y=byiy −1 y=ayiy
(4.34)
where ix = 1(1)mx , iy = 2(1)my and q = 1(1)Ny , are added to the given problem. All of the sub-problems (4.30)-(4.34) can be formulated by the Tau-Collocation Method for 2-dimensional PDEs, however, there are some modifications of the formulation process. Note that the number of conditions of every sub-problem is as same as the number of conditions of the given problem. And, the number of redundant linear algebraic equations inside every sub-problem is Nx Ny when the TauCollocation Method is applied. Therefore, there are totally mx my Nx Ny redundant linear algebraic equations. Refer to Section 3.2.2, equation (4.30) can be written in matrix form Γix iy vec Aix iy ,nx ny = F ,
ix = 1(1)mx and iy = 1(1)my
,
(4.35)
where Aix iy ,nx ny and Γix iy are the corresponding unknown coefficients Anx ny and matrix Γ in equation (3.34) respectively of (ix , iy ) th sub-problem, ix = 1(1)mx and iy = 1(1)my . Also, conditions (4.31) can be written as Γyix iy vec Aix iy ,nx ny = Fyix iy ,
ix = 1(1)mx and iy = 1(1)my
,
(4.36)
113
y u1m u2m y
y
...
um m
x y
... u22
...
um 2
u11
u21
...
um 1
...
... u12
x
x
x Figure 8: The geometrical presentation of sub-problems given from a segmented 2-dimensional PDE.
where Aix iy ,nx ny , Γyix iy and Fyix iy are the corresponding matrices Anx ny , Γy and Fy in equation (3.36) respectively of (ix , iy ) th sub-problem, ix = 1(1)mx and iy = 1(1)my . Similarly, conditions (4.32) can be written as Γxix iy vec Aix iy ,nx ny = Fxix iy ,
ix = 1(1)mx and iy = 1(1)my
,
(4.37)
where Aix iy ,nx ny , Γxix iy and Fxix iy are the corresponding matrices Anx ny , Γx and Fx in equation (3.38) respectively of (ix , iy ) th sub-problem, ix = 1(1)mx and iy = 1(1)my . Let si,n = (1, s i1 , s 2i , · · · , s in ), where si is a dummy variable and let Aix iy ,nx ny be the matrix of unknown coefficients Anx ny of (ix , iy ) th sub-problem, ix = 1(1)mx and iy = 1(1)my . By Lemma 3.3 in Section 3.2.2, additional condition (4.33) can be written as
yiy ,ny ⊗ bxix −1,nx (η np−1 )0 vec Aix −1iy ,nx ny x 0 − yiy ,ny ⊗ axix ,nx (η np−1 ) vec A ix iy ,nx ny x
= 0 ,
(4.38)
where p = 1(1)Nx , ix = 2(1)mx and iy = 1(1)my . Also, by Lemma 3.3, additional
114
condition (4.34) can be written as 0 byiy −1,ny (η nq−1 ) ⊗ x vec A ix ,nx ix iy −1,nx ny y 0 ) ⊗ x vec A − ayiy ,ny (η nq−1 ix ,nx ix iy ,nx ny y
= 0 ,
(4.39)
where q = 1(1)Ny , ix = 1(1)mx and iy = 2(1)my . Since equations (4.38) and (4.39) contain unknown variables y and x respectively, a set of linear algebraic equations can be obtained by comparing coefficients of unknown variables y and x for equations (4.38) and (4.39) respectively. Let ein be a row vector with N + 1 elements, where the vaule of (i + 1) th element of ein is 1 and the values of all other elements are zero. Let Cy1 ix iy ,ip = and
Cy2 ix iy ,ip =
Also, let Cx1 ix iy ,iq = and
Cx2 ix iy ,iq =
0 einy ⊗ bxix −1,nx (η np−1 ) vec A i −1i ,n n x y x y x 0 . − einy ⊗ axix ,nx (η np−1 ) vec A i i ,n n x y x y x
0 byiy −1,ny (η np−1 ) ⊗ e vec A inx ix iy −1,nx ny y 0 . − ayiy ,ny (η np−1 ) ⊗ e vec A inx ix iy ,nx ny y
By comparing coefficients with respect to y, equation (4.38) becomes ny + 1 linear algebraic equations
C y∗1 ix iy ,p |C y∗2 ix iy ,p
0 vec Aix −1iy ,nx ny |vec Aix iy ,nx ny = O ny
, (4.40)
0 where C y∗1 ix iy ,p = Cy1 ix iy ,0p |Cy1 ix iy ,1p | · · · |Cy1 ix iy ,ny p , 0 C y∗2 ix iy ,p = Cy2 ix iy ,0p |Cy2 ix iy ,1p | · · · |Cy2 ix iy ,ny p , for p = 1(1)Nx , ix = 2(1)mx and
iy = 1(1)my , and Ony is a column vector with ny + 1 zero elements. Similarly,
by comparing coefficients with respect to x, equation (4.39) becomes nx + 1 linear algebraic equations
C x∗1 ix iy ,q |C x∗2 ix iy ,q
0 vec Aix iy −1,nx ny |vec Aix iy ,nx ny = O nx
, (4.41)
0 where C x∗1 ix iy ,q = Cx1 ix iy ,0q |Cx1 ix iy ,1q | · · · |Cx1 ix iy ,nx q , 0 C x∗2 ix iy ,q = Cx2 ix iy ,0q |Cx2 ix iy ,1q | · · · |Cx2 ix iy ,nx q , for q = 1(1)Ny , ix = 1(1)mx and
115
iy = 2(1)my , and Onx is a column vector with nx +1 zero elements. For simplification purpose, let
Γ i∗∗x iy
Γ ix iy = Γyix iy Γxix iy
Equations (4.35)-(4.37) become
F = Fyix iy and F i∗∗ x iy Fxix iy
Γ ∗∗ ix iy vec Aix iy ,nx ny
= F i∗∗ x iy
.
,
(4.42)
where ix = 1(1)mx and iy = 1(1)my . And, let ∗ C y1 ix iy ,1 C y∗2 ix iy ,1 C y∗2 ix iy ,2 C y∗1 ix iy ,2 ∗∗ ∗∗ and C y i i = C y 1 ix iy = 2 x y .. .. . . C y∗1 ix iy ,Nx C y∗2 ix iy ,Nx Similarly, let
C x∗∗1 ix iy
C x∗1 ix iy ,1
C x∗1 ix iy ,2 = .. . C x∗1 ix iy ,Ny
C x∗2 ix iy ,1
C x∗2 ix iy ,2 ∗∗ and C x i i = 2 x y .. . C x∗2 ix iy ,Ny
.
.
Also, let O n∗∗y be a column vector with (ny + 1)Nx zero elements and let O n∗∗x be a column vector with (nx + 1)Ny zero elements. Equation (4.40) becomes
C y∗∗1 ix iy |C y∗∗2 ix iy
0 vec Aix −1iy ,nx ny |vec Aix iy ,nx ny = O n∗∗y
,
(4.43)
,
(4.44)
where ix = 2(1)mx and iy = 1(1)my , and equation (4.41) becomes
C x∗∗1 ix iy |C x∗∗2 ix iy
0 vec Aix iy −1,nx ny |vec Aix iy ,nx ny = O n∗∗x
where ix = 1(1)mx and iy = 2(1)my . In order to find out the unknown coefficients Aix iy ,nx ny , for ix = 1(1)mx and iy = 1(1)my , of every sub-problem, equations (4.35),
116
(4.36), (4.37), (4.40) and (4.41) should be combined into one system linear algebraic equations. By equations (4.42)-(4.44), let ∗∗
Γ 11
G ∗∗
=
Γ ∗∗ 12
.
.
O
. Γ ∗∗ 1my −1 Γ ∗∗ 1my Γ ∗∗ 21 Γ ∗∗ 22
O
.
.
. Γ ∗∗ mx my
∗∗ Cy 21
∗∗ Cy 21
1
2
∗∗ Cy 22
∗∗ Cy 22
1
2
.
.
. .
.
. .
∗∗ Cy 1 2my −1 ∗∗ Cy 2m 1
.
.
y ∗∗ Cy 31 1
∗∗ Cy 32 1
O
.
.
. ∗∗ Cy 2 mx my
∗∗ Cx 12 1
∗∗ Cx 12 2
∗∗ Cx 13 1
.
.
O
. ∗∗ Cx 1my −1 2
∗∗ Cx 1m 1
y
∗∗ Cx 1m 2
y ∗∗ Cx 22 1
∗∗ Cx 22 2
∗∗ Cx 23 1
O
.
.
. ∗∗ Cx mx my 2
where O is zero matrix. Then, equations (4.35), (4.36), (4.37), (4.40) and (4.41) can be formulated as G ∗∗ A ∗∗ = F ∗∗
(4.45)
117
where
A ∗∗
vec A11,nx ny
vec A12,nx ny .. . = vec A1my ,nx ny vec A21,nx ny .. . vec Amx my ,nx ny
and
F ∗∗
=
∗∗ F 11 ∗∗ F 12
.. .
∗∗ F 1m y ∗∗ F 21 .. .
F m∗∗x my O n∗∗y O n∗∗y .. . O n∗∗y O n∗∗x O n∗∗x .. . O n∗∗x
.
Note that A ∗∗ is a mx my (nx + 1)(ny + 1) × 1 matrix. Finally, the piecewise polynomial approximate solutions unx ny ,ix iy (x, y), for ix = 1(1)mx and iy = 1(1)my , to the problem (4.30)-(4.32) can be obtained by solving equation (4.45) with usual method. The following examples show the results obtained from the segmented Tau-Collocation Method.
Example 4.8 Consider a 2-dimensional Poisson equation ∂2 ∂2 u(x, y) + u(x, y) = 2π 2 sin(πx) sin(πy) , ∂x 2 ∂y 2
(x, y) ∈ [−1, 1] × [−1, 1]
with boundary conditions u(−1, y) = u(1, y) = 0 ,
y ∈ [0, 1]
,
u(x, −1) = u(x, 1) = 0 ,
x ∈ [0, 1]
.
,
118
The exact solution of this problem is u(x, y) = sin(πx) sin(πy) , see Khalifa et al. [17]. In this problem, we take partitions x = {−1 < 0 < 1} and y = {−1 < 0 < 1}. The additional conditions are u11 (0, y) − u21 (0, y) = 0
,
u12 (0, y) − u22 (0, y) = 0
,
u11 (x, 0) − u12 (x, 0) = 0
,
u21 (x, 0) − u22 (x, 0) = 0
,
∂ u (x, y) x=0 ∂x 11
∂ u (x, y) x=0 ∂x 12
− −
∂ u (x, y) − ∂y 11 y=0 ∂ u (x, y) − ∂y 21 y=0
∂ u (x, y) x=0 ∂x 21
= 0
,
= 0 ∂ u (x, y) = 0 ∂y 12 y=0 ∂ u (x, y) = 0 ∂y 22
,
∂ u (x, y) x=0 ∂x 22
y=0
, .
In Table 22 we give the maximum absolute error E := max|u(x, y) − unx ny (x, y)| for Example 4.8 obtained from the segmented Tau-Collocation Method (STC) with degrees nx = ny = 8(2)16 of the approximate solutions and compared with the results obtained from the Tau-Collocation Method without segmentations (TC) and the Chebyshev expansion method (CEM) [17]. Better results can be obtained from the segmented Tau-Collocation Method with lower degrees.
Example 4.9 Consider a first order wave equation ∂ ∂ u(x, t) + u(x, t) = 0 , ∂t ∂x
(x, y) ∈ [0, 1] × [0, 20]
with boundary condition u(0, t) = e −t ,
t ∈ [0, 20]
,
u(x, 0) = e x ,
x ∈ [0, 1]
.
and initial condition
,
119
Table 22: The maximum absolute error E := max|u(x, y) − unx ny (x, y)| for Example 4.8 obtained from the segmented Tau-Collocation Method (STC), Tau-Collocation Method without segmentations (TC) and Chebyshev expansion method (CEM) with degrees nx = ny = 8(2)16 of the approximate solutions. nx , ny
E (STC)
E (TC)
E (CEM)
8,8
4.4443 × 10 −7
2.0429 × 10 −3
−
10,10
1.7675 × 10 −9
4.7470 × 10 −5
−
12,12
5.9240 × 10 −12
8.0051 × 10 −7
7.49 × 10 −8
14,14
1.4558 × 10 −13
9.0797 × 10 −9
7.62 × 10 −10
16,16
−
7.8145 × 10 −11
6.25 × 10 −12
The exact solution of this problem is u(x, t) = e x−t
,
see Akram et al. [3]. Since the problem is defined on long interval t ∈ [0, 20], the segmented Tau-Collocation Method is used in this problem. We take two set of partitions of this problem. The first set is x = {0 < 1} and t = {0 < 2 < 4 < 6 < 8 < 10 < 12 < 14 < 16 < 18 < 20}. The another set is x = {0 < 1} and t = {0 < 1 < 2 < 3 < 4 < 5 < 6 < 7 < 8 < 9 < 10 < 11 < 12 < 13 < 14 < 15 < 16 < 17 < 18 < 19 < 20}. In Table 23 we give the maximum absolute error E := max|u(x, t) − unx nt (x, t)| for Example 4.9 obtained from the segmented Tau-Collocation Method with the first set of partitions (STC-1 × 10) and second set of partition (STC-1 × 20) with Tau degrees nx = nt = 4(1)8. Similar accuracy of results can be obtained from the STC-1 × 10 with degrees nx = nt = 7, the STC-1 × 20 with degrees nx = nt = 6 and the third-order finite differences method [3] with width h = 0.0125 and length l = 0.01. For similar accuracy of the approximate solutions, the segmented Tau-Collocation Method gives a smaller resultant matrix than the third-order finite differences Method. The size of resultant matrix,
120
included the redundant linear algebraic equations, formulated by STC-1 × 10 with Tau degrees nx = nt = 7 is 650 × 640 and STC-1 × 20 with degrees nx = nt = 6 is 1000 × 980. The graphs of the error surface defined on (x, t) ∈ [0, 1] × [0, 5] of the approximate solutions for Example 4.9 obtained from STC-1 × 10 with Tau degrees nx = nt = 7 and STC-1 × 10 with degre nx = nt = 6 are shown in Figure 9. It shows a remarkable equioscillatory behavior.
Table 23: The maximum absolute error E := max|u(x, t) − unx nt (x, t)| for Example 4.9 obtained from the segmented Tau-Collocation Method with 1 × 10 uniform partitions (STC-1 × 10) and 1 × 20 uniform partitions (STC-1 × 20) with Tau degrees nx = nt = 4(1)8. nx , nt
E (STC-1 × 10)
E (STC-1 × 20)
4,4
1.5159 × 10 −3
1.2987 × 10 −4
5,5
1.1238 × 10 −4
5.1690 × 10 −6
6,6
7.2507 × 10 −6
1.7420 × 10 −7
7,7
4.5998 × 10 −7
5.1097 × 10 −9
8,8
2.5443 × 10 −8
−
Example 4.10 Consider a nonlinear hyperbolic PDE ∂ ∂ u(x, t) + u(x, t) u(x, t) = 0 , ∂t ∂x
(x, y) ∈ [0, 1] × [0, 15]
with boundary condition u(0, t) =
1 , 1+t
t ∈ [0, 15]
,
x ∈ [0, 1]
.
and initial condition u(x, 0) = 1 + x ,
,
121
2E-7
4E-7
1E-7 2E-7 5E-8 0
0 -5E-8
-2E-7
-4E-7 0
1
2 t
3
0 0.2 0.4 0.6 x 0.8 4
5
-1E-7 0
1
STC-1 × 10 with nx = nt = 7
2 t
1
3
0 0.2 0.4 0.6 x 0.8 4
5
1
STC-1 × 20 with nx = nt = 6
Figure 9: The error surface defined on (x, t) ∈ [0, 1] × [0, 5] of the approximate solution for Example 4.9 obtained from the segmented Tau-Collocation Method with 1 × 10 uniform partitions (STC-1 × 10) with degree nx = nt = 7 and 1 × 20 uniform partitions (STC-1 × 20) with degree nx = nt = 6.
The exact solution of this problem is u(x, t) =
1+x 1+t
,
see Akram et al. [3]. Since the problem contains nonlinear term and is defined on long interval t ∈ [0, 15], the segmented Tau-Collocation Method incorporated with the quadratic iterative scheme is used in this problem. We take partition x = {0 < 1} and t = {0 < 0.4 < 0.8 < 1.2 < 1.6 < 2 < 4 < 6 < 9 < 12 < 15}. An initial gauss u(x, t) = 1 + x is used in the first stage of every sub-problem so as to start the iteration process. Note that the results of all the sub-problems at k th iteration are used in activating the (k + 1) th iteration process. And, the iteration process is ended when the maximum absolute value of the difference between two consecutive estimates of all sub-problems are less than the tolerance error ε. In Table 24 we give the maximum absolute error E := max|u(x, t) − unx nt (x, t)| and number of it-
122
erations K for Example 4.10 obtained from the segmented Tau-Collocation Method incorporated with the quadratic iterative scheme with Tau degrees nx = nt = 4(1)8 and ε = 1 × 10 −10 . Similar accuracy of results can be obtained from the suggested method with Tau degrees nx = nt = 6 and the third-order finite differences method [3] with width h = 0.2 and length l = 0.1. For similar accuracy of the approximate solutions, the suggested method gives a smaller resultant matrix than the third-order finite differences method. The size of resultant matrix, included the redundant linear algebraic equations, formulated by the suggested method with Tau degrees nx = nt = 6 is 500 × 490.
Table 24: The maximum absolute error E := max|u(x, t) − unx nt (x, t)| and number of iterations K for Example 4.10 obtained from the segmented Tau-Collocation Method with incorporation of quadratic iterative scheme with Tau degrees nx = nt = 4(1)7 and ε = 1 × 10 −10 . nx , nt
K
4,4 5,5
E
n x , nt
K
E
14 1.0855 × 10 −4
6,6
29 1.4157 × 10 −6
14 1.2280 × 10 −5
7,7
41 1.9181 × 10 −7
123
Chapter 5
Applications to Linear Elastic Fracture Mechanics
The segmented Tau-Collocation Method, discussed in Section 4.2, with incorporation of the singularity subtraction technique can be used in the real life problems of the linear elastic fracture mechanics. The singularity subtraction technique is first applied into the given linear elastic facture mechanics problem in order to remove the boundary singularities of the given problem and the segmented Tau-Collocation Method, with some modifications in formulation process, can then be applied to find out the stress intensity factor K of the given problem. In our research, we tried to tackle two linear elastic fracture mechanics problems by the suggested method. The problems are the Motz’ Problem and the biharmonic equation for mode I crack. Both problems are 2-dimensional PDE with boundary singularity. These matters are our research results and are first purposed in this thesis.
5.1
Brief Introduction in Linear Elastic Fracture Mechanics
Many failures of engineering structures are due to fracture. However, cracks in a material cannot be eliminated and is present as small flaws, which may be presented in both external and internal of the material, during the manufacturing stage. The cracks arise during fabrication and may become the result of damage to the completed structure. As the cracks must be existed, it is better to devise some rules to characterize cracks, determine their effects and predict the safety of the structures during the operational service life. Such rules, determination and prediction become the science titled linear elastic fracture mechanics since about 1948-1957. The first systematic investigation of the linear elastic fracture mechanics was carried out by A.A. Griffith in 1920. Nowadays, there are three main streams in the linear elastic fracture mechanics, the theoretical, experimental and computational developments.
124
In our research, we only focus on the computational development of linear elastic fracture mechanics. The computational development of the linear elastic fracture mechanics aims to predict the safety of the structures during the operational service life by finding out the stress intensity factor K from mathematical models. The stress intensity factor K is a parameter used in specifying the stress intensity at crack tip and is expressed in terms of the applied stresses near the crack tip in usual. Note that the applied stresses can be described in three modes. They are tension or opening (mode I), inplane shear or sliding (mode II) and out-of-plane shear or tearing (mode III). Once the stress intensity factor K of a given problem is determined, maximum stress near the crack tip can also determined. To ensure the safety of the structures, the computed stress intensity factor K should not be exceeded the fracture toughness which is the critical stress intensity factor Kc for each material. The fracture toughness of the material is obtained by performing a series of experiments, which are done by theoretical and experimental development of linear elastic fracture mechanics.
5.2
Singularity Subtraction Technique
When a given differential problem suffered form singular points on the boundary or in the interior of the domain of the given problem, usual numerical methods, such as the finite different method, the finite element method and the spectral methods, can hardly to get an accuracy numerical result near the singular points. The error of the result near the singular point obtained by the usual numerical method is very large, even extremely small gird size or extremely high degree is used in the approximation process. In order to obtain an accuracy result near the singular points with normal approximation setting and procedures of usual numerical methods, some methods, for examples, the conformal transformation methods, the Motz’ technique, the relaxation technique and the nonconforming combined method, should be adopted first to remove the singularities. The singularity subtraction
125
technique is one of the methods that are used in removing the singularities before the numerical method is applied to the differential problem with singularities. The idea of the singularity subtraction technique is to subtract off the singularity part
of the
solution
of
given
differential
problem
and
transform
the
given problem into a well behaved problem defined on the whole problem domain.
Taking 2-dimensional PDEs with singularities into consideration.
Let u(x, y) = u ¯(x, y) + u ∗ (x, y) be the solution with singularities of given PDEs, where u ¯(x, y) is the smooth part and u ∗ (x, y) is the singular part of the solution. By substituting u ¯(x, y) = u(x, y) − u ∗ (x, y) into the given problem, the transferred problem become a smooth problem over the problem domain. Then, an accuracy numerical result of the transferred problem, i.e. u ¯(x, y), can be obtained and so as the u(x, y).
5.3
Two Mathematical Models in Linear Elastic Fracture Mechanics
Two linear elastic fracture mechanics problems will be discussed in this thesis. The problems are Motz’ problem and the biharmonic equation for mode I crack. Two problems are 2-dimensional PDE with boundary singularity. The problems are solved by the segmented Tau-Collocation Method with incorporation with singularity subtraction technique successfully. The details of ideas and formulation processes will be discussed in this section. These matters are our research results and are first proposed in this thesis.
5.3.1 Motz’ Problem The Motz’ problem was first proposed by H. Motz [29] for studying about the elliptic boundary value problems with boundary singularities. It is a benchmark problem for many mathematicians to approximate the results of elliptic boundary
126
value problems with boundary singularities by different numerical methods. The Motz’ problem is a Laplace equation ∂2 ∂2 φ(x, y) + φ(x, y) = 0 ∂x 2 ∂y 2
(5.1)
defined on a rectangular domain ΩM := OABCDEF GO with a crack AO, see Figure 10(a), with boundary conditions ∂ φ(x, y) = 0 on BC, CE, EF, AO and GO, ∂n φ(x, y) = 1000 on AB, φ(x, y) = 0 where
∂ ∂n
(5.2)
on FG,
is outward direction normal to the boundary. Since function φ(x, y) − 500
is anti-symmetric about AOD, the problem (5.1)-(5.2) can be reduced to a problem defined on the upper half domain ΩM1 := OABCDO with a crack AO, see Figure 10(b), with one more boundary condition φ(x, y) = 500
on DO.
(5.3)
Note that a singularity point exist in O, i.e. between the conditions DO and AO, so that the Motz’ problem contains boundary singularity. The analytical local series solution in polar coordinates (r, θ), where (r, θ) ∈ ΩM1 , of the Motz’ problem near the singular point O is φ(r, θ) = 500 +
∞ X i=1
λi r
i− 21
1 cos i − θ 2
,
(5.4)
where λ1 is the stress intensity factor and λi , i = 2(1)∞, are higher order factors. Note that λi , i = 1(1)∞, are unknowns parameters to be determined and the results of λi , i = 1(1)∞, depends on the setting of the domain, i.e. the length and width of domain and crack length. λ1 is a very important parameter with physical meaning and is aimed to find out its value.
127
C
B
M
C
A G
D O
B
M1
A
D E
F
O
(a) Domain of Motz’ problem
(b) Upper half domain of Motz’ problem
Figure 10: The geometrical presentation of the domain of Motz’ problem .
5.3.1.1 Subtracted Off the Boundary Singularities by the Singularity Subtraction Technique Since the Motz’ problem suffers from a boundary singularity at O, the singularity subtraction technique can be applied to the Motz’ problem so as to remove the singularity. Let ∗
φ (r, θ) =
m X i=1
λi r
i− 21
1 cos i − θ 2
(5.5)
be the singular part of solution of the Motz’ problem (5.4) defined on upper half domain ΩM1 , where m is number of subtraction terms. Also, let ¯ θ) := φ(r, θ) − φ ∗ (r, θ) φ(r,
(5.6)
¯ θ) is a m times continuously difbe the smooth part of solution (5.4), where φ(r, ferentiable function on the upper half domain ΩM1 . Since solution φ(r, θ) and 1 r i− 2 cos i − 12 θ, i = 1(1)m, are harmonic, function φ ∗ (r, θ) satisfies the Laplace
¯ θ) into equation (5.1)-(5.3), we have equation. By substituting function φ(r, ∂2 ¯ ∂2 ¯ φ(r, θ) + φ(r, θ) = 0 ∂x 2 ∂y 2
(5.7)
128
defined on a rectangular domain ΩM1 with a crack AO with boundary conditions ∂ ¯ φ(r, θ) ∂y m X 1 i− 21 ¯ φ(r, θ) + λi r cos i − θ 2 i=1 m ∂ ¯ 1 θ X 1 3 − 12 i− 32 sin i − φ(r, θ) + λ1 r sin − λi i − r θ ∂y 2 2 i=2 2 2 m 1 θ X 1 3 ∂ ¯ − 21 i− 32 cos i − φ(r, θ) + λ1 r cos + λi i − r θ ∂x 2 2 i=2 2 2 ¯ θ) φ(r,
= 0 on AO, = 1000 on AB, = 0 on BC, (5.8) = 0 on CD, = 500 on DO.
The Motz’ problem becomes a smooth problem (5.7)-(5.8) after the singularity subtraction technique is applied. Then, the approximate solution of the smooth problem and the stress intensive factor λ1 can be obtained from the segmented TauCollocation Method. 5.3.1.2 Formulation by the Segmented Tau-Collocation Method Since smooth problem (5.7)-(5.8) for Motz’ problem contains some unknown parameters λi , i = 1(1)m, there are some modifications should be done when the segmented Tau-Collocation Method is applied. Taking smooth problem (5.7)-(5.8) into consideration, the conditions defined on DO and OA are different. Therefore, the smooth problem should be divided into at least two rectangular segments OZCD and ABZO, where Z is a point on CB with same horizontal position as the crack tip O. The segmented smooth problem can be formulated by the segmented TauCollocation Method with some modifications on the Cartesian coordinates. The ¯ θ) part and the λi formulation can be conceptually divided into two parts, the φ(r, ¯ θ) part, with absent from all unknown parameters λi , part, i = 1(1)m. The φ(r, i = 1(1)m, can be formulated normally by the segmented Tau-Collocation Method discussed in Section 4.2.2.2. And, the λi part, i = 1(1)m, can be formulated by ¯ θ) part into the coefficients of λi , collocating as same collocation points as the φ(r, i = 1(1)m. By the resultant system of linear algebraic equations (4.45) in Section
129
4.2.2.2, the segmented smooth problem becomes λ 0 1 λ2 0 . . Λ1 |Λ2 | · · · |Λm |G ∗∗ .. = .. λm 0 ∗∗ ∗∗ A F
,
(5.9)
¯ θ) part where G ∗∗ , A ∗∗ and F ∗∗ are the corresponding resultant matrices of the φ(r, and Λi , i = 1(1)m, are column vector of λi part, i = 1(1)m, obtained by collocating ¯ θ) part into the coefficients of λi , i = 1(1)m. as same collocation points as the φ(r, Note that there are no touching between two conditions defined on DO and OA, therefore, there are two extra linear independent algebraic equations inside matrix G ∗∗ in system (5.9), as well as system (5.9) itself, due to the singularity structure of the Motz’ problem . Then, system (5.9) becomes a overdetermined system of linear algebraic equations. Nevertheless, these two extra linear independent algebraic equations can be used in finding out the unknown parameters λi , i = 1(1)m. Since there are two extra linear independent algebraic equations inside system (5.9), the number of subtraction terms m should be selected greater than or equal to 2. In the case that m = 2, system (5.9) becomes a balanced system of linear algebraic equations and can be solved by usual method. Then, the unknown parameters λi , i = 1(1)2, can be obtained and so as the stress intensive factor λ1 . In the case that m > 2, system (5.9) becomes an underdetermined system of linear algebraic equations. In order to balance the system (5.9), additional conditions are introduced. ¯ θ) The additional conditions are generated by subtracting two smooth solutions φ(r, in equation (5.6) with points evaluation on the neighborhood of singular point O ¯ θ) and neglect of the higher order terms. For example, into smooth function φ(r, when m = 3, the additional conditions can be formulated as √ √ 1 11 13 π 1 ¯ ¯ √ φ(r, 0) − φ(r, ) = 500 √ − 1 + 2 λ6 r 2 + 2 λ7 r 2 + · · · 2 2 2
. (5.10)
Since the value of free parameters λ1 < λ2 < λ3 < · · ·, and the value of λi become very small as i increase, the higher order factors in equation (5.10) can be ignored.
130
Then equation (5.10) becomes 1 1 ¯ π ¯ √ φ(r, 0) − φ(r, ) ≈ 500 √ − 1 2 2 2
.
(5.11)
By evaluating a point h to equation (5.11), an additional condition with error of 11
O(h 2 ) can be obtained. Note that the conditions can be formulated as one linear algebraic equations and this linear algebraic equations can be appended into the bottom of system (5.9) so as to balance the whole system. In Table 25 we give some instances of additional conditions for number of subtraction terms m = 3, 4 and 5. Finally, the whole system can be solved by usual method and then the unknown parameters λi , i = 1(1)m, and the approximate solution to the smooth problem (5.7)-(5.8) can be obtained. By equations (5.5)-(5.6), the approximate solution to the Motz’ problem (5.1)-(5.2) can also be obtained. 5.3.1.3 Numerical Examples In our research, we successfully apply the segmented Tau-Collocation Method with incorporation of the singularity subtraction technique to solve the Motz’ problem. A very accuracy results of the stress intensity factor λ1 can be obtained from the suggested method. The setting of the Motz’ problem for the following example is defined as BC = CE = EF = 14 units and AO = GO = AB = F G = 7 units. It is convenience to use such setting for comparing the results with other literatures.
Example 5.1 Consider the Motz’ problem (5.1)-(5.3). The additional conditions used in the case m = 3 is π 1 1 ¯ ¯ √ φ(1, 0) − φ(1, ) ≈ 500 √ − 1 2 2 2
,
m = 4 are √ 29 − 1 1 ¯ π 1 ¯ √ φ(1, 0) − φ(1, ) ≈ 500 √ − 1 2 2 2
√ and
¯ 0) − φ(2, ¯ 0) ≈ 500 2 9 φ(1,
131
Table 25: Some instances of additional conditions for number of subtraction terms m = 3, 4 and 5. m 3
4
5
Additional Condtions ¯ 0) − φ(h, ¯ π ) ≈ 500 √1 − 1 √1 φ(h, 2 2 2 11 ¯ 0) − φ(h, ¯ π) (2h) 2 √12 φ(h, 11 11 2 √1 − 1 2 − h 2 ≈ 500 (2h) 11 2 π ¯ ¯ − h 2 √12 φ(2h, ) 0) − φ(2h, 2 7 7 7 ¯ 0) − h 27 φ(2h, ¯ 0) ≈ 500 (2h) 2 − h 2 (2h) 2 φ(h, 9 9 9 ¯ 0) − h 29 φ(2h, ¯ (2h) 2 φ(h, 0) ≈ 500 (2h) 2 − h 2 9 1 ¯ π ¯ √ 2 φ(h, 0) − φ(h, 2 ) (2h) 9 9 2 √1 + 1 2 − h 2 (2h) ≈ 500 9 2 π ¯ ¯ − h 2 √12 φ(2h, 0) − φ(2h, ) 2 ¯ 0) − φ(h, ¯ π ) ≈ 500 √1 − 1 √1 φ(h, 2 2 2 9 9 9 9 ¯ ¯ (3h) 2 φ(2h, 0) − (2h) 2 φ(3h, 0) ≈ 500 (3h) 2 − (2h) 2 11 11 π π ¯ ¯ 2 (2h) φ(h, 2 ) − h φ(2h, 2 ) ≈ 500 (2h) 2 11 11 ¯ 0) − h 112 φ(2h, ¯ 0) ≈ 500 (2h) 2 (2h) 2 φ(h, 11 11 11 ¯ ¯ 2 2 (3h) φ(2h, 0) − (2h) φ(3h, 0) ≈ 500 (3h) 2 11 11 11 ¯ ¯ (4h) 2 φ(3h, 0) − (3h) 2 φ(4h, 0) ≈ 500 (4h) 2 11 11 11 ¯ ¯ 2 2 (5h) φ(4h, 0) − (4h) φ(5h, 0) ≈ 500 (5h) 2 11 2
−h
11 2
−h
11 2
− (2h)
11 2
− (3h)
11 2
− (4h)
11 2
132
and m = 5 are √ ¯ π ) − φ(2, ¯ π ) ≈ 500 2 11 φ(1, 2 11 − 1 , 2 2 √ √ ¯ 0) − φ(2, ¯ 0) ≈ 500 2 11 φ(1, 2 11 − 1 , √ √ √ √ ¯ 0) − 2 11 φ(3, ¯ 0) ≈ 500 3 11 φ(2, 3 11 − 2 11 √
and
.
In Table 27 we give the results of unknown parameters λi , for i = 1(1)5, for Example 5.1 obtained from the segmented Tau-Collocation Method incorporated of the singularity subtraction technique with Tau degrees nx = ny = 6(1)8 with number of subtraction terms m = 2(1)5. In Table 26 we give the results of unknown parameters λi , for i = 1(1)5, for Example 5.1 obtained from the segmented Tau-Collocation Method incorporated of the singularity subtraction technique (STC-SST) with Tau degrees nx = ny = 9 with number of subtraction terms m = 5 and compared the results of the Tau Method incorporated of singularity subtraction technique with Tau degrees n = 14 with number of subtraction terms m = 5 (Tau-SST) [43], the power series solution (PSS) [45], the boundary integral equation method (BIEM) [55], the penalty-combined approaches to the Ritz-Galerkin and finite element method (RGFEM) [20] and the finite difference methods (FDM) [23]. Similar results can be obtained from the mentioned methods. Note that the size of resultant matrix formulated by the segmented Tau-Collocation Method incorporated of the singularity subtraction technique is 211 × 200 which is much less than other methods. 5.3.2 Biharmonic Equation for Mode I Crack Problem The biharmonic equation for mode I crack describes a rectangular plate, with length 2l and width 2w, contained a crack with length w subjected to a uniform normal stress σ perpendicular to the crack direction over the two edges of plate. This problem can be written in term of the airy stress function Ψ(x, y) [15, 54] and it becomes a biharmonic equation ∂4 ∂4 ∂4 Ψ(x, y) + 2 Ψ(x, y) + Ψ(x, y) = 0 ∂x 4 ∂x 2 ∂y 2 ∂y 4
(5.12)
133
Table 26: The results of unknown parameters λi , for i = 1(1)5, for Example 5.1 obtained from the segmented Tau-Collocation Method incorporated of the singularity subtraction technique with Tau degrees nx = ny = 6(1)8 with number of subtraction terms m = 2(1)5. m
n x , ny
λ1
λ3
λ4
λ5
2
5,5
151.6306 4.7010 −
−
−
6,6,
151.6253 4.7193 −
−
−
7,7
151.6250 4.7235 −
−
−
8,8
151.6252 4.7261 −
−
−
5,5
151.6308 4.7291 1.1634 × 10 −1
−
−
6,6
151.6253 4.7313 1.1602 × 10 −1
−
−
7,7
151.6251 4.7318 1.1578 × 10 −1
−
−
8,8
151.6252 4.7321 1.1569 × 10 −1
−
−
5,5
151.6283 4.7317 1.3205 × 10 −1
−8.3156 × 10 −3
−
6,6
151.6247 4.7331 1.3289 × 10 −1
−8.7679 × 10 −3
−
7,7
151.6249 4.7330 1.3291 × 10 −1
−8.7866 × 10 −3
−
8,8
151.6251 4.7330 1.3286 × 10 −1
−8.7632 × 10 −3
−
5,5
151.6266 4.7318 1.3189 × 10 −1
−8.5765 × 10 −3
8.7972 × 10 −4
6,6
151.6247 4.7331 1.3303 × 10 −1
−8.9447 × 10 −3
1.9919 × 10 −4
7,7
151.6249 4.7330 1.3304 × 10 −1
−8.9660 × 10 −3
2.5071 × 10 −4
8,8
151.6251 4.7329 1.3298 × 10 −1
−8.9364 × 10 −3
2.4871 × 10 −4
3
4
5
λ2
134
Table 27: The results of unknown parameters λi , for i = 1(1)5, for Example 5.1 obtained from the segmented Tau-Collocation Method incorporated of the singularity subtraction technique (STC-SST) with Tau degrees nx = ny = 9 with number of subtraction terms m = 5 and compared the results of the Tau Method incorporated of singularity subtraction technique with Tau degrees n = 14 with number of subtraction terms m = 5 (Tau-SST), the power series solution (PSS), the boundary integral equation method (BIEM), the penalty-combined approaches to the RitzGalerkin and finite element method (RG-FEM) and the finite difference methods (FDM). Methods
λ1
λ2
λ3
λ4
λ5
STC-SST
151.6251612 4.732974 0.132996 -0.0089441 0.0002476
Tau-SST
151.6251554 4.732974 0.132964 -0.0088943 0.0002264
PSS
151.6251553 4.732975 0.132966 -0.0088940 −
BIEM
151.63
4.73
0.133
-0.009
0.0002
RG-FEM
151.560
4.733
0.134
-0.0094
0.00026
FDM
151.634
4.729
0.134
-0.009
0.0002
defined on a rectangular domain ΩB := OABCDEF GO with a crack AO, see Figure 11(a). The problem is symmetric about AOD and can be reduced to a problem defined on the upper half domain ΩB1 := OABCDO with a crack AO, see Figure 11(b), with boundary conditions ∂ Ψ(x, y) = 0 on AO, ∂y ∂ Ψ(x, y) = 0 , Ψ(x, y) = 0 on AB, ∂x 2 x w2 ∂ Ψ(x, y) = σ + wx + , Ψ(x, y) = 0 on BC, 2 2 ∂y ∂ Ψ(x, y) = 2σw on CD, Ψ(x, y) = 2σw 2 , ∂x Ψ(x, y) = 0 ,
(5.13)
135
∂3 ∂ Ψ(x, y) = 0 , Ψ(x, y) = 0 ∂y ∂y 3
on OD.
Note that a singularity point exist in O, i.e. between the conditions DO and AO, so that the biharmonic equation contains boundary singularity. The analytical local series solution in polar coordinates (r, θ), where (r, θ) ∈ ΩB1 , of the biharmonic equation near the singular point O is ∞ X 1 3 i−1 i+ 12 2i − 3 φ(r, θ) = (−1) λ2i−1 r cos i + θ − cos i − θ 2i + 1 2 2 i=1 i i+1 + (−1) λ2i r [cos (i + 1) θ − cos (i − 1) θ] (5.14) see Williams [54], where λi , i = 1(1)∞, are unknowns parameters to be determined and the results of λi , i = 1(1)∞, depends on the setting of the domain, i.e. length 2l and width 2w of domain and crack length w. λ1 is a very important parameter and reflects the stress intensity factor KI of the problem. And, the stress intensity factor KI can be determined by the formula √ KI = − 2π λ1
.
(5.15)
5.3.2.1 Subtracted Off the Boundary Singularities by the Singularity Subtraction Technique Since the biharmonic equation suffers from a boundary singularity at O, the singularity subtraction technique can be applied to the biharmonic equation so as to remove the singularity. Let m X 1 3 ∗ i−1 i+ 12 2i − 3 φ (r, θ) = (−1) λ2i−1 r cos i + θ − cos i − θ 2i + 1 2 2 i=1 i i+1 + (−1) λ2i r [cos (i + 1) θ − cos (i − 1) θ] (5.16) be the singular part of the solution of the biharmonic equation defined on upper half domain ΩB1 , where m is number of subtraction terms. Also, let ¯ θ) := Ψ(r, θ) − Ψ ∗ (r, θ) Ψ(r,
(5.17)
136
B
C
B
A G
O
F
D B
C
E
B1
D
A O
(a) Domain of biharmonic equation
(b) Upper half domain of biharmonic equation
Figure 11: The geometrical presentation of the domain of the biharmonic equation for mode I crack.
¯ θ) is a m times continuously differbe the smooth part of the solution, where Ψ(r, ¯ θ) entiable function on the upper half domain ΩB1 . By substituting function Ψ(r, into equation (5.12)-(5.13), we have ∂4 ¯ ∂4 ¯ ∂4 ¯ Ψ(r, θ) + 2 Ψ(r, θ) + Ψ(r, θ) = 0 ∂x 4 ∂x 2 ∂y 2 ∂y 4
(5.18)
defined on a rectangular domain ΩB1 with a crack AO with boundary conditions ¯ θ) + Ψ ∗ (r, θ) = 0 , Ψ(r, ¯ θ) + Ψ ∗ (r, θ) = 0 , Ψ(r, 2 x w2 ∗ ¯ Ψ(r, θ) + Ψ (r, θ) = σ + wx + , 2 2 ¯ θ) + Ψ ∗ (r, θ) = 2σw 2 , Ψ(r, ∂ ∂ ∗ Ψ(r, θ) + Ψ (r, θ) = 0 , ∂y ∂y Note that
∂ ∂ Ψ ∗ (r, θ), ∂y Ψ ∗ (r, θ) ∂x
and
∂ ¯ ∂ ∗ Ψ(r, θ) + Ψ (r, θ) = 0 on AO, ∂y ∂y ∂ ¯ ∂ ∗ Ψ(r, θ) + Ψ (r, θ) = 0 on AB, ∂x ∂x ∂ ¯ ∂ ∗ Ψ(r, θ) + Ψ (r, θ) = 0 on BC, (5.19) ∂y ∂y ∂ ¯ ∂ ∗ Ψ(r, θ) + Ψ (r, θ) = 2σw on CD, ∂x ∂x ∂3 ¯ ∂3 ∗ Ψ(r, θ) + Ψ (r, θ) = 0 on OD. ∂y 3 ∂y 3
∂3 Ψ ∗ (r, θ) ∂y 3
in conditions (5.19) can be ob-
137
tained by taking differentiation on equation (5.16) by some symbolic computation tools, such as Maple. The biharmonic equation becomes a smooth problem (5.18)(5.19) after the singularity subtraction technique is applied. Then, the approximate solution of the smooth problem and the important unknown parameter λ1 can be obtained from segmented Tau-Collocation Method.
5.3.2.2 Formulation by the Segmented Tau-Collocation Method Smooth problem (5.18)-(5.19) can be formulated by the segmented Tau-Collocation Method. As same as the formulation of the smooth problem for Motz’ problem in Section 5.3.1.2, smooth problem (5.18)-(5.19) is first divided into at least two rectangular segments. Then, the segmented smooth problem is formulated by the segment Tau-Collocation Method. Note that in order to fix the point O during the computation process, conditions (5.19) are formulated by collocating uniform mesh points instead of the Chebyshev zeros. Besides, there are no touching between four conditions defined on DO and OA, therefore, there are eight extra linear independent algebraic equations inside the resultant matrices. In order to balance the resultant matrices without adding additional equations, number of subtraction terms m is set as 4 in usual, i.e. eight unknown parameters λi , i = 1(1)8, are generated. The coefficients of unknown parameters λi , i = 1(1)8, are formed as eight column vectors and these eight column vectors are appended to the resultant matrices. Finally, the whole system can be solved by usual method and then the unknown parameters λi , i = 1(1)8 and the approximate solution to the smooth problem (5.18)-(5.19) can be obtained. By equations (5.16)-(5.17), the approximate solution to the biharmonic equation for mode I crack (5.12)-(5.13) can also be obtained. 5.3.2.3 Numerical Examples In our research, we successfully apply the segmented Tau-Collocation Method with incorporation of the singularity subtraction technique to solve the biharmonic
138
equation. A very accuracy results of the stress intensity factor K can be obtained from the suggested method. The setting of the biharmonic equation for following example is defined as length l = 0.8 units, width w = 0.7 units and uniform normal stress σ = 1 unit. It is convenience to use such setting for comparing the results with other literatures.
Example 5.2 Consider the biharmonic equation (5.12)-(5.13). In Table 27 we give the results of √ unknown parameters λ1 , λ2 and stress intensive factor KI = − 2π λ1 for Example 5.2 obtained from the segmented Tau-Collocation Method incorporated of the singularity subtraction technique with Tau degrees nx = ny = 11(2)15 with number of subtraction terms m = 8. In Table 28 we give the results of unknown param√ eters λ1 and stress intensive factor KI = − 2π λ1 for Example 5.2 obtained from the segmented Tau-Collocation Method incorporated of the singularity subtraction technique (STC-SST) with Tau degrees nx = ny = 11 with number of subtraction terms m = 8 and compared the results of the Tau Method incorporated of singularity subtraction technique with Tau degrees n = 13 with number of subtraction terms m = 2 (Tau-SST) [43], the collocation plus linear programming (CPLP) [53], the Motz’ technique with 4 near and far points (MT-4) and 10 near and far points (MT10) [6] and the finite element method (FEM) [49]. Similar results can be obtained from the mentioned methods. Note that the size of resultant matrix formulated by the segmented Tau-Collocation Method incorporated of the singularity subtraction technique is 320 × 288 which is much less than other methods.
139
Table 28: The results of unknown parameters λi , for i = 1(1)2, and stress in√ tensive factor KI = − 2π λ1 for Example 5.2 obtained from the segmented TauCollocation Method incorporated of the singularity subtraction technique with Tau degrees nx = ny = 11(2)15 with number of subtraction terms m = 8. nx , ny
λ1
λ2
KI
11,11
-1.2629 1.6958 × 10 −1
3.1656
13,13
-1.2046 1.0868 × 10 −1
3.0195
15,15
-1.2100 4.6999 × 10 −2
3.0330
Table 29: The results of unknown parameters λ1 and stress intensive factor √ KI = − 2π λ1 for Example 5.2 obtained from the segmented Tau-Collocation Method incorporated of the singularity subtraction technique (STC-SST) with Tau degrees nx = ny = 11 with number of subtraction terms m = 8 and compared the results of the Tau Method incorporated of singularity subtraction technique with Tau degrees n = 13 with number of subtraction terms m = 2 (Tau-SST) the collocation plus linear programming (CPLP), the Motz’ technique with 4 near and far points (MT-4) and 10 near and far points (MT-10) and the finite element method (FEM). Methods
λ1
STC-SST
KI
Methods
λ1
KI
-1.2629 3.1656
MT-4
-1.244 3.1182
Tau-SST
-1.2654 3.1719
MT-10
-1.279 3.2059
CPLP
-1.2651 3.1711
FEM
-1.28
3.2061
140
Chapter 6
Conclusions and Discussions
In our research, we put great afford on developing both the theoretical and application level of the Tau-Collocation Method and the Tau Method. Our research results are discussed in this thesis and sent to some public journals. This chapter is to conclude our research works on the Tau-Collocation Method and suggest the further development of the Tau-Collocation Method.
6.1
Conclusions
The Tau-Collocation Method, based on the Tau Method, is a very useful and powerful numerical method for accuracy approximate solution of linear ordinary and partial differential equations. It can be directly applied to different types of linear differential equations, with any linear differential operators with variable coefficients, any orders of the given equation and any types of conditions. Also, the formulation and the computer programming implementation of the Tau-Collocation Method are very simple compared with other spectral methods, such as the Galerkin method and even the operational approach of the Tau Method. No integration process and basis changing process should be done when the Tau-Collocation Method is applied. The approximate solutions obtained by the Tau-Collocation Method are very accuracy and converged of infinite order. More accuracy results can be simply obtained by increase the Tau degrees. When the Tau-Collocation Method is applied, the perturbation term of the Tau problem becomes explicit and changeable so that the Tau-Collocation Method can be used in simulating other numerical methods, such as the simulation of the recursive formulation of the Tau Method discussed in Section 3.1.3. However, there are some drawbacks of the Tau-Collocation Method. The operations of the Tau-Collocation Method is O(N 3 ). Although the resultant matrix formulated by the Tau-Collocation Method is very small in usual, it will be-
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comes very large for the segmented Tau-Collocation Method, the Tau-Collocation Method for 3-dimensional PDEs and the problem with large Tau degrees, such as Tau degree > 30. Also, the results obtained by the Tau-Collocation Method may not be as accuracy as the results obtained by the operational approach of the Tau Method in general because the perturbation term used in the Tau-Collocation Method is not targeted to minimize the error of approximations. Nevertheless, same decimal digits of the error of approximations can be obtained by both methods with same Tau degrees. With incorporation of other methods or techniques, the applicability of the TauCollocation Method can be extended to wider type of differential problems. The Tau-Collocation Method incorporated of the linear iterative scheme , the quadratic iterative scheme and the Adomian’s polynomials can be used in approximating nonlinear ordinary and partial differential equations. Since the Adomian’s polynomials can be formulated independent of the dimension of the problems and can be used in nonlinear differential equations with highly nonlinearity, the Tau-Collocation Method with incorporated of Adomian’s polynomials inherits such benefits. Moreover, the segmented Tau-Collocation Method, i.e. the Tau-Collocation Method incorporated of segmentation techniques, is developed for the numerical solution of the differential problems defined on a large domain or the solution of the differential problems experienced with a sharp change over the domain. Besides, the segmented Tau-Collocation Method can also be incorporated of the quadratic iterative scheme for the solution of nonlinear PDEs defined on a long interval. Moreover, the segmented Tau-Collocation Method with incorporation of singularity subtraction technique can be applied to PDEs with boundary singularities with application to the problems in linear elastic fracture mechanics. The formulation of the problems in linear elastic fracture mechanics becomes simpler when the suggested method is applied. The equations for finding out the unknown parameters λi , i = 1(0)m, of the given problem are compensated by the linear algebraic equations generated by the non-touching conditions so that the procedures for building up the additional conditions can be neglected. In fact, the Tau-Collocation Method is not perfect
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enough. Some improvement can be done on it.
6.2
Further Developments
The materials discussed in this thesis are, mostly, the basic ideas to the TauCollocation Method. The Tau-Collocation Method can be enhanced to a more powerful, simple and accuracy numerical method by modifying the formulation process and incorporating of other methods or techniques. Since the operations for the formulation of the Tau-Collocation Method are expensive, more attention should be paid on modifying the formulation process. There are two suggestions on this topic. The first one is starting from the polynomial approximate solutions yn (x), unx ny (x, y) and unx ny nz (x, y, z), i.e. equations (2.4), (3.28) and (3.58) respectively, used in the Tau-Collocation Method. All of the polynomial approximate solutions are in x basis. In fact, the polynomial approximate solutions may select as the basis other than x basis. We tested that exactly same results can be obtained by both x basis and the Chebyshev basis. It is worth to study the selection of the basis so that the operations of the formulation can be reduced by some special basis. For example, if the polynomial approximate solution for the P ODE case is written as nk=0 e ikx , the discrete Fourier transform maybe helpful to improve the operations of formulation. The second is to use some better methods to
do the inversion of the resultant matrix generated by the Tau-Collocation Method. An iterative method with per-conditioner may be a workable idea, see Ghoreishi and Hosseini [13]. Besides, some fast matrix inversion method for full matrix discussed by Miller [28] may also be a workable idea for reducing the operations of formulation. Furthermore, more attention should be paid on increasing the accuracy of the approximate solution. Since the perturbation term is explicit and changeable when the Tau-Collocation Method is applied, it is worth to study how to choose a perturbation term which can give more accuracy results and do not break the simply
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formulation process. For example, we tried to use the perturbation term x [a,b] Hn (x) := gn (x; τ ) 1 − T n−ν+1 0.85 for the ODEs case, where the symbols here follow the symbols defined on Section 3.1.1. In order to make the value of the perturbation term becomes zero, the 0.85 [a,b]
times the maxima of the shifted Chebyshev Polynomial T n−ν+1 (x) are used as the collocation points. The following example shows the results obtained by the TauCollocation Method with the suggested perturbation term.
Example 6.1 Consider a ODEs discussed in Section 3.1.2 Example 3.1. In Table 30 we give the maximum absolute error E := max|y(x) − yn (x)| for Example 6.1 in obtained from the Tau-Collocation Method with new perturbation term (TC-P) with degrees n = 4(1)8 of the approximate polynomial solutions. The results are compared with the results obtained from the traditional Tau-Collocation Method (TC) and the recursive formulation of the Tau Method (Tau-Re). More accuracy result can be obtained by the Tau-Collocation Method with new perturbation term. The higher accuracy can also be obtained when applying the Tau-Collocation Method with new perturbation term to other problems.
In additions, the Tau-Collocation Method can be extended to handle the system of differential equations directly. Also, the Tau-Collocation Method for 3dimensional PDEs can be extended to incorporate of the Adomian’s polynomials for the numerical solution of nonlinear 3-dimensional PDEs and incorporate of the segmentation techniques for the numerical solution of 3-dimensional PDEs defined on a large domain or the solution of 3-dimensional PDEs experienced with a sharp change over the domain. In the application level, it is good to try to tackle some 3dimensional crack problems by the Tau-Collocation Methodfor 3-dimensional PDEs. In fact, the Tau-Collocation Method can be extended to approximate the solution of 4-dimensional PDEs. We have tried for these matters. However, the resultant
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Table 30: The maximum absolute error E := max|y(x) − yn (x)| for Example 6.1 obtained from the Tau-Collocation Method with new perturbation term (TC-P), the traditional Tau-Collocation Method (TC) and the recursive formulation of the Tau Method (Tau-Re) with degrees n = 4(1)8 of approximate polynomial solutions. n
E (TC-P)
E (TC)
E (Tau-Re)
4
3.6564 × 10 −4
3.6397 × 10 −4
3.8106 × 10 −4
5
1.3882 × 10 −5
3.0562 × 10 −5
3.1631 × 10 −5
6
5.6423 × 10 −7
1.6506 × 10 −6
1.6337 × 10 −6
7
3.6636 × 10 −8
9.5120 × 10 −8
8.8297 × 10 −8
8
1.5713 × 10 −9
4.3420 × 10 −9
4.2741 × 10 −9
matrix is too large for us to obtain the approximated solution with Tau degree > 7 by nowadays personal computer. The Tau-Collocation Method for 4-dimensional PDEs will become more workable if the size of resultant matrix can be reduced or the operations for the formulation can be reduced. The Tau-Collocation Method still has great potential for further development in both theoretical and application level. These matters are opened for someone who is interested in the Tau-Collocation Method and Tau Method.
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