Hindawi Publishing Corporation International Journal of Engineering Mathematics Volume 2015, Article ID 650425, 14 pages http://dx.doi.org/10.1155/2015/650425
Research Article Block Backward Differentiation Formulas for Fractional Differential Equations T. A. Biala1 and S. N. Jator2 1
Department of Mathematics and Computer Science, Sule Lamido University, Kafin Hausa, PMB 048, Kafin Hausa, Nigeria Department of Mathematics and Statistics, Austin Peay State University, Clarksville, TN 37044, USA
2
Correspondence should be addressed to S. N. Jator;
[email protected] Received 20 May 2015; Accepted 14 July 2015 Academic Editor: Yurong Liu Copyright Β© 2015 T. A. Biala and S. N. Jator. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper concerns the numerical approximation of Fractional Initial Value Problems (FIVPs). This is achieved by constructing π-step continuous BDFs. These continuous schemes are developed via the interpolation and collocation approach and are used to obtain the discrete π-step BDF and (π β 1) additional methods which are applied as numerical integrators in a block-by-block mode for the integration of FIVP. The properties of the methods are established and regions of absolute stability of the methods are plotted in the complex plane. Numerical tests including large systems arising form the semidiscretization of one-dimensional fractional Burgerβs equation show that the methods are highly accurate and efficient.
1. Introduction In what follows, we consider the FIVP of the following form: π
π·π₯πΌ0 π¦ (π₯) = π (π₯, π¦ (π₯)) , π¦ (π₯0 ) = π¦0 ,
(1)
where 0 < πΌ < 1 is the fractional order and π π·πΌπ₯0 (in the sequel we will simply use π·πΌ ) denotes the Caputo πΌ derivative operator which is defined as π·πΌ π¦ (π₯) =
π₯ 1 β« (π₯ β π )βπΌ π¦σΈ (π ) ππ . Ξ (1 β πΌ) π₯0
(2)
The mathematical modeling of several physical phenomena results in fractional differential equations of form (1) and it plays an important role in various branches of science and engineering. Applications of FDEs are found in chemistry, electronics, circuit theory, seismology, signal processing, control theory, and so on. Also, these FDEs serve as a generalization of their corresponding ordinary differential equations (ODEs). For a brief history and introduction to fractional calculus, we refer the reader to [1β3].
We have adopted the Caputoβs definition of derivatives of noninteger order (which is a modification of the RiemannLiouville definition) since it can be coupled with initial conditions having a clear physical meaning. The existence and the uniqueness of the solution of (1) have been given in Diethelm and Ford [4]. The development of well suited methods for numerically approximating FDEs has received great attention over the past few decades. This is due to the occurence of FDEs in several models. Several methods have been proposed and analyzed for the numerical approximation of this important class of problems (see Lubich [5β7], Garrappa [8], Galeone and Garrappa [9β11], and the references therein). These authors have independently developed Fractional Linear Multistep Methods (FLMMs) using convolution quadratures. Lubich [6] proposed formulas of the following form: π
π
π=0
π=0
(πΌ) π (π‘π , π¦π ) + βπΌ βπππ π (π‘π , π¦π ) , π¦π = π (π‘π ) + βπΌ β ππβπ
(3)
πβ β I, where πππΌ
and πππ are the convolution and starting quadrature weights, respectively, and are independent of the stepsize β.
2
International Journal of Engineering Mathematics
One major difficulty in the FLMMs (3) is in evaluating the convolution weights πππΌ . Most of the methods rely on the J.C.P Miller formula for the computation of these weights. In order to avoid this major drawback, we give a different approach in the construction of the FLMMs. This approach is based on interpolation and collocation as was discussed by Onumanyi et al. [12]. The main aim of this paper is to present and investigate a class of fractional BDF methods which generalize the BDF methods for ODEs. The fractional BDF methods are developed using the interpolation and collocation approach and the regions of absolute stability of the methods are plotted via the boundary locus method. The paper is organized as follows: in Section 2, we discuss the development of the fractional BDF methods. Section 3 details the convergence of the methods while, in Section 4, we give the stability properties and implementation of the methods. In Section 5, we give five numerical examples to elucidate our theoretical results. Finally, we give some concluding remarks in Section 6.
ππ is obtained by replacing the πth column of π by π where π denotes the transpose, ππ (π₯) = π₯π , π = 0(1)π, are basis functions, and π is a vector given by
2. Fractional BDFs
which may be written as
We will construct a π-step Continuous Fractional BDF (CFBDF) which will be used to obtain the discrete fractional BDF (FBDF). The CFBDF has the following general form:
π
π = (π¦π , π¦π+1 , π¦π+2 , . . . , π¦π+πβ1 , ππ+π ) .
Proof. We require that method (4) be defined by the assumed polynomial basis functions: π
πΎπ (π₯) = βπΎπ+1,π ππ (π₯) ,
π = 0 (1) (π β 1) ,
π=0
(9)
π
πΌ
πΌ
β π½π (π₯) = ββ π½π+1,π ππ (π₯) , π=0
πΌ
where πΎπ+1,π and β π½π+1,π are coefficients to be determined. Substituting (9) into (4), we have π πβ1
π
π=0 π=0
π=0
π (π₯) = β β πΎπ+1,π ππ (π₯) π¦π+π + ββπΌ π½π+1,π ππ (π₯) ππ+π
π {πβ1 } π (π₯) = β { β πΎπ+1,π π¦π+π + βπΌ π½π+1,π ππ+π } ππ (π₯) π=0 π=0 { }
(10)
(11)
and expressed as
πβ1
π (π₯) = β πΎπ (π₯) π¦π+π + βπΌ π½π (π₯) ππ+π ,
(4)
π
π=0
π (π₯) = ββπ ππ (π₯) ,
where πΎπ (π₯) and π½π (π₯) are continuous coefficients. We assume that π¦π+π = π(π₯π + πβ) is the numerical approximation to the analytical solution π¦(π₯π+π ) and ππ+π = π·πΌ π(π₯π + πβ) is an approximation to π·πΌ π¦(π₯π+π ). The CFBDF is constructed from its equivalent form by requiring that the exact solution π¦(π₯) is locally approximated by function (4) on the interval [π₯π , π₯π+π ]. Next, we discuss the construction of the CFBDF in the following theorem. Theorem 1. Let (4) satisfy the following conditions: π (π₯π+π ) = π¦π+π ,
π = 0 (1) (π β 1) ,
π·πΌ π (π₯π+π ) = ππ+π .
(5)
π
π (π₯) = β
det (ππ )
π=0
det (π)
ππ (π₯) ,
where πβ1
βπ = β πΎπ+1,π π¦π+π + βπΌ π½π+1,π ππ+π .
By imposing condition (5) on (12), we obtain a system of (π + 1) equations, which can be expressed as π = πΏπ where πΏ = (β0 , β1 , . . . , βπ )π is a vector of (π + 1) undetermined coefficients. Using Crammerβs rule, the elements of πΏ can be obtained and are given by βπ =
det (ππ ) det (π)
(6)
π
π (π₯) = β
where we define matrix π as β
β
β
ππ (π₯π )
π0 (π₯π+1 )
β
β
β
ππ (π₯π+1 )
.. .
.. .
.. .
) ). )
π0 (π₯π+πβ1 ) β
β
β
ππ (π₯π+πβ1 ) πΌ πΌ (π· π0 (π₯π+π ) β
β
β
π· ππ (π₯π+π ))
(13)
π=0
π=0
π0 (π₯π )
(12)
π=0
,
π = 0 (1) π,
(14)
where ππ is obtained by replacing the πth column of π by π. We rewrite (12) using the newly found elements of πΏ as
Then continuous representation (4) is equivalent to
( π=( (
(8)
(7)
det (ππ ) det (π)
ππ (π₯) .
(15)
Remark 2. Continuous scheme (4) which is equivalent to (6) is evaluated at π₯π+π to obtain the π-step FBDF of the following form: πβ1
π¦π+π β β πΎππ π¦π+π = βπΌ π½ππ ππ+π . π=0
(16)
International Journal of Engineering Mathematics
3
πβ1
where πΏ(β) is the truncation error vector of the formula in (18), π = [π¦(π₯π+1 ), π¦(π₯π+2 ), . . . , π¦(π₯π+π )]π , and πΉ(π) = [π(π₯π+1 , π¦(π₯π+1 )), π(π₯π+2 , π¦(π₯π+2 )), . . . , π(π₯π+π , π¦(π₯π+π ))]π . The approximate form of the system is given by
π=0
π΄π β βπΌ π΅πΉ (π) + πΆ = 0,
Also, we emphasize that continuous scheme (6) is used to obtain πσΈ (π₯) and evaluated at π₯π+π , π = 1(1)(π β 1), to obtain βπΌ ππ+1 β β πΎ1π π¦π+π = βπΌ π½1π ππ+π πβ1
βπΌ ππ+2 β β πΎ2π π¦π+π = βπΌ π½2π ππ+π π=0
(17) .. .
where π is the approximate solution of vector π. Subtracting (20) from (21), we obtain the following error system: π΄πΈ = βπΌ π΅ [πΉ (π) β πΉ (π)] + πΏ (β) ,
πβ1 π=0
π΄πΈ β€ βπΌ π΅ β
πΎ1 β
πΈ + πΏ (β) , β1
which, together with (16), forms the Block FBDF (BFBDF) which may be written in the following form: π΄π β β π΅πΉ (π) + πΆ = 0,
πΈ β€ (π΄ β βπΌ π΅ β
πΎ1 ) πΏ (β) .
(18)
Remark 3. We note that it is possible to construct method (6) using other bases such exponential and trigonometric functions. However, (6) is constructed using polynomial basis functions, since methods produced via polynomial basis functions are easier to analyze. In fact, using other bases for the construction of (6) has the disadvantage of introducing additional parameters which makes the analysis of the methods produced more cumbersome. Moreover, the polynomial basis functions are appropriate for the construction of this class of methods since other bases can be written as polynomials in π₯ via the Taylor series expansion.
πΈ β€ (π΄ + π)β1 πΏ (β) = π ((πβ)πΌβ1 ) β
πΎβπ .
Definition 5. BFAMM (18) is said to be (a) stable if and only if, for any π¦0 β Rπ , there exists πΎ2 > 0 such that the solution of (1) satisfies βπ¦(π₯π )β β€ πΎ2 for π β₯ 1, (b) asymptotically stable if and only if, for any π¦0 β Rπ , the solution of (1) satisfies βπ¦(π₯π )β β 0 as π β β.
4. Stability Properties of the Methods To study the stability properties of BFBDF (18), we consider the following linear test problem: π·πΌ π¦ (π₯) = ππ¦ (π₯) ,
Theorem 4. Let π(π₯, π¦) be Lipschitz continuous with respect to y in a region D defined by π β€ π₯ β€ π and ββ < π¦ < β, where π and π are finite. Let (16) and (17) be constructed in such a way that π½ππ = π(ππΌβ1 ); then (18) is said to be convergent if, for all initial value problems (1), we have that (19)
for all π₯ β [π, π] and where constant πΎ does not depend on β and π is the order of the method and β > 0 is sufficiently small. Proof. The exact form of the system formed by (16) and (17) is given by πΌ
π β C, 0 < πΌ < 1,
π¦ (π₯0 ) = π¦0 ,
In this section, we will discuss the convergence of the methods in the following theorem.
π΄π β β π΅πΉ (π) + πΆ + πΏ (β) = 0,
(20)
(24)
((π΄ + π)β1 exists since it is a monotone matrix (see [13])).
3. Convergence of Methods
(π₯ = πβ) .
(23)
Let π = ββπΌ π΅ β
πΎ1 , so that
where π = [π¦π+1 , π¦π+2 , . . . , π¦π+π ]π , πΉ(π) = [ππ+1 , ππ+2 , . . ., ππ+π ]π , π΄ and π΅ are the coefficients of the formulas in (16) and (17), and πΆ is a vector of initial conditions.
σ΅¨ σ΅¨σ΅¨ σ΅¨σ΅¨π β πσ΅¨σ΅¨σ΅¨ β€ πΎ β
π₯πΌβ1 βπ , σ΅¨ σ΅¨
(22)
where πΈ = π β π. Let πΎ1 be the Lipschitz constant of π; then
βπΌ ππ+πβ1 β β πΎ(πβ1)π π¦π+π = βπΌ π½(πβ1)π ππ+π
πΌ
(21)
(25)
whose exact solution can be expressed in terms of the Mittagπ Leffler function πΈπΌ (π₯) = ββ π=0 (π₯ /Ξ(πΌπ + 1)), as π¦(π₯) = πΌ πΈπΌ (π(π₯ β π₯0 ) )π¦0 . We rewrite (18) in the following form: π΄ 0 ππΎ + π΄ 1 ππΎβ1 = π΅0 πΉπΎ ,
(26)
where ππΎ = [π¦π+1 , π¦π+2 , . . . , π¦π+π ]π , ππΎβ1 = [π¦πβπβ1 , . . . , π¦πβ1 , π¦π ]π , and πΉπΎ = [ππ+1 , ππ+2 , . . . , ππ+π ]π . Applying (26) to the linear test equation, we have the following linear recurrence relation: ππΎ = π (π) ππΎβ1 , π = πβπΌ ,
(27)
where β1
π (π) = (π΄ 0 β ππ΅0 ) π΄ 1 ,
(28)
4
International Journal of Engineering Mathematics Absolute stability region Im
Absolute stability region Im
Re
Re
(a) πΌ = 0.25
(b) πΌ = 0.5
Absolute stability region Im
Re
(c) πΌ = 0.75
Figure 1: The region of stability of the BFBDF for π = 1 is to the left of the dividing line and is symmetric about the real axis.
where π(π) is the amplification matrix which determines the stability of the method. We are interested in investigating the values of βπΌ π for which the numerical solution of (1) given by (18) asymptotically vanishes as the true solution. We give below the following definitions. Definition 6. Stability domain Ξ© of BFBDF (18) is the set of all π = βπΌ π β C such that linear recurrence (18) is asymptotically stable. Definition 7. The stability domain of BFBDF (18) is given by the following set: σ΅¨ σ΅¨ Ξ© = {π : σ΅¨σ΅¨σ΅¨π (π)σ΅¨σ΅¨σ΅¨ < 1} , (29) where π(π) is the spectral radius of π(π). Figures 1, 2, 3, and 4 show the plot of the stability domain Ξ© of BFBDF (18) for some πΌ for π = 1 to 4. Remark 8. The 1-step BFBDF is π΄-stable for all values of πΌ β (0, 1) while the 2-step and 3-step BFBDF are π΄-stable for values of πΌ β₯ 0.75. Also, the 4-step BFBDF is π΄-stable for πΌ β₯ 0.80. 4.1. Implementation. Conventionally, FBDF (16) requires π initial conditions which are usually provided using a onestep method (like the Runge-Kutta method). However, in
this paper, we construct additional methods from continuous scheme (15) which are implemented together with FBDF (16) to obtain approximate solutions to the exact solution of (1) without requiring starting values and predictors. For instance, if π = 0 and π = 2, then (π¦1 , π¦2 )π are simultaneously obtained over the interval [π₯0 , π₯2 ] as π¦0 is known from the IVP. Similarly, if π = 1 and π = 3, then (π¦4 , π¦5 , π¦6 )π are simultaneously obtained over the interval [π₯3 , π₯6 ] and π¦3 is known from the previous block and so on, until we reach the final subinterval [π₯πβπ , π₯π].
5. Numerical Examples We validate our theoretical results from the previous sections by considering the following examples, which were solved using the BFBDF using a written code in Mathematica. The maximum errors are obtained for different step sizes in the interval of integration. We have solved two scalar examples, one system, one linear, and one nonlinear heat-type fractional differential equation. Example 1. We consider the following problem given in [14]: π·πΌ π¦ (π₯) = βπ¦ + π₯2 β π₯ +
2π₯2βπΌ π₯1βπΌ + , Ξ (3 β πΌ) Ξ (2 β πΌ) 0 < πΌ < 1, 0 β€ π₯ β€ 1,
International Journal of Engineering Mathematics
5
Absolute stability region Im Absolute stability region Im Re Re
(a) πΌ = 0.25
(b) πΌ = 0.5
Absolute stability region Im
Re
(c) πΌ = 0.75
Figure 2: The region of stability of the BFBDF for π = 2 is to the left of the dividing line and is symmetric about the real axis; the square and plus symbols represent the zeros and poles of the spectral radius of π(π), respectively. Absolute stability region Im
Absolute stability region Im
Absolute stability region Im
Re
Re
Re
(a) πΌ = 0.25
(b) πΌ = 0.5
(c) πΌ = 0.75
Figure 3: The region of stability of the BFBDF for π = 3 is to the left of the dividing line and is symmetric about the real axis; the square and plus symbols represent the zeros and poles of the spectral radius of π(π), respectively.
π¦ (0) = 0, Exact: π¦ (π₯) = π₯2 β π₯. (30)
Tables 1, 2, and 3 show the numerical results for Example 1 using different values of πΌ. It is evident from the table that the BFBDF performs favorably well with smaller number of steps.
6
International Journal of Engineering Mathematics Absolute stability region Im Absolute stability region Im
Re Re
(a) πΌ = 0.25
(b) πΌ = 0.5
Absolute stability region
Absolute stability region
Im
Im
Re
Re
(c) πΌ = 0.75
(d) πΌ = 0.80
Absolute stability region Im
Re
(e) πΌ = 0.85
Figure 4: The region of stability of the BFBDF for π = 4 is to the left of the dividing line and is symmetric about the real axis; the square and plus symbols represent the zeros and poles of spectral radius of π(π), respectively.
Example 2. We also consider the following FIVP with variable coefficients: π·πΌ π¦ (π₯) β π₯π¦ =
Ξ (2πΌ + 1) πΌ π₯ β π₯ β π₯2πΌ+1 , Ξ (πΌ + 1) 0 < πΌ < 1, 0 β€ π₯ β€ 1,
show the numerical results for Example 2 using different values of πΌ. It is evident from the table that the BFBDF performs favorably well with smaller number of steps. Example 3. We also consider the system of FIVP given in [15]:
(31)
π¦ (0) = 1,
π·πΌ π¦1 (π₯) = π¦1 (π₯) + π¦2 (π₯) , π·πΌ π¦2 (π₯) = βπ¦1 (π₯) + π¦2 (π₯) ,
Exact: π¦ (π₯) = 1 + π₯2πΌ . This example was chosen to show the performance of the BFBDF on FIVP with variable coefficients. Tables 4, 5, and 6
0 < πΌ < 1, 0 β€ π₯ β€ 1, π¦1 (0) = 0, π¦2 (0) = 1,
International Journal of Engineering Mathematics
7
Table 1: Maximum errors using the BFBDF π = 2 for Example 1. π 10 20 40 80
πΌ = 0.25 76.661π β 16 4.441π β 16 9.437π β 16 2.220π β 15
πΌ = 0.50 8.882π β 16 9.992π β 16 1.499π β 15 1.499π β 15
πΌ = 0.75 3.1091π β 15 1.998π β 15 9.992π β 16 3.386π β 15
Table 6: Maximum errors using the BFBDF π = 4 for Example 2. π 10 20 40 80
πΌ = 0.25 3.761π β 01 9.320π β 02 1.123π β 01 1.242π β 01
πΌ = 0.50 1.165π β 11 1.075π β 11 5.349π β 11 2.826π β 10
πΌ = 0.75 5.805π β 03 5.008π β 03 5.606π β 03 5.740π β 03
Table 2: Maximum errors using the BFBDF π = 3 for Example 1.
Table 7: Maximum errors using the BFBDF πΌ = 1 for Example 3.
π 10 20 40 80
π 5 10 20 40 80
πΌ = 0.25 8.882π β 16 2.648π β 15 1.724π β 15 3.109π β 15
πΌ = 0.50 3.071π β 15 2.387π β 15 6.085π β 15 1.263π β 14
πΌ = 0.75 2.442π β 15 2.054π β 15 2.866π β 15 1.379π β 14
π=2 6.647π β 02 5.276π β 03 9.839 β 04 2.073π β 04 4.717π β 05
π=3 6.033π β 03 7.498π β 04 9.430π β 05 1.174π β 05 1.446π β 06
π=4 3.991π β 03 1.382π β 04 2.814π β 06 1.727π β 07 6.489π β 06
Table 3: Maximum errors using the BFDBF π = 4 for Example 1. π 10 20 40 80
πΌ = 0.25 3.997π β 15 3.425π β 15 8.527π β 14 1.337π β 12
πΌ = 0.50 9.869π β 15 8.040π β 14 5.887π β 13 4.041π β 12
πΌ = 0.75 2.663π β 12 2.988π β 12 3.077π β 12 9.291π β 12
Table 4: Maximum errors using the BFDBF π = 2 for Example 2. π 10 20 40 80
πΌ = 0.25 4.723π β 01 5.122π β 01 5.281π β 01 5.320π β 02
πΌ = 0.50 1.243π β 15 6.661π β 15 8.660π β 15 1.776π β 14
πΌ = 0.75 2.563π β 02 2.900π β 02 2.931π β 02 2.894π β 02
Table 5: Maximum errors using the BFBDF π = 3 for Example 2. π 10 20 40 80
πΌ = 0.25 8.047π β 02 1.322π β 01 1.587π β 01 1.796π β 01
πΌ = 0.50 1.688π β 14 5.240π β 14 6.239π β 14 4.352π β 13
πΌ = 0.75 7.124π β 03 9.719π β 03 1.036π β 02 1.074π β 02
Exact: π¦1 (π₯) = ππ₯ sin (π₯) , π¦2 (π₯) = ππ₯ cos (π₯) for πΌ = 1. (32) This example was chosen to show that the BFBDF performs well on a system as demonstrated for πΌ = 1. We note that for πΌ = 0.85 the solutions produced by the BFBDF are in agreement with the expected behavior of the exact solution. Details of the results are displayed in Table 7 and Figures 5 and 6. Example 4. We also consider the following one-dimensional fractional heat-like problem given in [16]: ππΌ π’ π₯2 π2 π’ = , ππ‘πΌ 2 ππ₯2
0 < πΌ < 1, 0 β€ π₯ β€ 1
(33)
subject to the initial/boundary conditions π’(π₯, 0) = π₯2 , π’(0, π‘) = 0, π’(1, π‘) = ππ‘ , and π‘ β₯ 0. The exact solution π’(π₯, π‘) = π₯2 (1+π‘πΌ /Ξ(πΌ+1)+π‘2πΌ /Ξ(2πΌ+1)+π‘3πΌ /Ξ(3πΌ+1)+β
β
β
). In order to solve this PDE using the BFBDF, we carry out the semidiscretization of the spatial variable π₯ using the second-order finite difference method to obtain the following first-order system in the second variable π‘: β 2π’π + π’πβ1 ) (π’ ππΌ π’π 2 β π₯π ( π+1 ) = ππ , πΌ ππ‘ (Ξπ₯)2 π = 1, . . . , π β 1,
(34)
2 , π’ (π₯π , 0) = π₯π
where Ξπ₯ = (π β π)/π, π₯π = π + πΞπ₯, π = 0, 1, . . . , π, u = [π’1 (π‘), . . . , π’π(π‘)]π , g = [π1 (π‘), . . . , ππ (π‘)]π , π’π (π‘) β π’(π₯π , π‘), and ππ (π‘) β π(π₯π , π‘) = 0, which can be written in the following form: uπΌ = f (t, u) ,
(35)
subject to the boundary conditions u(π‘0 ) = u0 , where f(t,u) = Au + g and A is π Γ π matrix arising from the semidiscretized system and g is a vector of constants. This example was chosen to show that the BFBDF performs well on a one-dimensional fractional heat-like problem with variable coefficients as demonstrated for πΌ = 1. We note that for πΌ = 0.75 the solutions produced by the BFBDF are in agreement with the expected behavior of the exact solution. Details of the results are displayed in Figures 7 and 8. Example 5. We consider the following fractional Burgerβs equation given in [17]: ππΌ π’ ππ’2 π2 π’ + = 0, β ππ‘πΌ ππ₯ ππ₯2
0 < πΌ β€ 1, 0 < π₯ < 1
(36)
subject to the initial/boundary conditions π’(π₯, 0) = π₯, π’(0, π‘) = 0, π’(1, π‘) = 1/(1 + π‘), and π‘ β₯ 0. The exact solution π’(π₯, π‘) = π₯/(1 + π‘), πΌ = 1.
8
International Journal of Engineering Mathematics 2.0 1.5
1.0
y(x)
y(x)
1.5
0.5
1.0 0.5
0.2
0.4
0.6
0.2
0.8
0.4
0.6
0.8
x
x y1 y2
y1 y2 (a) Exact for πΌ = 1
(b) BFBDF for πΌ = 1, π = 2
2.0 1.5
y(x)
y(x)
1.5 1.0
1.0
0.5
0.5
0.2
0.4
0.6
0.2
0.8
0.4
0.6
0.8
x
x y1 y2
y1 y2 (c) BFBDF for πΌ = 1, π = 3
(d) BFBDF for πΌ = 1, π = 4
Figure 5: Graphical evidence for Example 3, π = 10.
In order to solve this PDE using the BFBDF, we carry out the semidiscretization of the spatial variable π₯ using the second-order finite difference method to obtain the following first-order system in the second variable π‘: β 2π’π + π’πβ1 ) (π’ ππΌ π’π 2 β π₯π ( π+1 ) = ππ , πΌ ππ‘ (Ξπ₯)2 π = 1, . . . , π β 1,
(37)
2 π’ (π₯π , 0) = π₯π ,
where Ξπ₯ = (π β π)/π, π₯π = π + πΞπ₯, π = 0, 1, . . . , π, u = [π’1 (π‘), . . . , π’π(π‘)]π , g = [π1 (π‘), . . . , ππ (π‘)]π , π’π (π‘) β π’(π₯π , π‘), and ππ (π‘) β π(π₯π , π‘) = 0, which can be written in the following form: uπΌ = f (t, u) ,
(38)
subject to the boundary conditions u(π‘0 ) = u0 , where f(t,u) = Au+g and A is πΓπ matrix arising from the semidiscretized system and g is a vector of constants. This example was chosen to show that the BFBDF performs well on the Burgerβs equation as demonstrated for πΌ = 1. We note that for πΌ = 0.75 the solutions produced by the BFBDF are in agreement with the expected behavior of the exact solution. Details of the results are displayed in Figures 9 and 10. 5.1. Block versus Predictor-Corrector Implementations. In order to demonstrate the superiority of the block implementation of the BFBDF over its predictor-corrector (PC) implementation, we have solved Example 1 for π = 1, 2, 3, 4 and the results are displayed in Table 8. We constructed Fractional Explicit Adams Methods and used them as predictors for the FBDF. In Table 8, we observe that as the step length decreases the PC mode gives inaccurate approximations, while the BFBDF still retains its high accuracy. With these observations, we conclude that the BFBDF can be used as
International Journal of Engineering Mathematics
9
Table 8: Comparison of exact errors for Example 1 (πΌ = 0.75). π 10 20 40 80
π=1 Block 9.434π β 02 1.209π β 01 1.344π β 01 1.412π β 01
π=2
PC 8.054π β 02 1.143π β 01 1.306π β 01 1.389π β 01
Block 3.107π β 15 1.887π β 15 1.110π β 15 3.776π β 15
π=3 PC 1.611π β 12 4.733π β 10 7.388π β 09 3.203π β 06
Block 2.442π β 15 2.054π β 15 2.866π β 15 1.379π β 14
π=4 PC 6.158π β 16 1.409π β 15 5.560π β 14 4.492π β 13
Block 2.663π β 12 2.988π β 12 3.077π β 12 7.365π β 12
PC 5.480π β 10 4.590π β 07 6.973π β 05 5.914π β 03
2.0
2.0
1.5
1.5 y(x)
y(x)
2.5
1.0 0.5
1.0 0.5
0.2
0.4
0.6
0.2
0.8
0.4
0.6
0.8
x
x y1 y2
y1 y2 (a) BFBDF for πΌ = 0.85, π = 2
(b) BFBDF for πΌ = 0.85, π = 3
2.0
y(x)
1.5 1.0 0.5
0.2
0.4
0.6
0.8
x y1 y2 (c) BFBDF for πΌ = 0.85, π = 4
Figure 6: Graphical evidence for Example 3, π = 10.
a general purpose method for the integration of fractional differential equations.
6. Conclusion We have constructed a family of fractional linear multistep methods via the interpolation and collocation techniques.
The methods developed are implemented as block methods for the numerical approximation of FIVPs. The stability properties of the methods are discussed and numerical examples are given to show that the methods are accurate and efficient, even when applied to large systems arising from the semidiscretization of one-dimensional fractional heatlike partial differential equations.
10
International Journal of Engineering Mathematics
Error
Analytic solution
0.010
2
1.0
0.005
1.0
1
0.000 0.0
0.5
0 t
0.5
0.0
t
0.5 x
0.5 x
1.0 0.0
1.0 0.0 (a) Error for πΌ = 1, π = 2
(b) Exact for πΌ = 1
Numerical solution
Analytic solution
3
2
1.0
1 0
1.0
2 1
0.5
0.0
0 t
0.5
0.0
0.5 x
0.5 x 1.0 0.0
1.0 0.0
(c) BFBDF for πΌ = 1, π = 2
(d) Exact for πΌ = 0.75 Numerical solution
2
1.0
1 0
0.5
0.0
t
0.5 x 1.0 0.0 (e) BFBDF for πΌ = 0.75, π = 2
Figure 7: Graphical evidence for Example 4, π = 10.
t
International Journal of Engineering Mathematics
11
Error
Analytic solution
0.010
2
1.0
1.0
1
0.005 0.000 0.0
0.5
0 t
0.5
0.0
t
0.5 x
0.5 x 1.0 0.0
1.0 0.0
(a) Error for πΌ = 1, π = 3
(b) Exact for πΌ = 1
Numerical solution
Analytic solution
3
2
1.0
1 0
1.0
2 1
0.5
0.0
0 t
0.5
0.0
0.5 x
0.5 x 1.0 0.0
1.0 0.0
(c) BFBDF for πΌ = 1, π = 3
(d) Exact for πΌ = 0.75 Numerical solution
2
1.0
1 0
0.5
0.0
t
0.5 x 1.0 0.0 (e) BFBDF for πΌ = 0.75, π = 3
Figure 8: Graphical evidence for Example 4, π = 10.
t
12
International Journal of Engineering Mathematics
Error
Analytic solution
1.0 0.0002
1.0
0.0001 0.0000 0.0
0.5
1.0
0.5 0.0
t
0.5
0.0
0.5 x
t
0.5 x 1.0 0.0
1.0 0.0
(a) Error for πΌ = 1, π = 2
(b) Exact for πΌ = 1
Numerical solution
Analytic solution
1.0
1.0 1.0
0.5 0.0 0.0
0.5
t
1.0
0.5 0.0 0.0
0.5
0.5 x
0.5 x 1.0 0.0
1.0 0.0
(c) BFBDF for πΌ = 1, π = 2
(d) BFBDF for πΌ = 0.75, π = 1 Numerical solution
1.0 1.0
0.5 0.0
0.5
0.0
t
0.5 x 1.0 0.0 (e) BFBDF for πΌ = 0.75, π = 2
Figure 9: Graphical evidence for Example 5, π = 10.
t
International Journal of Engineering Mathematics
13
Error
Analytic solution
1.0 0.00003 0.00002 0.00001 0
1.0
0.5
0.0
1.0
0.5 0.0
t
0.5
0.0
t
0.5 x
0.5 x
1.0 0.0
1.0 0.0 (a) Error for πΌ = 1, π = 3
(b) Exact for πΌ = 1
Numerical solution
Analytic solution
1.0
1.0 1.0
0.5 0.0 0.0
0.5
t
1.0
0.5 0.0 0.0
0.5
0.5 x
0.5 x 1.0 0.0
1.0 0.0
(c) BFBDF for πΌ = 1, π = 3
(d) BFBDF for πΌ = 0.75, π = 3 Numerical solution
1.0 1.0
0.5 0.0 0.0
0.5
t
0.5 x 1.0 0.0 (e) BFBDF for πΌ = 0.75, π = 4
Figure 10: Graphical evidence for Example 5, π = 10.
t
14
Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment The authors are very grateful to the referee whose useful suggestions greatly improved the quality of the paper.
References [1] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. [2] K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974. [3] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. [4] K. Diethelm and N. J. Ford, βAnalysis of fractional differential equations,β Journal of Mathematical Analysis and Applications, vol. 265, no. 2, pp. 229β248, 2002. [5] C. Lubich, βFractional linear multistep methods for AbelVolterra integral equations of the second kind,β Mathematics of Computation, vol. 45, no. 172, pp. 463β469, 1985. [6] C. Lubich, βDiscretized fractional calculus,β SIAM Journal on Mathematical Analysis, vol. 17, no. 3, pp. 704β719, 1986. [7] C. Lubich, βA stability analysis of convolution quadratures for Abel-Volterra integral equations,β IMA Journal of Numerical Analysis, vol. 6, no. 1, pp. 87β101, 1986. [8] R. Garrappa, βOn some explicit Adams multistep methods for fractional differential equations,β Journal of Computational and Applied Mathematics, vol. 229, no. 2, pp. 392β399, 2009. [9] L. Galeone and R. Garrappa, βExplicit methods for fractional differential equations and their stability properties,β Journal of Computational and Applied Mathematics, vol. 228, no. 2, pp. 548β560, 2009. [10] L. Galeone and R. Garrappa, βOn multistep methods for differential equations of fractional order,β Mediterranean Journal of Mathematics, vol. 3, no. 3-4, pp. 565β580, 2006. [11] L. Galeone and R. Garrappa, βSecond order multistep methods for fractional differential equations,β Tech. Rep. 20/2007, Department of Mathematics, University of Bari, 2007. [12] P. Onumanyi, U. W. Sirisena, and S. N. Jator, βContinuous finite difference approximations for solving differential equations,β International Journal of Computer Mathematics, vol. 72, no. 1, pp. 15β27, 1999. [13] M. K. Jain and T. Aziz, βCubic spline solution of two-point boundary value problems with significant first derivatives,β Computer Methods in Applied Mechanics and Engineering, vol. 39, no. 1, pp. 83β91, 1983. [14] S. Kazem, βExact solution of some linear fractional differential equations by Laplace transform,β International Journal of Nonlinear Science, vol. 16, no. 1, pp. 3β11, 2013. [15] V. S. Erturka and S. Momani, βSolving systems of fractional differential equations using differential transform method,β Journal of Computational and Applied Mathematics, vol. 215, no. 1, pp. 142β151, 2008.
International Journal of Engineering Mathematics [16] S. Momani, βAnalytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method,β Applied Mathematics and Computation, vol. 165, no. 2, pp. 459β472, 2005. [17] M. Dehghan, J. Manafian, and A. Saadatmandi, βSolving nonlinear fractional partial differential equations using the homotopy analysis method,β Numerical Methods for Partial Differential Equations, vol. 26, no. 2, pp. 448β479, 2010.
Advances in
Operations Research Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Advances in
Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Applied Mathematics
Algebra
Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Probability and Statistics Volume 2014
The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of
Differential Equations Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Submit your manuscripts at http://www.hindawi.com International Journal of
Advances in
Combinatorics Hindawi Publishing Corporation http://www.hindawi.com
Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Journal of
Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of Mathematics and Mathematical Sciences
Mathematical Problems in Engineering
Journal of
Mathematics Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Discrete Mathematics
Journal of
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Discrete Dynamics in Nature and Society
Journal of
Function Spaces Hindawi Publishing Corporation http://www.hindawi.com
Abstract and Applied Analysis
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
International Journal of
Journal of
Stochastic Analysis
Optimization
Hindawi Publishing Corporation http://www.hindawi.com
Hindawi Publishing Corporation http://www.hindawi.com
Volume 2014
Volume 2014