Numerical Methods for Solving Abel Integral Equation ...

World Academy of Science, Engineering and Technology 65 2012

Numerical Methods for Solving Abel Integral Equation and Fredholm Integration Equation Yucheng Liu and easily approximated the kernel. Besides the traditional numerical methods, a relatively new approach, wavelet, was employed to solve such integral equation. S.A. Yousefi [6] investigated the properties of Legendre wavelet and used it to approximate the function in Abel integral equation. By using Legendre wavelet, the integral equations were easily reduced to a system of algebraic equations. In this paper, another wavelet, Chebyshev wavelet is applied for the solution of Abel integral equation. From the numerical examples, it is concluded that Chebyshev wavelets can provide a good approximation for the functions in Abel integral equation and are especially useful in the solution of this integral equation.

Abstract— This paper presents methods of solving Abel integral equation as well as Fredholm integral equation of the second kind. In solving Abel integral equation, the functions in Abel integral equation are approximated based on Chebyshev wavelet and therefore, the solving of Abel integral equation is reduced to the solving of linear algebraic equations. In solving Fredholm integral equation, Chebyshev polynomials are applied to approximate a solution for an unknown function in the Fredholm integral equation and convert this equation to system of linear equations. Convergence of this method and rate of convergence are discussed. These presented methods are demonstrated and validated through several numerical examples.

Keywords— Abel integral equation, Chebyshev polynomials, Chebyshev wavelets, Fredholm integral equation, linear algebraic equations I.

II. CHEBYSHEV WAVELETS Wavelets have been extensively applied in many scientific and engineering fields [6, 7]. A wavelet is a kind of mathematical function used to divide a single function or continuous signal into different frequency components and investigate each component with a proper resolution. The wavelets are a family of dilated and transformed functions (daughter wavelets) of a given function known as the mother wavelet, ψ(t). For the given mother wavelet ψ(t) being dilated by a factor of a and transformed by a factor of be, the family of continuous wavelets are given as [8]

INTRODUCTION OF ABEL INTEGRAL EQUATION

Abel integral equation was derived by Abel when he was generalizing and solving the tautochrone problem. It allows users to compute the total time required for a particle to fall along a given curve. This integral equation has two forms [1] First kind:

∫

x

0

y (t ) x −t

dt = f ( x) x

(1)

y (t )

y ( x) = ∫ dt + f ( x ) 0 x−t (2) Second kind: Where f(x) is a given function and y(t) is an unknown function. The Abel problem is to find a path y(x) that with a specified total time of descent from a given initial height, f(x), a particle will follow if it moves without initial velocity and only under the influence of gravity. Abel integral equation has aroused many researchers’ interests and several numerical methods for the solution of this equation were developed. Piessens and Verbaeten [2] proposed a method of using Chebyshev polynomials to approximate the given function f(x), and expressed the solution as a sum of generalized hypergeometric functions and then evaluate them using a simple recurrence relation. Capobianco [3] presented a solution for the first-kind integral equation using a collocation method and a trigonometric polynomial. Mandal et. al. [4] employed the ideas of fractional integral operators to solve a system of generalized Abel integral equations. Liu and Tao [5] developed a high accuracy combination algorithm for solving the first kind Abel integral equations. In their works, the first kind Abel integral equation was transformed to the second kind Volterra integral equation

ψ a ,b (t ) = a

−1 / 2

⎛t −b⎞ ⎟ ⎝ a ⎠ , a, b ∈ ℜ , a ≠ 0

ψ⎜

(3) Chebyshev wavelet is an important type of wavelets and has been used for solving diverse differential and integral equations such as Volterra integral equations [9, 10]. From Babolian and Fattahzadeh [10], the Chebyshev wavelets ψn,m(t) = ψ(k, n, m, t) have four arguments: dilation argument k can assume any positive integer; translation argument n = 1, 2, …, 2k-1; m is the order of Chebyshev polynomials of the first kind; t is the normalized time. The wavelets are defined as: n −1 n ⎧ k/2 ~ k 2 Tm (2 t − 2n + 1), k −1 ≤ t < k −1 ⎪⎪ 2 2 ψ nm (t ) = ⎨ ⎪ ⎪⎩ 0, otherwise

(4)

Where ⎧ 1 ,m = 0 π ⎪ ~ ⎪ Tm (t ) = ⎨ ⎪ 2 T (t ), m > 0 ⎩⎪ π m

(5) and m = 0, 1, …, M – 1, n = 1, 2, …, 2k-1. In Eq. (5), Tm(t) are Chebyshev polynomials of the first kind with order m. Based on the definition, Chebyshev polynomials are orthogonal with respect to the weight function ω(t) = (1 – t2)1/2 within the interval [-1, 1]. The recursive equations are:

Dr. Yucheng Liu is with the University of Louisiana, Lafayette, LA 70504 USA (phone: 337-482-5822; fax: 337-482-1129; e-mail: yxl5763@ Louisiana.edu).

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∫

A function f(t) defined through [0, 1] may be expanded as: ∞

∞

(8)

n =1 m = 0

Second kind:

f (t )ψ nm (t ) dt ϖ n (t ) , in which (.,.)

1

∫

0

ψ (t )

0

III. FUNCTION APPROXIMATION USING CHEBYSHEV WAVELETS f (t ) = ∑ ∑ c nmψ nm (t )

x

dt = Pψ ( x) x−t (14) where P is the 2k-1M × 2k-1M operational matrix and can be determined from Eq. (4), (5), and (13) based on given k and M. The detailed approach of obtaining the Chebyshev wavelet operational matrix P was thoroughly demonstrated by Babolian and Fattahzadeh [9]. In this paper, P will be listed for each numerical example. With the calculated P, the unknown coefficient vector YT can be calculated from Eq. (11), (12), and (14): Y T Pψ ( x) = C Tψ ( x) ⇒ Y T = C T ⋅ P −1 (15) First kind:

T0(t) = 1, T1(t) = t, Tm+1(t) = 2tTm(t) – Tm-1(t), m = 1, 2, .. (6) It needs to be noticed that in Chebyshev wavelets, in order to obtain the orthogonal wavelets, the weight functions have to be dilated and translated as: ωn(t) = ω(2kt – 2n + 1) = (1 – (2kt – 2n + 1)2)-1/2 (7)

Y Tψ ( x ) = Y T P ψ ( x ) + C T ψ ( x ) ⇒ Y T = Y T P + C T ⇒ Y T = C T ⋅ ( I − P ) − 1

(16) The unknown function y(x) is finally solved as y(x) = YTΨ(t) (Eq. (10).

where cnm = (f(t), ψnm(t)) = denotes the inner product. If the infinite series in Eq. (8) is truncated, then we have 2 k −1 M −1

V. NUMERICAL EXAMPLES

f (t ) ≈ ∑ ∑ c nmψ nm (t )

The presented method of using Chebyshev wavelets to solve Abel integral equation is validated through following examples, which has been solved by Yousefi [6] by using Legendre wavelets.

n =1 m = 0 = CTΨ(t) (9) Where C and Ψ(t) are 2k-1M × 1 vectors given by: C = [c10, c11, …, c1(M – 1), c20, c21, …, c2(M – 1), …, c2k-1 0, c2k-1 1, …, c2k-1 (M – 1)]T Ψ(t) = [ψ10(t), ψ11(t), …, ψ1(M – 1)(t), ψ20(t), ψ21(t), …, ψ2(M – 1)(t), …, ψ2k-1 0(t), ψ2k-1 1(t), …, ψ2k-1 (M – 1)(t)]T

Example 1 Solve the Abel integral equation: 5

y ( x) = x 2 +

IV. SOLUTION OF ABEL INTEGRAL EQUATION In this section, Abel integral equations (1) and (2) are solved by using Chebyshev wavelets. At first, the functions f(x) and y(x) are approximated with Chebyshev wavelets as: y(x) = YTΨ(t) and f(x) = CTΨ(t) (10) where the approximation of f(x) is demonstrated in above section with given CT and Ψ(t), and YT is an unknown 2k-1M × 1 vector. Substituting Eq. (10) into (1) and (2), the integral equations are transformed as:

∫

x

Y Tψ (t )

0

First kind:

x−t

dt = C ψ ( x)

Second kind:

which has the exact solution y(x) = x2. Solution Expanding f(x) = x2 + (16/15)x(5/2) using the Chebyshev wavelets with M = 2 and k = 1 and we have: From Eq. (8), f ( x) = c10ψ 10 ( x) + c11ψ 11 ( x) + c12ψ 12 ( x) From Eq. (4) and (5) we obtain: 2(2 x − 1) 2(8 x 2 − 8 x + 1) ψ 11 ( x) = ψ 12 ( x) = π , π , and π CT is calculated as [0.92, 0.9, 0.26]. In calculating P, from Eq. (13) we obtain

ψ 10 ( x) =

T

Y ψ ( x) = ∫ T

x

0

(11)

Y Tψ (t ) x−t

dt + C ψ ( x) T

∫

x

Y Tψ (t )

(12)

∫

integral

∫

x

0

tn x−t

∫

x

0

dt

dt

∫

x−t

dt =

πx

Based on Eq. (13), the integral

x

0

ψ (t )

x −t

dt

dt =

2 x1/ 2 0.5!

,

x

ψ 11 (t )

0

x−t

∫

⎛ 2 x 3 / 2 x1 / 2 ⎞ ⎟ dt = 2⎜⎜ − 0.5! ⎟⎠ ⎝ 1.5! ,

and

ψ 12 (t )

⎛ 16 x 5 / 2 8 x 3 / 2 x 1 / 2 ⎞ ⎟ − + dt = 2⎜⎜ 1.5! 0.5! ⎟⎠ x−t ⎝ 2.5!

Substituting P and CT into Eq. (16) and the y(x) can be evaluated as: y( x) = 0.4684ψ 10 ( x) + 0.4419ψ 11 ( x) + 0.1099ψ 12 ( x) = x 2 (13)

Example 2 Solve the Abel integral equation:

where Γ(n) = (n – 1)! for a positive integer n.

∫

x−t x

0

1 ( +n) 2

Γ(n + 1) 3 Γ(n + ) 2

ψ 10 (t )

2

The operational matrix P is then calculated from Eq. (14): ⎡ 1.27 0.60 − 0.12⎤ P = ⎢⎢− 0.2 0.51 0.36 ⎥⎥ ⎣⎢ − 0.5 − 0.44 0.43 ⎦⎥

. In [6], the basis integral is solved as:

tn

x

0

x − t . In that Now we need to calculate the integral integral, YT is an unknown coefficients and the Chebyshev wavelets Ψ(t) are a combination of fundamental polynomials aiti (see Eq. (4) and (5)). The calculation of the integral therefore depends on how to calculate basis 0

x y (t ) 16 2 x −∫ dt 0 15 x−t

x y (t ) 2 x (105 − 56 x 2 + 48 x 3 ) = ∫ dt 0 105 x−t

can be obtained as:

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which has the exact solution y(x) = x3 – x2 + 1. Solution

n

∑ a C (s) = f (s) + ∫ i

2 x (105 − 56 x 2 + 48 x 3 ) = 105 using

have

∫

x

0

π

Rn ( s ) =

we

the matrix P is

(20)

∫

b

a

n

K ( s, t )

∑ a C (t )dt − f (s) i

i

i =0

j (b − a ) , j = 0, … , n n

∫

b

a

bn = [ f ( s j )], j = 0, … , n X T = [ a i ]in=0

(23) Theorem1. Let χ be a Banach space, let К be a bounded operator from χ into χ, and assume λ – К: χ → χ is one to one and onto, then we have || κ − Tnκ ||→ 0 as n → ∞ (24) Furthermore, for all sufficiently large n, say n ≥ N, the operator (λ – TnК)-1 exists as a bounded operator from χ to χ and is uniformly bounded

Given Fredholm integral equation of the second kind: b ϕ ( s ) = f ( s ) + K ( s, t )ϕ (t )dt , – ∞ < a ≤ s ≤ b < ∞ (17)

∫

a

in solving the integral equation with given kernel K(s, t) and the function f(s), the problem is typically to find the function φ(t). Maleknejad et al. [11] estimated φ(t) using Legendre polynomials and solve the integral equation. In this study, Chebyshev polynomials are employed to approximate the φ(t) and the effects of the Chebyshev polynomials on the accuracy of the estimation of φ(t) are compared through several numerical problems.

sup || (λ − Tn κ ) −1 ||< ∞

(25) For the solution of (λ – TnК)xn = Tny, xn ∈ χ and (λ – К)x = y we can have (Proof [14]) n≥ N

x − x n = λ (λ − Tnκ ) −1 ( x − Tn ( x)) |λ| || x − Tn ( x) ||≤|| x − x n ||≤| λ ||| (λ − Tnκ ) −1 |||| x − Tn ( x) || || λ − Tnκ ||

(26) From Proposition 1, it is concluded that approximation rate of Chebyshev polynomials is m-k, and Theorem 1 indicates that ||x – xn|| converge to zero at exactly the same speed as ||x – Tn(x)||.

VII. DISCRETIZATION OF INTEGRAL EQUATION In this section, eq. (17) is discretized and converted to system of linear equations. Chebyshev polynomials are chosen as basis functions to estimate the solution of the integral equation, φ(t), together with collocation method. The Chebyshev polynomials with the interval of orthogonality [-1, 1] are defined as [12]: Cn+1(x) = 2xCn(x) – Cn-1(x), and C0(x) = 1; C1(x) = x (18)

∑

i

An = [C i ( s j ) − K ( s j , t )C i (t )dt ] nj =1 , i = 0, … , n

VI. INTRODUCTION OF FREDHOLM INTEGRAL EQUATION

Tn ( x(t )) =

i

i =0

(22) Thus, this integral equation (eq. (20)) can be converted to a system of linear equations AnX = bn where

+ 0.0275ψ 13 ( x) = x3 − x 2 + 1

Let

∑

ai C i (s) −

sj =a +

calculated as: Substituting P and CT into Eq. (15) and the y(x) can be solved as: y ( x) = 1.1801ψ 10 ( x) − 0.0351ψ 11 ( x) + 0.0548ψ 12 ( x)

1.

∑ a C (t )dt

(21) The unknown coefficients ai are defined by selecting several collocation points sj so that Rn(sj) = 0 for j from 0 to n. In this study the collocation points are evenly selected from the space [a, b] that

.

0 .6 − 0.12 0.051 ⎤ ⎡ 1.27 ⎢ − 0.2 0.51 0.36 − 0.094⎥⎥ P=⎢ ⎢ − 0.5 − 0.44 0.43 0.28 ⎥ ⎢ ⎥ 0.37 ⎦ ⎣0.061 − 0.22 − 0.34

Proposition

n

i =0

ψ 13 (t )

⎛ 192 x 7 / 2 96 x 5 / 2 18 x 3 / 2 x 1 / 2 ⎞ ⎟ dt = 2⎜⎜ − + − 2.5! 1.5! 0.5! ⎟⎠ x−t ⎝ 3.5! ,

n

K ( s, t )

Hence the residual equation is defined as

2(4(2 x − 1) 3 − 3( 2 x − 1))

ψ12 were solved in example 1 and CT is calculated as [1.48, 0.66, -0.14, 0.089]. Similarly from Eq. (13)

b

a

i =0

Expanding f(x) the Chebyshev wavelets with M = 3 and k = 1 and we have f ( x) = c10ψ 10 ( x) + c11ψ 11 ( x) + c12ψ 12 ( x) + c13ψ 13 ( x) . ψ10, ψ11, and ψ 13 ( x) =

i

x(t ) ∈ H ( −1,1) (Sobolev k

VIII. NUMERICAL EXAMPLES In this section, several numerical examples of the Fredholm integral equation (eq. (17)) are considered to show the accuracy of presented method. In this study, all examples are solved using the method stated in section 2, and the integral equations are converted to systems of linear equations following eq. (22) and (23). All calculations are performed using Maple 11 and MatLab; the detailed steps are: 1. Construct n × n square matrix Am (from eq. (23)). 2. Build up n × 1 vector bm (from eq. (23)). 3. Calculate X using X = Am-1bm and determine all ai. 4. Estimate φ(t) based on the ai using eq. (19). 5. Compare the approximated φ(t) to the exact one and show in figures. Numerical results are compared with the exact solutions and plotted in following figures to illustrate the efficiency of the proposed method. (In all figures the exact solutions are

space),

n

a C (t ) i =0 i i

be the best approximation polynomial of x(t) in L2-norm: || x(t ) − Tn ( x(t )) || L [ −1,1] ≤ C 0 m − k || x(t ) || k , H ( −1,1) 2 where C0 is a positive constant, which depends on the selected norm and is independent of x(t) and m. (Proof [13]) At first, we estimate the unknown function φ(t) with the Chebyshev polynomials as n

ϕ (t ) ≈ Tn (ϕ (t )) = ∑ ai C i (t ) i =0

(19)

Substituting (19) into (17) and get

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represented by solid lines while the numerical solutions are plotted with dashed lines) Example 1. Solve eq. (17) with a = -1, b = 1 and

K ( s, t ) = e

5 (2s− t ) 3

1 (2s+ )

f (s) = e 3 where the exact solution is φ(t) = e2t and results are shown in figure 1 and 2 with different n.

Fig. 3. Result of example 2 for n = 3



Example 3. Solve eq. (17) with a = -1, b = 1 and

K ( s, t ) = 0.5s 2 t 2 f ( s ) = 0.9 s 2 where the exact solution is φ(t) = t2 and results are shown in figure 5 and 6 for different n. Fig. 2. Result of example 1 for n = 3

Example 2. Solve eq. (17) with a = -1, b = 1 and K ( s, t ) =

(s − t ) 3 s 2 (1 + t 2 )

f (s) = 1 + s 2 −

3( 2 − arcsin h(1)) − 2 s arcsin h(1) s

where the exact solution is φ(t) = (1+t2)1/2 and results are shown in figure 3 and 4 with different n.


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[7] [8] [9]

[10]

[11] [12] [13] [14]


M. Razzaghi, S.A. Yousefi, “Legendre wavelets direct method for variational problems”, Mathematics and Computers in Simulation, 53 (2000) 185 – 192. I. Daubechies, “Ten lectures on wavelets”, SIAM, Philadelphia, PA, 1992 E. Babolian, F. Fattahzadeh, “Numerical solution of differential equations by using Chebyshev wavelet operational matrix of integration”, Applied Mathematics and Computation, 188(1) (2007) 417 – 426. E. Babolian, F. Fattahzadeh, “Numerical computation method in solving integral equations by using Chebyshev wavelet operational matrix of integration”, Applied Mathematics and Computation, 188(1) (2007) 1016 – 1022. K. Maleknejad, K. Nouri, and M. Yousefi, “Discussion on convergence of Legendre polynomial for numerical solution of integral equations”, Applied Mathematics and Computation, 193 (2007) 335 – 339. T.S. Chihara, “An introduction to orthogonal polynomials”, Gordon and Breach, New Yors, 1978. C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zhang, “Spectral methods in fluid dynamics”, Springer-Verlag, New York, 1988. K.E. Atkinson, “A survey of numerical methods for the solution of Fredholm integral equations of the second kind”, Society for Industrial and Applied Mathematics, 1976.

IX. CONCLUSION The first part of this paper presents a method of using Chebyshev wavelets for solving Abel integral equation. In this method, the functions in the integral equations are approximated with Chebyshev wavelets and the original integral equations are transformed to a system of linear algebraic equations. Two numerical examples are discussed to validate the presented method. From the examples, it is concluded that Chebyshev wavelets can be used for solving Abel integral equations efficiently (with lower order M less than 3) and accurately. In the second part of this paper, the Fredholm integral equation of the second kind is solved by employing Chebyshev polynomials and collocation method. Convergence of the presented method and its convergence rate are proved in Proposition 1 and Theorem 2. As shown in the figures, the proposed method provides good efficiency that in order to acquire enough accuracy, we only need to convert the integral equation to the system of linear equations by order less than five. Besides the Legendre method presented by Maleknejad et al. [11], this paper proves that the Chebyshev polynomials can also be used to solve the Fredholm integral equation of the second kind with high accuracy and efficiency.

Yucheng Liu PhD, PE is an Assistant Professor of Mechanical Engineering at the University of Louisiana at Lafayette. Dr. Liu’s research interests focus on numerical analysis and methods, computer modeling and simulation, applied mathematics, structural mechanics, and crashworthiness analysis. To date, he has authored and co-authored over 80 publications in his areas and has been the PI or Co-PI of numerous research projects with total funds over 2 million dollars. Dr. Liu serves as the editorial board member of an international journal and has reviewed proposals and papers for US Department of Energy, 19 peer reviewed journals, and 5 international conferences. Dr. Liu is a registered Professional Engineer and holds active membership in ASME, ASEE, and SAE.

REFERENCES [1] [2] [3] [4] [5]

[6]

R. Gorenflo, S. Vessella, “Abel integral equations: analysis and applications”, Springer-Verlag, Berlin-New York, 1991. R. Piessens, P. Verbaeten, “Numerical solution of the Abel integral equation”, BIT Numerical Mathematics, 13(4) (1973) 451 – 457. M.R. Capobianco, “A method for the numerical resolution of Abel-type equations of the first kind”, Journal of Computational and Applied Mathematics, 23(3) (1988) 281 – 304. N. Mandal, A. Chakrabarti, B.N. Mandal, “Solution of a system of generalized Abel integral equations using fractional calculus”, Applied Mathematics Letters, 9(5) (1996) 1 – 4. Y.-P. Liu, L. Tao, “High accuracy combination algorithm and a posteriori error estimation for solving the first kind Abel integral equations”, Applied Mathematics and Computation, 178(2) (2006) 441 – 451. S.A. Yousefi, “Numerical solution of Abel’s integral equation by using Legendre wavelets”, Applied Mathematics and Computation, 175(1) (2006) 574 – 580.

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