A Jacobi Spectral Collocation Scheme for Solving Abel s Integral Equations

Transcription

1 Progr Fract Differ Appl, o 3, 87-2 (25) 87 Progress in Fractional Differentiation and Applications An International Journal A Jacobi Spectral Collocation Scheme for Solving Abel s Integral Equations Mohamed A Abdelawy,, Samer S Ezz-Eldien 2 and Ahmad Z M Amin Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt 2 Department of Mathematics, Faculty of Science, Assiut University, ew Valley Branch, El-Kharga, Egypt Received: 4 Jan 25, Revised: 27 Feb 25, Accepted: 28 Feb 25 Published online: Jul 25 Abstract: In this article, we construct a numerical technique for solving the first and second inds of Abel s integral equations Using the spectral collocation method, the properties of fractional calculus and the Gauss-quadrature formula, we can reduce such problems into those of a system of algebraic equations which greatly simplifies the problem The proposed numerical technique is based on the shifted Jacobi polynomials and the fractional integral is described in the sense of Riemann-Liouville In addition, our numerical technique is applied also to solve the system of generalized Abel s integral equations For testing the accuracy and validity of the proposed numerical techniques, we apply them to solve several numerical eamples Keywords: Abel s integral equation; Spectral collocation method; Jacobi polynomials; Gauss quadrature; Riemann-Liouville integral Introduction The Abel s integral equation, a special case of the Volterra integral equations of the first ind, is considered as one of the most important integral equations as it can be derived directly from a mechanical or physical problem, that gives it the ability to describe many engineering and physical phenomena accurately, such as simultaneous dual relations [], stellar winds [2], water wave [3], spectroscopic data [4] and others [5,6,7], more about the properties of this equation and its different inds can be found in [8, 9] Finding numerical techniques for solving Abel s integral equations has become an active research undertaing Liu and Tao [, ] introduced a mechanical quadrature technique for approimating the solution of the first ind Abel s integral equation Also, in [2], the authors used the Bernstein polynomials for constructing a numerical solution of the Abel s integral equation, while in [3], the authors introduced the Miusinsis operator of fractional order for solving it numerically Recently, Jahanshahi et al [4] used the properties of the fractional calculus definitions for solving the Abel s integral equation of first ind Another numerical techniqus have been applied for solving the first ind Abel s integral equation, see [5, 6, 7] On the other hand, Yousefi [8] applied the Legendre wavelets method for solving the second ind Abel s integral equation, while Khan and Gondal [9] applied the two-step Laplace decomposition method for approimating its numerical solution Recently, Kumar et al [2] applied the homotopy perturbation transform method for finding a numerical solution of the second ind Abel s integral equation Meanwhile, many researchers have interested in studying the system of generalized Abel s integral equations, see [2] Kumar et al [22] applied the homotopy perturbation method for solving the system of Abel s integral equations, while in [23], the authors constructed a numerical method based on the Laguerre polynomials for approimating its solution Fractional calculus, the theory of derivatives and integrals with any non-integer arbitrary order, has become the focus of many researchers in recent years due to its high accuracy in modeling many engineering and physical phenomena, such as economics [24], anomalous transport [25], Bioengineering [26] and others [27, 28, 29] Therefore, studying the properties of the fractional differential equations and finding effective analytical and numerical techniques for them has become very important topic to be studied, such as the waveform relaation method [3], the alternating-direction finite difference method [3], the Haar wavelet method [32], the differential transform method [33] and others [34,35] Corresponding author melawy@yahoocom c 25 SP atural Sciences Publishing Cor

2 88 M A Abdelawy et al : A Jacobi Spectral Collocation Scheme for Solving One of the best methods for approimating the solution of different inds of differential equations, is the spectral collocation method [36, 37, 38] Besides, the spectral collocation method has high accuracy; it also has eponential rates of convergence By using the spectral collocation method and in contrast to finite difference and finite element methods, we have numerical solution of better accuracy with far fewer nodes and thus less computational time The spectral collocation method is used for obtaining the approimate solution of some types of fractional differential equations, such as the fractional Langevin equations [39] and the generalized fractional Pantograph equations [4] An important role of the Jacobi polynomials has been played in the implementation of spectral methods Using the Jacobi polynomials in terms of the Jacobi parameters α and β, we have the advantage of obtaining the approimate solution in terms of these parameters For that reason, instead of finding the approimate solution for each pair of these parameters, we can use the Jacobi polynomials in obtaining our approimate solution Recently, the Jacobi polynomials have been used as basis functions of the spectral collocation method for approimating the solution of different types of differential equations such as the generalized Foer-Planc type equations [4], the nonlinear comple generalized Zaharov system [42] and the nonlinear Schrödinger equation [43] On the other hand, the shifted Jacobi polynomials are used as basis functions for solving types of fractional differential equations such as ordinary fractional differential equations [44], fractional advection-dispersion equations [45] and the multi-term time-space fractional partial differential equations [46] Our main aim in this article is to construct a numerical method for solving the first ind, the second type and the system of Abel s integral equations Using the spectral collocation method with some properties of the fractional calculus, we reduce such problems into those consisting of a system of algebraic equations which greatly simplifies the problem This article is organized as follows In the net section, we present some properties of the Jacobi polynomials and shifted Jacobi polynomials In Sections 3 and 4, the spectral collocation method is used based on the shifted Jacobi polynomials to solve the first and second inds of Abel s integral equations, respectively, while in Section 5, we apply our technique for approimating the solution of the system of Abel s integral equations In Section 6, several numerical eamples with their approimate solutions obtained using our technique are introduced to show the efficiency of our technique Concluding remars are given in Section 7 2 Properties of shifted Jacobi polynomials In this section, we reprise some basic properties of orthogonal shifted Jacobi polynomials that are most relevant to spectral approimations A basic property of the Jacobi polynomials is that they are the eigenfunctions to the singular Sturm- Liouville problem ( 2 )ψ ()+[ϑ θ+(θ+ ϑ+ 2)]ψ ()+(+θ+ ϑ+ )ψ() () The Jacobi polynomials are generated from the three-term recurrence relation: where θ, ϑ >, [,] and + ()(a(θ,ϑ) b (θ,ϑ) ) () c (θ,ϑ) (), () 2 (θ + ϑ+ 2)+ (θ ϑ), 2 a (θ,ϑ) b (θ,ϑ) The Jacobi polynomials satisfy the relations (2+θ+ ϑ+ )(2+θ+ ϑ+ 2), 2(+)(+θ+ ϑ+ ) (ϑ 2 θ 2 )(2+θ+ ϑ+ ) 2(+)(+θ+ ϑ+ )(2+θ+ ϑ), c (θ,ϑ) (+θ)(+ϑ)(2+θ+ ϑ+ 2) (+)(+θ+ ϑ+ )(2+θ+ ϑ) Moreover, the qth derivative of j (), can be obtained from (),, ( )( ) (), ( ) ( ) Γ(+ϑ+ ) (2)!Γ(ϑ + ) D q Γ( j+ θ+ ϑ+ q+) j () 2 q Γ(j+ θ+ ϑ+ ) P(θ+q,ϑ+q) j q () (3) c 25 SP atural Sciences Publishing Cor

3 Progr Fract Differ Appl, o 3, 87-2 (25) / wwwnaturalspublishingcom/journalsasp 89 Denoting by L, () ( 2 L ), L>, the shifted Jacobi polynomial of degree The eplicit analytic form of the shifted Jacobi polynomials P (α,β) L, () of degree is given by L, () j j ( ) j Γ(+ϑ+ )Γ( j+ +θ+ ϑ+ ) Γ(j+ ϑ+ )Γ(+θ+ ϑ+ )( j)! j!l j j Γ(+θ+ )Γ(+ j+ θ+ ϑ+ ) j!( j)!γ( j+ θ+ )Γ(+θ+ ϑ+ )L j( L) j, (4) and this in turn, enables one to get which will be of important use later In virtue of (2) and (3), we can deduce that et, let w (θ,ϑ) L product and norm D q L, L, ()( ) Γ(+ϑ+ ) Γ(ϑ+ )!, Γ(+θ+ ) L, (L) Γ(θ+ )!, () ( ) q Γ(+ϑ+ )(+θ+ ϑ+ ) q L q, (5) Γ( q+)γ(q+ϑ+ ) D q L, (L) Γ(+θ+ )(+θ+ ϑ+ ) q L q Γ( q+)γ(q+θ+ ), (6) D m Γ(m++θ+ ϑ+ ) L, () L m Γ(+θ+ ϑ+ ) P(θ+m,ϑ+m) L, m () (7) ()(L ) θ ϑ, then we define the weighted space L 2 [,L] in the usual way, with the following inner w (θ,ϑ) (u,v) w (θ,ϑ) L L u()v()w (θ,ϑ) L ()d, v (θ,ϑ) w L L (v,v) 2 w (θ,ϑ) L (8) The set of shifted Jacobi polynomials forms a complete L 2 [, L]-orthogonal system Moreover, and due to (8), we have w (θ,ϑ) L L, 2 w (θ,ϑ) L ( L 2 ) θ+ϑ+ h (θ,ϑ) h (θ,ϑ) L, (9) We denote by (θ,ϑ), j, j, the nodes of the standard Jacobi-Gauss interpolation on the interval [,], their corresponding Christoffel numbers are ϖ (θ,ϑ), j, j The nodes of the shifted Jacobi-Gauss interpolation on the interval[,l] are the zeros of L,+ (), which we denote by (θ,ϑ) L,, j, j Clearly (θ,ϑ) L,, j L 2 ((θ,ϑ), j + ), and their corresponding Christoffel numbers are ϖ (θ,ϑ) L,, j ( L 2 )θ+ϑ+ ϖ (θ,ϑ), j, j Let S [,L] be the set of polynomials of degree at most Thans to the property of the standard Jacobi-Gauss quadrature, it follows that for any φ S 2+ [,L], we have L (L ) θ ϑ φ()d ( ) L θ+ϑ+ ( ) L ( ) θ (+) ϑ φ 2 2 (+) d ( L θ+ϑ+ ϖ 2) (θ,ϑ), j φ j j ( ϖ (θ,ϑ) L,, j φ (θ,ϑ) L,, j ) ( L ) 2 ((θ,ϑ), j + ) () c 25 SP atural Sciences Publishing Cor

4 9 M A Abdelawy et al : A Jacobi Spectral Collocation Scheme for Solving Consequently, the qth-order derivative of shifted Jacobi polynomial can be written in terms of the shifted Jacobi polynomials themselves as D q q L, () C q (,i,θ,ϑ) (), () where C q (,i,θ,ϑ) (+λ) q(+λ+ q) i (i+θ+ q+) i q Γ(i+λ) L q ( i q)! Γ(2i+λ) 3 F 2 +i+q, +i+λ+ q, i+θ+ ;, i+θ+ q+, 2i+λ+ and for the general definition of a generalized hypergeometric series and special 3 F 2, see [47], p 4 and pp 3-4, respectively (2) 3 First ind Abel s integral equation In this section, we use the spectral collocation method to solve the first ind Abel s integral equation in the following form [, ] dt g(), L, < µ <, (3) ( t) µ where g() is a given function, and is the unnown function First, we state the Riemann-Liouville integral definition as the following form: Definition 3 The integral of order µ (fractional) according to Riemann-Liouville is given by J µ f() ( t) µ f(t)dt, Γ(µ) J f() f(), where Γ(µ) µ e d is the gamma function The operator J µ satisfies the following properties µ >, >, (4) J µ J ν f()j ν J µ f()j µ+ν f(), J µ λ Γ(λ + ) Γ(λ + µ+ ) λ+µ (5) Using definition (4), the Abel s integral equation (3) is transformed to the fractional integral equation in the form: Γ( µ)j µ u()g(), L, < µ < (6) In order to use the spectral collocation method based on the shifted Jacobi polynomials, we approimate u() by the shifted Jacobi polynomials as as u () a i () (7) In virtue of (4) and (5), we can epress the fractional integration of order µ of any shifted Jacobi polynomial () J µ( () ) P (θ,ϑ,µ) () i ( ) i Γ(i+ϑ+ )Γ(i++θ+ ϑ+ ) Γ(i +)Γ(+ µ+ )Γ(+ϑ+ )Γ(i+θ+ ϑ+ )L +µ, (8) c 25 SP atural Sciences Publishing Cor

5 Progr Fract Differ Appl, o 3, 87-2 (25) / wwwnaturalspublishingcom/journalsasp 9 therefore, easily we can write J µ u () Therefore, adopting (6) with (9), we can write (3) in the form: Γ( µ) a i P (θ,ϑ, µ) () (9) a i P (θ,ϑ, µ) ()g(), L, < µ < (2) Here, we apply the spectral collocation method by setting the residual of the previous equation to be zero at the + collocation points as following: where Γ( µ) a i P (θ,ϑ, µ) L,, )g((θ,ϑ) L,, ), < µ <,,, (2) The previous equation alternatively may be written in the matri form: P ( µ) Γ( µ)p ( µ) AG, (22) P (θ,ϑ, µ) L, L,, ) P(θ,ϑ, µ) P (θ,ϑ, µ) L, L,, ) P(θ,ϑ, µ) L, L,, L, L,, P (θ,ϑ, µ) L, L,,i ) P(θ,ϑ, µ) L, L,,i P (θ,ϑ, µ) L, L,, ) P(θ,ϑ, µ) L, L,, P (θ,ϑ, µ) L, L,, ) P(θ,ϑ, µ) A a a a i a a L, L,,, G ) P(θ,ϑ, µ) L, ) P(θ,ϑ, µ) L, L,, ) L,, ) ) P(θ,ϑ, µ) L, ) P(θ,ϑ, µ) L, ) P(θ,ϑ, µ) g L,, ) g L,, ) g L,,i ) g L,, ) g L,, ) L, L,,i ) L,, ) L,, ) The previous system of algebraic equations can be solved using the ewton s iterative method, 4 Second ind Abel s integral equation In this section, we apply our technique to solve the second ind Abel s integral equation in the form: u()+ dt g(), L, < µ <, (23) ( t) µ where g() is a given function, while is an unnown function As in the previous section, the second ind Abel s integral equation (23) may be transformed into the fractional integral equation: u()+ Γ( µ)j µ u()g(), L, < µ < (24) c 25 SP atural Sciences Publishing Cor

6 92 M A Abdelawy et al : A Jacobi Spectral Collocation Scheme for Solving After epressing u() by shifted Jacobi polynomials as in (7) and approimating the fractional integral of order µ of u() as in (9), we can write (24) as in the form: a i ()+ Γ( µ) a i P (θ,ϑ, µ) ()g(), L, < µ <, (25) where P (θ,ϑ, µ) () is defined as in (8) ow, we collocate the previous equation at the + collocation points as following a i ( L,, )+ Γ( µ)p(θ,ϑ, µ) L,, ) ) g L,, ), < µ <,,, (26) The previous system may be rewritten in a matri form as ( P+Γ( µ)p ( µ)) AG, (27) where P ( µ), A and G are given as in (22) and P L, L,, ) P(θ,ϑ) L, L,, L, L,, ) P(θ,ϑ) L, L,, L, L,,i ) P(θ,ϑ) L, L,,i L, L,, ) P(θ,ϑ) L, L,, L, L,, ) P(θ,ϑ) L, L,, L, ((θ,ϑ) L,, ) L, ((θ,ϑ) L,, ) L, ((θ,ϑ) L,,i ) P(θ,ϑ) L, ((θ,ϑ) P(θ,ϑ) L, L,, ) ((θ,ϑ) L,, ) The previous system of algebraic equations can be solved using ewton s iterative method 5 System of generalized Abel s integral equations In the current section, we apply our technique to solve the second ind Abel s integral equation in the form a () a 2 () ( t) µ dt+ a 2() (t ) µ dt+ a 22() v(t) (t ) µ dt g (), v(t) ( t) µ dt g 2(), where a (), a 2 (), a 2 (), a 22 (), g () and g 2 () are given functions, while and v(t) are unnown functions Definition 5 The left- and right-sided Riemann-Liouville fractional integrals of order µ of any function f() for [,L] are defined, respectively, as J µ + f() Γ(µ) J µ f() Γ(µ) J ± f() f() L ( t) µ f(t)dt, µ >, (t ) µ f(t)dt, µ >, The left- and right-sided Riemann-Liouville fractional integrals operators satisfy the following properties J µ + β Γ(β + ) Γ(β + + µ) β+µ, J µ ( L) β ( )β Γ(β + ) Γ(β + + µ) (L )β+µ (28) (29) (3) c 25 SP atural Sciences Publishing Cor

7 Progr Fract Differ Appl, o 3, 87-2 (25) / wwwnaturalspublishingcom/journalsasp 93 The system of generalized Abel s integral equations (28) can be transformed into the following system of fractional integral equations a ()Γ( µ)j µ + u()+a 2 ()Γ( µ)j µ v()g (), a 2 ()Γ( µ)j µ u()+a 22()Γ( µ)j µ + v()g 2() First, we epress u() by shifted Jacobi polynomials as in (7) and epress v() as v () (3) b i () (32) The left- and right-sided Riemann-Liouville fractional integrals of shifted Jacobi polynomials may be given by J µ ( (θ,ϑ) + P () ) P (θ,ϑ,µ) () J µ ( P (θ,ϑ) then, easily we can write i () ) P (θ,ϑ,µ) i ( ) i Γ(i+ϑ+ )Γ(i++θ+ ϑ+ ) Γ(i +)Γ(+ µ+ )Γ(+ϑ+ )Γ(i+θ+ ϑ+ )L +µ, () ( ) Γ(i+θ+ )Γ(i++θ+ ϑ+ ) Γ(i +)Γ(+ µ+ )Γ(+θ+ )Γ(i+θ+ ϑ+ )L ( L)+µ, J µ + u () J µ u () J µ + v () J µ v () a i P (θ,ϑ, µ) (), a i P (θ,ϑ, µ) (), b i P (θ,ϑ, µ) (), b i P (θ,ϑ, µ) () Therefore, adopting (32)-(34), the system (3) may be written in the form: a ()Γ( µ) a 2 ()Γ( µ) a i P (θ,ϑ, µ) a i P (θ,ϑ, µ) ()+ Γ( µ)a 2 () ()+ Γ( µ)a 22 () L, < µ < b i P (θ,ϑ, µ) ()g (), b i P (θ,ϑ, µ) ()g 2 (), As in the previous section, we collocate the system (35) at + collocation points as a L,, )Γ( µ) a i P (θ,ϑ, µ) L,, )+ Γ( µ)a 2 L,, ) (33) (34) (35) b i P (θ,ϑ, µ) L,, )g L,, ), a 2 L,, )Γ( µ) a i P (θ,ϑ, µ) L,, )+ Γ( µ)a 22 L,, ) (36) < µ <,,, b i P (θ,ϑ, µ) L,, )g 2 L,, ), The previous system of algebraic equations can be solved using ewton s iterative method c 25 SP atural Sciences Publishing Cor

8 94 M A Abdelawy et al : A Jacobi Spectral Collocation Scheme for Solving u u Fig : The eact and numerical solutions curves for Eample 6 with θ, ϑ 2 and 6 umerical results For clarifying the validity and accuracy of the presented algorithm, we have applied it to solve numerical eamples of the first ind, the second ind and the system of generalized Abel s integral equations 6 Eample Firstly, we introduce the first ind Abel s integral equation in the following form [5] and the eact solution is given by u() 9 Using the technique discussed in Section 3 with different choice of θ, ϑ dt 5 3,, (37) ( t) 3 ( eg θ ϑ (shifted Legendre-Gauss collocation method), θ ϑ 2 (first and second ind shifted Chebyshev-Gauss collocation), θ, ϑ 2 ) and θ, ϑ 2 at, we obtained the eact solution u 9 In the case of θ, ϑ 2 and, the numerical and eact solutions curves for problem (37) by using our method are shown in Fig c 25 SP atural Sciences Publishing Cor

9 Progr Fract Differ Appl, o 3, 87-2 (25) / wwwnaturalspublishingcom/journalsasp 95 u 3 u Fig 2: The eact and numerical solutions curves for Eample 62 with θ, ϑ 2 and 3 62 Eample 2 Here, we consider the problem ( t) 2 dt 4 5 3/2( ),, (38) The eact solution of this problem is given by u() 3 Also, the approimate solution for this problem gives the eact solution for different choice of θ, ϑ at 3 The approimate solution obtained using our method at θ, ϑ 2 and 3 for (38) shown in Figure (2) to mae it easier to compare with the eact solution 63 Eample 3 Consider the following Abel s integral equation studied in [, ] 2 t 2 + t 4 + ( t) 4 which having an eact solution given by u() 2 By applying the technique described in Section 3, we obtain that ( 2 t 2 + t 4 + )( a i ( t) 4 dt ,, (39) (t)) dt ,, (4) c 25 SP atural Sciences Publishing Cor

10 96 M A Abdelawy et al : A Jacobi Spectral Collocation Scheme for Solving yields the following system of algebraic equations L, (( L, )2 t 2 + t 4 + )( ( L, t) 4 a i (t)) dt (L, ) (L, ) (L, ) 4,,,, Despite that the best results for the maimum absolute errors achieved by using the numerical methods (mid-point rectangular, trapezoidal quadrature, Asymptotic epansion, combination algorithm, a posteriori estimate and mechanical quadrature methods) given in [,] with 4 or 8 steps are bounded between and , we obtained the eact solution using 2 and different choice of θ, ϑ (4) 64 Eample 4 Let us consider the following second ind Abel s integral [2] u() dt [,], (42) ( t) 2 Applying the numerical technique discussed in Section 4 we obtain that u (),, which is the eact solution of the mentioned problem Although we got the eact solution using a small number of nodes ( and different choice of θ, ϑ) where the best value of the maimum absolute errors obtained in [2] was 5 6 at Eample 5 Here, we tested the following second ind Abel s integral equation [2] u() dt,, (43) ( t) 2 Kumar et al [2] introduced this problem and applied the homotopy perturbation transform method, the maimum absolute error achieved in [2] was 5 7 at 25 ( Taing 2, we used the technique presented in Section 4 with different choices of θ, ϑ eg θ ϑ (shifted Legendre-Gauss collocation ) method), θ ϑ 2 (first and second ind shifted Chebyshev-Gauss collocation), θ, ϑ 2 and θ, ϑ 2, the numerical approimation u () is equal to the eact solution u() 2 66 Eample 6 As a system of Abel s integral equations, we consider this problem [23] 2 dt+ ( t) 2 4 (t ) 2 dt+ 3 v(t) (t ) 2 v(t) ( t) 2 dt ( ) ( ) ( ) ( ) ( ) ( ) ( ) 5 2, dt 5 ( ) ( ) ( ) , where the eact solution u() 2 and v() Setia and Pandey [23] introduced this problem and used the Laguerre polynomials for approimating its numerical solution In order to show that our technique discussed in Section 5 is accurate than this introduced in [23], in Fig 3, we plot the eact and numerical solutions of u() and v(t) at 3 with different choice of θ and ϑ the numerical approimation u () is equal to the eact solution u() 2 and v() c 25 SP atural Sciences Publishing Cor

11 Progr Fract Differ Appl, o 3, 87-2 (25) / wwwnaturalspublishingcom/journalsasp 97 2 u 5 u v v Fig 3: The eact and numerical solutions curves for Eample 66 with θ, ϑ 2 and 3 67 Eample 7 ow, we consider the following system of Abel integral equations [23] ( 2 + ) 2 2 ( t) 2 (t ) 2 dt+ + 4 dt+(2 ) v(t) (t ) 2 v(t) ( t) 2 dt , dt , with eact solution given by u() 2 and v() 3 Using the method presented in Section 5, we plot the graph of the numerical and eact solutions of u() and v() at 3 with different choice of θ, ϑ in Fig 4 7 Conclusion In this article, a new fast numerical technique is constructed to introduce an approimate solution of the first and second inds of Abel s integral equations Our numerical approach is consisting of transforming such problems into a fractional c 25 SP atural Sciences Publishing Cor

12 98 M A Abdelawy et al : A Jacobi Spectral Collocation Scheme for Solving u u v v Fig 4: The eact and numerical solutions curves for Eample 67 with θ, ϑ 2 and 3 integral equation (described in the Riemann-Liouville sense) Using the shifted Jacobi polynomials as basis functions of the spectral collocation method and the Gauss-quadrature formula, the fractional integral equation is reduced into a problem consisting of system of algebraic equations that can be solved using any standard iteration method The system of generalized Abel s integral equations is investigated also using the proposed technique The numerical results we have obtained demonstrate the high accuracy of our technique, only a small number of shifted Jacobi polynomials are needed to obtain a satisfactory result References [] J Walton, Systems of generalized Abel integral equations with applications to simultaneous dual relations, SIAM J Math Anal, (979) [2] O Knill, R Dgani and M Vogel, A new approach to Abel s integral operator and its application to stellar winds, Astronom, Astrophys 274, 2-8 (993) [3] S De, B Mandal and A Charabarti, Use of Abel s integral equations in water wave scattering by two surface-piercing barriers, Wave Motion 47, (2) [4] C J Cremers and RC Bireba, Application of the Abel integral equation to spectroscopic data, Appl Opt 5, (966) [5] R Gorenflo and S Vessella, Abel Integral Equations: Analysis and Applications, Lecture otes Math, 46, Springer, Berlin, 99 [6] L E Kosarev, Applications of integral equations of the first ind in eperiment physics, Comput Phys Commun 2, (98) [7] H Brunner, : One hundred years of Volterra integral equation of the first ind, Appl umer Math 24, (997) [8] A M Wazwaz, A First Course in Integral Equations, World Scientific Publishing, Singapore, 997 [9] A D Polyanin and AV Manzhirov, Handboo of Integral Equations, CRC Press, Boca Raton, 998 [] Y-p Liu and L Tao, High accuracy combination algorithm and a posteriori error estimation for solving the first ind Abel integral equations, Appl Math Comput 78, (26) c 25 SP atural Sciences Publishing Cor

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