NUMERICAL SOLUTION FOR CLASS OF ONE DIMENSIONAL PARABOLIC PARTIAL INTEGRO DIFFERENTIAL EQUATIONS VIA LEGENDRE SPECTRAL COLLOCATION METHOD

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1 Journal of Fractional Calculus and Applications, Vol. 5(3S) No., pp (6th. Symp. Frac. Calc. Appl. August, 01). ISSN: NUMERICAL SOLUTION FOR CLASS OF ONE DIMENSIONAL PARABOLIC PARTIAL INTEGRO DIFFERENTIAL EQUATIONS VIA LEGENDRE SPECTRAL COLLOCATION METHOD G. I. EL BAGHDADY, M. S. EL AZAB Abstract. Partial integro-differential equations occur in many fields of science and engineering. In this work, we apply the Legendre spectral collocation method to obtain approximate solutions for some types of parabolic partial integro differential equations (PPIDEs). In the first approach, we converted our equation into two coupled equivalent linear Volterra equations of the second kind. In the second approach, we approximate the integration by replacing it by using Gauss quadrature formulas to obtain a linear algebraic system that can be solved by direct or iterative methods. Finally, some numerical examples are included to illustrate the validity and applicability of the proposed technique. 1. Introduction Many problems in science and engineering arise in the system mixed of a partial differential equation and the integral terms involving the unknown function. We are concerned with an efficient numerical approximation scheme of the mathematical model of a physical phenomenon involving contaminant transport process with memory term. Mathematically, the simplest example of such a problem would be represented by the one dimensional parabolic partial integro differential equation of the form: t u(x, t) u(x, t) + p(x) u(x, t) = t 0 k(x, t s)u(x, s)ds + f(x, t), (1) associated with the initial condition and Dirichlet boundary conditions: u(x, 0) = u 0 (x), x Ω = [, 1], () u(0, t) = 0, u(1, t) = 0, t I = [0, T ], (3) where and denote the Laplace and gradient operators, respectively. Here (x, t) Q Ω I, where I [0, T ], T > 0 is a time interval and Ω R 010 Mathematics Subject Classification. Primary 35R09, 5K05, 97N50, 5D05. Key words and phrases. Spectral collocation method, Legendre polynomials, Legendre Differentiation matrix, Gauss quadrature formulas, parabolic partial integro differential equations. 1

2 G. I. EL BAGHDADY, M. S. EL AZAB JFCA-01/5(3S) [, 1] is a bounded domain with boundary Ω {, 1}. For implementation of high-order methods such as spectral methods, we assume that the functions p(x), f(x, t), k(t s) and u 0 (x) are smooth enough on Q. Problems of type (1) (3) represent the mathematical model of several physical phenomena involving process with memory term. For example, they describe compression of poro-viscoelastic media [1, ], heat conduction through materials with memory term [, 3], the nonlocal reactive flows in porous media [, 5], and non- Fickian flow of fluid in porous media [6]. Many authors have considered the numerical solution of parabolic partial integro differential equations by various methods. The method of lines used in [7] and the discontinuous Galerkin method in [8]. In [9], they use Galerkin mixed finite element methods and ADI orthogonal spline collocation [10, 11] and the references therein. Also in [1, 13] they introduce some solutions to a class of (PPIDEs). In [1, 15] the method of radial basis functions was used to nonlinear and linear equations respectively. Finally, F. Fakhar-Izadi and M. Dehghan proposed spectral method for parabolic Volterra integro-differential equations based on Legendre collocation scheme [16]. In recent years, the scientists have paid much attention to spectral method due to its high accuracy (see for instance, [17, 18, 19, 0, 1, 38] and the references therein). Spectral methods are powerful approaches for the numerical solution for ordinary or partial differential equations on a simple domain and if the data defining the problem are smooth [, 3]. Volterra integral and ordinary Volterra integro-differential equations have a wide interest by using many methods of spectral type. In [] H. Brunner used Collocation methods for second-order Volterra integro-differential equations. Multistep collocation method is also used for Volterra integral equations in [5]. Chebyshev spectral collocation method for the solution of Volterra integral and ordinary Volterra integro-differential equations are discussed in [6]. In [7] Tang introduces Legendre-spectral method with its error analysis for ordinary Volterra integro-differential equation of the second kind. Another spectral method using Legendre spectral Galerkin method was introduced for second-kind Volterra integral equations [8]. Also some authors [9, 31, 30] introduce Chebyshev spectral collocation method for Volterra integral equations. All of these papers that mentioned before were devoted to VIEs and ordinary VIDEs, but in this article we considered the solution to parabolic partial integro differential ones (PPIDEs). Our goal in this paper is to apply Legendre spectral collocation method for both space and time variables that are an extension of the method presented in [7, 3, 16]. The organization of this work is as follows. In Section, we present the Legendre spectral collocation method for discretizing the introduced problem. As a result a set of algebraic linear equations are formed and a solution of the considered problem is discussed. In Section 3, we present some numerical examples to illustrate the effectiveness of the proposed method. Finally, in Section we review some concluding remarks.. Legendre Spectral Collocation Method In this section, we apply the Legendre spectral-collocation method to problem (1) (3). Before making this, we will introduce some basic properties of the most commonly used set of orthogonal polynomials[33, 3]; Legendre polynomials.

3 JFCA-01/5(3S) LEGENDRE SPECTRAL COLLOCATION METHOD 3.1. Legendre polynomials. The Legendre polynomials L n (x), n = 0, 1,..., are the Eigenfunctions of the singular Sturm Liouville problem ( d (1 x ) dl ) n(x) + n(n + 1)L n (x) = 0, dx dx they are mutually orthogonal with respect to L ω inner product on the interval [, 1] with the weight function ω(x) = 1, this imply to 1 L n (x)l m (x)dx = n + 1 δ nm, where δ nm is the Kronecker delta. The Legendre polynomials satisfy the following three term recurrence relations L 0 (x) = 1, L 1 (x) = x, (n + 1)L n+1 (x) = (n + 1)xL n (x) nl n (x), n 1, () and 1 ( L n (x) = L (n + 1) n+1 (x) L n(x) ), n 1. Unfortunately there are no explicit forms for the Legendre points and their quadrature weights like Chebyshev points and corresponding weights. They are found in expressions related to Legendre polynomials, for example the Legendre-Gauss- Lobatto (LGL) points are zeros of (1 x )L N (x), and the corresponding quadrature weights are 1 ω n = N(N + 1) [L N (x n )], 0 n N... Legendre spectral collocation method. Now, consider problem (1) (3) on bounded domain Q. Because of the orthogonally property of the Legendre polynomials on the interval [, 1], we transfer equation (1) from [0, T ] to an equivalent problem defined in the interval [, 1], by using the substitution t = T (τ + 1), τ [, 1], then equation (1) becomes, for all x, τ [, 1] U U + p(x) U = T τ in which U := U(x, τ) = u T (τ+1) 0 k (x, T (τ + 1) s ) u(x, s)ds + g(x, τ), (5) (x, T ) (τ + 1), g(x, τ) = f (x, T ) (τ + 1). By using the following linear transformation in equation (5) s = T (ρ + 1), ρ [, 1], by converting the integration interval from [0, T (τ + 1)/] to the interval [, τ], so that equation (5) becomes U T τ U + p(x) U = T τ K(x, τ ρ)u(x, ρ)dρ + g(x, τ), (6)

4 G. I. EL BAGHDADY, M. S. EL AZAB JFCA-01/5(3S) where K(x, τ ρ) = k(x, T (τ ρ)/). Define an auxiliary function Φ(x, τ) = g(x, τ) + T τ K(x, τ ρ)u(x, ρ)dρ. (7) which we will use it in the next approximation. In order to approximate problem (1) (3) by spectral methods, we restart equation (6) as two equivalent Volterra integral equations of the second kind by using the aid of equation (7) as the following τ Φ(x, τ) = g(x, τ) + T K(x, τ ρ)u(x, ρ)dρ, Φ(x, τ) = (8) U T τ U + p(x) U. by integrating both sides in the second part of equation (8) over the interval [, τ] with respect to τ, we get the required two linear equations of (VIEs) as the following Φ(x, τ) = g(x, τ) + T U(x, τ) = u 0 (x) + T τ τ K(x, τ ρ)u(x, ρ)dρ, [ ] Φ(x, ξ) + U(x, ξ) p(x) U(x, ξ) dξ. Let the collocation points{(x i, τ j )} i,j be the set of (N + 1)(M + 1) points, in which {x i } N i=0 are the Legendre Gauss Lobatto points (LGL nodes), and {τ j} M j=0 are the Legendre Gauss points (LG nodes). Equation (9) holds at the collocation points such that, for all 0 j M yields Φ(x i, τ j ) = g(x i, τ j ) + T τj K(x i, τ j ρ)u(x i, ρ)dρ, 0 i N, U(x i, τ j ) = u 0 (x i ) + T τj [ ] (10) Φ(x i, ξ) + U(x i, ξ) p(x i ) U(x i, ξ) dξ, 1 i N 1. For approximating the integral terms in equation (10), first the integral interval will transfer from [, τ] to a fixed one [, 1] by using the following change of variables ξ(τ j, θ) = ρ(τ j, θ) = τ j + 1 θ + τ j 1, θ [, 1], where {θ} p are the roots of the (p+1) th Legendre polynomials. Then (10) becomes: Φ(x i, τ j ) = g(x i, τ j ) + T (τ j + 1) 1 ( ) ( ). K x i, τ j ρ(τ j, θ) U x i, τ j ρ(τ j, θ) dθ, U(x i, τ j ) = u 0 (x i ) + T (τ j + 1) 1 ( ) Φ x i, ξ(τ j, θ) dθ + T (τ j + 1) (11) 1 [ ( ) ( )]. U x i, ξ(τ j, θ) p(x i ) U x i, ξ(τ j, θ) dθ. (9)

5 JFCA-01/5(3S) LEGENDRE SPECTRAL COLLOCATION METHOD 5 Now, to approximate the integration in equation (11), we use (p+1) point Legendre- Gauss quadrature formula with the corresponding Legendre weights {ω k } p. Equation (11) becomes Φ(x i, τ j ) g(x i, τ j ) + T (τ j + 1) ( ) ( ). ω k K x i, τ j ρ(τ j, θ k ) U x i, τ j ρ(τ j, θ k ), U(x i, τ j ) u 0 (x i ) + T (τ j + 1). ω k [ U ( x i, ξ(τ j, θ k ) ( ) ω k Φ x i, ξ(τ j, θ k ) + T (τ j + 1) ) ( p(x i ) U x i, ξ(τ j, θ k ) where {θ k } p are roots of the (p + 1) th Legendre polynomial. Now, we use Lagrange interpolation polynomials to expand the functions Φ and U as Φ M N (x, σ) U M N (x, σ) N n=0 m=0 N n=0 m=0 l n (x)f m (σ)φ(x n, τ m ), l n (x)f m (σ)u(x n, τ m ), where l n (x) and F m (σ)are the n th and m th Lagrange basis functions corresponding to non uniform meshes of {x i } and {τ j } respectively. After enforcing the homogeneous boundary conditions at x 0 = and x N = 1 the first and the last terms in the interpolation polynomial of U are omitted. Therefore, we have U M N (x, σ) N )], (1) (13) l n (x)f m (σ)u(x n, τ m ). (1) Now we can approximate U and U in equation (1) by using the interpolation polynomial of U from the previous equation as the following (U M N ) = (U M N ) x (x, σ) N N (UN M ) = (UN M ) xx (x, σ) l n(x)f m (σ)u(x n, τ m ), l n(x)f m (σ)u(x n, τ m ), where l n(x) and l n(x) are the first and the second derivative of Lagrange interpolation basis polynomials l n (x) respectively, such that l n(x) P N and l n(x) P N. By replacing the interpolation polynomials of Φ, U, and approximation to U and U from equations (13), (1) and (15), respectively in equation (1) and write Φ(x i, τ j ) = Φ i,j, U(x i, τ j ) = U i,j, g(x i, τ j ) = g i,j, ρ(τ j, θ k ) = ρ i,k, ξ(τ j, θ k ) = ξ i,k. (15)

6 6 G. I. EL BAGHDADY, M. S. EL AZAB JFCA-01/5(3S) After replacing in equation (1), each equation in it can be reformulated respectively as Φ i,j g i,j + T (τ j + 1) { N l n (x i )U(x n, τ m ) }. ω k K(x i, τ j ρ j,k )F m (ρ j,k ), (16) U i,j u 0 (x i ) + T (τ j + 1) { N n=0 m=0 N p(x i ) N l n (x i )Φ(x n, τ m ) + l n(x i )U(x n, τ m ) } p l n(x i )U(x n, τ m ) ω k F m (ξ j,k ). (17) Now, we will define the differentiation matrices, which are an essential building block for collocation methods used in spectral schemes [38,, 3]. The differentiation matrices have a connection with the first and the second derivative of Lagrange interpolation basis polynomials. Denoting d i,n := l n(x i ), we introduce the second order differentiation matrix as the following D = (d i,k ) 0 i,k N which is the second order to differentiation matrix D. By writing the entries of D = Di,k for {x i} as ) 1 D i,k (D i,i, i k Di,k x = i x k N Di,l, i = k, l=0,i l where D i,k are the entries of the so called differentiation matrix of dimension (N + 1). The entries of the differentiation matrix can be defined for (LGL) points (cf. [38]) as the following N(N + 1), i = k = 0, L N (x i ) 1, i k, D i,k = L N (x k ) x i x k (18) 0, i, k [1, N 1], N(N + 1), i = k = N. Now rewrite equations (16) and (17) in the matrix form as { Φ(N+1)(M+1) = G (N+1)(M+1) + LU (N)(M+1), U (N)(M+1) = U + AΦ (N+1)(M+1) + (B + C)U (N)(M+1), where Φ, G, U and U represent vectors defined as in the following: Φ (N+1)(M+1) = vec [Φ i,j ], G (N+1)(M+1) = vec[g i,j ], 0 i N, U (N)(M+1) = vec[u i,j ], 1 i N, 0 j M, u 0 (x ) u 0 (x 3 ) u 0 (x N ) U = vec...., u 0 (x ) u 0 (x 3 ) u 0 (x N ) (19)

7 JFCA-01/5(3S) LEGENDRE SPECTRAL COLLOCATION METHOD 7 in which the (vec) operator reshapes any matrix into a vector by placing columns of the matrix below each other from the first to the last. For the other matrices each one can described as block ones as the following: A = (A i j ), A is a matrix with a dimension (N + 1)(N 1) (N + 1) in which the first and last (N + 1) columns are zeros, and each block matrix (A i j ) has a dimension of (N + 1) (N + 1). For each 0 m M, 0 j M, 1 i N 1, 1 n N 1, the other entries of (A i j ) can be given by the following formula (A i j) i,n+1 = T (τ j + 1) ω k F m (ξ j,k ), the shape of the global matrix A will be 0 A 1 j 0 0 A = A N j 0 L = (L i j ), L is a matrix with a dimension (N + 1) (N + 1)(N 1) in which the first and last (N + 1) rows are zeros, and each block matrix (L i j ) has a dimension of (N + 1) (N + 1). For each m, j, i, n, the other entries of (L i j ) can be given by the following formula (L i j) i+1,n = T (τ j + 1) ω k K(x i, τ j ρ j,k )F m (ρ j,k ), B = (Bj i ), B is a matrix with a dimension (N + 1)(N 1) (N + 1)(N 1) in which each block matrix (Bj i ) has a dimension of (N + 1) (N + 1). For each m, j, i, n, the entries of (Bj i ) can be found from the following relation (B i j) i,n = T (τ j + 1) ω k F m (ξ j,k )Di,n, Finally, C = (Cj i ) which is similar to the matrix B. For each m, j, i, n, the entries of (Cj i ) can be found from the following relation (C i j) i,n = T (τ j + 1) p(x i ) ω k F m (ξ j,k )D i,n. To solve the coupled equations system in equation (19), we convert them to a linear algebraic system as follows (I (B + C) AL) U (N)(M+1) = U + AG (N+1)(M+1). By solving the previous system, we obtain an approximation to U (N)(M+1) then the approximation to the original problem for all x (, 1) and t [0, T ] can obtain by u(x, t) N l n (x)f m ( T t 1)U(x n, τ m ).

8 8 G. I. EL BAGHDADY, M. S. EL AZAB JFCA-01/5(3S) 3. Numerical examples In order to test the utility of the proposed new method, we apply the new scheme to the following two examples whose exact solutions are provided in each case. For both examples, we take T = 1, p = N = M, and consider {θ k } N as the Legendre- Gauss points, and proceed as in Section. To show the efficiency of the present method for our problem in comparison with the exact solution, we calculate for different values of N the maximum error defined by E = max U i,j u(x i, T (τ j + 1) ). 1 i N 0 j M All the computations are carried out in double precision arithmetic using Matlab (R009b). To obtain sufficient accurate calculations, variable arithmetic precision (vpa) is employed with digit being assigned to be 3. The code was executed on a second generation Intel Core i5 10M,.3 Ghz Laptop. Example 3.1. [16] Consider the linear PPIDE (1)-(3) with the kernel k(x, t) = cos xt and the forcing function f(x, t) = ( 8x 3 +13x +p(x)(1 x 3 x ) ) e (x+t) + ( x )( xsin(xt) cos(xt) ) e x, so that the exact solution will be u(x, t) = (1 x )e (x+t). Table 1. Maximum L errors for Example 3.1. values of N. with different p 1 (x) p (x) N E E E E E E E E E E E E E E-13 In Table 1, we take p 1 (x) = x 3 e x and p (x) = x 3 sin(x)e x /(1 + x ). In Figure 1, the exact solution and numerical solution are plotted with p(x) = x 3 sin(x)e x /(1+x ) and the nonuniform set of collocation points (x i, t j ) at N = 16. Example 3.. [16] Consider the linear PPIDE (1)-(3) with the kernel k(x, t) = xt and the forcing function f(x, t) = ( x 6 1x 5 1 p(x)(1 x 6 ) ) sin(x + t) x(1 x 6 ) ( t sin(x) cos(x) ) + ( 1 + x + 30x x 6 x 7 6x 5 p(x) ) cos(x + t), so that the exact solution will be u(x, t) = (1 x 6 ) cos (x + t). In Table, we take p 1 (x) = cos(x )e x /(1 + x ) and p (x) = x 3 sin(x)e x /(1 + x ).

JFCA-01/5(3S) LEGENDRE SPECTRAL COLLOCATION METHOD 9 (a) Exact solution (b) Approximated solution Figure 1. Exact and Numerical solutions for p (x) with x [, 1] and t [0, T ] at N = 16 Table.

10633E-1.75E-1 (a) Exact solution (b) Approximated solution Figure. Exact and Approximated solutions for p (x) with x [, 1] and t [0, T ] at N = 16. See Example 3.

9 JFCA-01/5(3S) LEGENDRE SPECTRAL COLLOCATION METHOD 9 (a) Exact solution (b) Approximated solution Figure 1. Exact and Numerical solutions for p (x) with x [, 1] and t [0, T ] at N = 16 Table. Maximum L errors for Example 3.. values of N. with different p 1 (x) p (x) N E E E E E E E E E E E E E-1.75E-1 (a) Exact solution (b) Approximated solution Figure. Exact and Approximated solutions for p (x) with x [, 1] and t [0, T ] at N = 16. See Example 3.. In Figure, the exact solution and numerical solution are plotted with p(x) = x 3 sin(x)e x /(1+x ) and the nonuniform set of collocation points (x i, t j ) at N = 16.. Conclusion In this work, we used a new competitive numerical scheme based on the Legendre spectral collocation method to find the approximated solution of one dimensional parabolic partial integro differential equations of volterra type. we restate the original problem as two equivalent equations and use the Legendre-collocation methods

10 10 G. I. EL BAGHDADY, M. S. EL AZAB JFCA-01/5(3S) for each one which leads to conversion of the problem to solving a linear algebraic system. Moreover, we demonstrate the efficiency and accuracy of the proposed method with some numerical examples on bounded domain even with using a small number of collocation points. References [1] E. G. Yanik and G. Fairweather, Finite element methods for parabolic and hyperbolic partial integro differential equations, Nonlinear Anal. 1 (1988) pp [] M. Renardy, W. Hrusa and J. Nohel, Mathematical Problems in Viscoelasticity, (Pitman monographs and Surveys in pure Appl. Math., New York, wiley, ). [3] T. A. Burton, Volterra Integral and Differential Equations (Mathematics in Science and Engineering, Elsevier, nd edition, 005), 0. [] G. Dagan, The significance of heterogeneity of evolving scales to transport in porous formations, Water Resour. Res., 30 (199) pp [5] J. Cushman and T. Glinn, Nonlocal dispersion in media with continuously evolving scales of heterogeneity, Transport in Porous Media, 13 (1993) pp [6] R. Ewing, Y. Lin and J. Wang, A numerical approximation of non Fickian flows with mixing length growth in porous media, Acta. Math. Univ. Comenian., LXX (001) pp [7] J. P. Kauthen, The method of lines for parabolic partial integro differential equations, Journal of Integral equations and Application, (199), pp [8] S. Larsson, V. Thomée and L.B. Wahlbin, Numerical solution of parabolic integro differential equations by the discontinuous Galerkin method, Math. Comput., 67 (1998), pp [9] Amiya K. Pani and G. Fairweather, H 1 -Galerkin mixed finite element methods for parabolic partial integro differential equations, IMA Journal of Numerical Analysis, (00) pp [10] Amiya K. Pani, G. Fairweather and Ryan I. Fernandes, Alternating dirction implict orthogonal spline collocation methods for an evolution equation with positive type memory term, SIAM J. NUMER. ANAL., 6 (008) pp [11] Amiya K. Pani, G. Fairweather and Ryan I. Fernandes, ADI orthogonal spline collocation methods for parabolic partial integro differential equations, IMA Journal of Numerical Analysis, 30 (010) pp [1] R. Zacher, Boundedness of weak solutions to evolutionary partial integro differential equations with discontinuous coefficients, J. Math. Anal. Appl., 38 (008), pp [13] E.W. Sachs and A.K. Strauss, Efficient solution of a partial integro differential equation in finance, Appl. Numer. Math., 58 (008), pp [1] Z. Avazzadeh, M. Heydari, W. Chen and G. B. Loghmani, A numerical solution of nonlinear parabolic-type Volterra partial integro differential equations using radial basis functions, Eng. Anal. Boundary Elem., 36 (01) pp [15] Z. Avazzadeh, M. Heydari, W. Chen and G. B. Loghmani, Smooth solution of partial integro differential equations using radial basis functions, Journal of Applied Analysis and Computation, (01) pp [16] F. Fakhar-Izadi and M. Dehghan, The spectral methods for parabolic Volterra integro differential equations, Journal of Computational and Applied Mathematics, 35 (011) pp [17] C. Canuto, A. Quarteroni, M. Y. Hussaini and T. A. Zang, Spectral Methods Fundamentals in Single Domains, (Scientific Computation, Springer, New York, NY, USA, 006). [18] L. N. Trefethen, Spectral Methods in MATLAB, (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 000). [19] C. Canuto, A. Quarteroni, M. Y. Hussaini and T. A. Zang, Spectral Methods in Fluid Dynamics, (Springer Series in Computational Physics, Springer, New York, NY, USA, 1988). [0] W. Guo, G. Labrosse and R. Narayanan, The Application of the Chebyshev Spectral Method in Transport Phenomena, (Notes in Applied and Computational Mechanics, 68, Springer Verlag Berlin Heidelberg, 01). [1] J. P. Boyd, Chebyshev and Fourier Spectral Methods, (Dover, New York, NY, USA, nd edition, 000). [] J. Shen and T. Tang, Spectral and High Order Methods with applications, (Science Press, Beijing, China, 006).

11 JFCA-01/5(3S) LEGENDRE SPECTRAL COLLOCATION METHOD 11 [3] B. Y. Guo, Spectral Methods and their Applications, (World Scientific, River Edge, NJ, 1998.) [] M. Aguilar and H. Brunner, Collocation methods for second order Volterra integro differential equations, Appl. Numer. Math., (1988) pp [5] D. Conte and B. Paternoster, Multistep collocation methods for Volterra Integral Equations, Applied Numerical Mathematics, 59 (009) pp [6] Tobin A. Driscoll, Automatic spectral collocation for integral, integro differential, and integrally reformulated differential equations, Journal of Computational Physics, 9 (010) pp [7] T. Tang, On spectral methods for Volterra integral equations and the convergence analysis, J. Comput. Math., 6 (008) pp [8] Z. Wan, Y. Chen and Y. Huang, Legendre spectral Galerkin method for second kind Volterra integral equations, Front. Math. China, (009) pp [9] Foroogh Ghanei and Ali Salimi Shamloo, Numerical Solution of Volterra Integral Equations Using the Chebyshev-Collocation Spectral Methods, International Journal of Science and Engineering Investigations, 1 (01) pp [30] Zhendong Cu and Yanping Chen, Chebyshev spectral collocation method for Volterra integral equations, Recent Advances in Scientific Computing and Applications, Contemporary Mathematics, 586 (013) pp [31] Wu Hua and Zhang Jue, Chebyshev-Collocation Spectral Method for Volterra Integro- Differential equations, Journal of Shanghhai University (Natural Science), 17 (011) pp [3] Ying Jun jiang, On spectral methods for Volterra type integro Differential equations, Journal of Computational and Applied Mathematics, 30 (009) pp [33] W. Gautschi, Orthogonal Polynomials: Computation and Approximation, (Oxford University Press Inc., New York, NY, USA, 00). [3] G. Szegö, Orthogonal Polynomials, (AMS Providence, Rhode Island, USA, 5th edition, 1975). [35] B. Costa and W. S. Don, On the computation of high order pseudospectral derivatives, Appl. Numer. Math., 33 (000) pp [36] Elsayed M. E. Elbarbary and Salah M. El-Sayed, Higher order pseudospectral differentiation matrices, Appl. Numer. Math., 55 (005) pp [37] R. Baltensperger and M. R. Trummer, Spectral Differencing with a twist, J. Indus. and Appl. Math., (003) pp [38] J. S. Hesthaven, S. Gottlieb and D. Gottlieb, Spectral Methods for Time-Dependent Problems, (Cambridge University Press, 007). G. I. El Baghdady Faculty of Engineering, Mansoura University, Mansoura, Egypt address: amoun1973@yahoo.com M. S. El Azab Faculty of Engineering, Mansoura University, Mansoura, Egypt address: ms elazab@hotmail.com

1462. Jacobi pseudo-spectral Galerkin method for second kind Volterra integro-differential equations with a weakly singular kernel

1462. Jacobi pseudo-spectral Galerkin method for second kind Volterra integro-differential equations with a weakly singular kernel Xiaoyong Zhang 1, Junlin Li 2 1 Shanghai Maritime University, Shanghai,