Solving nonlinear Volterra integro-differential equation by using Legendre polynomial approximations

Document Type : Research Article

Authors

Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.

Abstract

‎In this paper‎, ‎we construct a new iterative method for solving nonlinear Volterra Integral Equation of second kind‎, ‎by approximating the Legendre polynomial basis‎. ‎Error analysis is worked using Banach fixed point theorem‎. ‎We compute the approximate solution without using numerical method‎. ‎Finally‎, ‎some examples are given to compare the results with some of the existing methods‎.

Keywords


1] Babolian, E., Masouri, Z. and Hatamzadeh-Varmazyar, S. New direct method to solve nonlinear Volterra-Fredholm integral and integro-differential equations using operational matrix with block-pulse functions, Prog. in Electromag. Research 8 (2008), 59-76.
[2] Babolian, E. and Davary, A. Numerical implementation of Adomian decomposition method for linear Volterra integral equations for the second kind, Appl. Math. Comput. 165 (2005) 223-227.
[3] Danfu, H. and Xufeng, S. Numerical solution of integro-differential equations by using CAS wavelet operational matrix of integration, Appl. Math. Comput.194 (2007) 460-466.
[4] Darania, P. and Ebadian, A. A method for the numerical solution of the integro-differential equations, Appl. Math. Comput. 188 (2007) 657-668.
[5] Elbarbary, E.M.E. Legendre expansion method for the solution of the second-and fourth-order elliptic equations, Math. Comput. Simul. 59 (2002) 389–399
[6] El-Mikkawy, M.E.A. and Cheon, G.S. Combinatorial and hypergeometric identities via the Legendre polynomials- A computational approach, Appl. Math. Comput. 166 (2005) 181-195.
[7] Fox, L. and Parker, I. Chebyshev polynomials in Numerical Analysis, Clarendon Press, Oxford, 1968.
[8] Ghasemi, M. and Kajani, C.M.T., Application of Hes homotopy perturbation method to nonlinear integrodifferential equations, Applied Mathematics and Computation, vol 188(1)(2007), 538–548
[9] Ghasemi, M. and Kajani, C.M.T.,Numerical solutions of the nonlinear integro-differential equations: Wavelet-Galerkin method and homotopy perturbation method, Applied Mathematics and Computation, vol 188(1)(2007), 450–455.
[10] Ghasemi, M. and Kajani, C.M.T.,Comparison bet ween the homotopy perturbation method and the sinecosine wavelet method for solving linear integro- differential equations, Computers Mathematics with Applications, vol 54(78)(2007), 1162–1168.
[11] Gillis, J., Jedwab J. and Zeilberger, D., A combinatorial interpretation of the integral of the product of Legendre polynomials, SIAM J. Math. Anal. 19 (6) (1988) 1455-1461.
[12] Gulsu, M. and Sezer, M. The approximate solution of high-order linear difference equation with variable coefficients in terms of Taylor polynomials, Appl. Math. Comput. 168 (2005) 76-88.
[13] Liu, Y. Application of Chebyshev polynomial in solving Fredholm integral equations, Math. Comput. Modelling 50 (2009) 465-469.
[14] Mahmoudi, Y. Wavelet Galerkin method for numerical solution of nonlinear integral equation, Appl. Math. Comput. 167 (2005) 1119-1129.
[15] Maleknejad,K., Tavassoli Kajani,M. Solving second kind integral equation by Galerkin methods with hybrid Legendre and Block-Pulse functions, Appl. Math. Comput. 145 (2003) 623–629.
[16] Marzban, H.R and Razzaghi, M. Optimal control of linear delay systems via hybrid of block-pulse and Legendre polnomials, J. Franklin Inst. 341 (2004) 279-293.
[17] Rashidinia, J. and Zarebnia, M. Solution of a Volterra integral equation by the Sinc-collocation method, Comput. Appl. Math. 206 (2007) 801-813.
[18] Streltsov, I.P. Approximation of Chebyshev and Legendre polynomials on discrete point set to function interpolation and solving Fredholm integral equations, Comput. Phys. Commun. 126 (2000) 178–181.
[19] U. Lepik, U. and Tamme, E. Solution of nonlinear Fredholm integral equations via the Haar wavelet method, Proc. Estonian Acad. Sci. Phys. Math. 56 (2007) 17-27
[20] Voigt, R.G., Gottlieb, D. and Hussaini, M.Y. Spectral Methods for Partial Differential Equations, SIAM, Philadelphia, PA, 1984.
[21] Yousefi,S. and Razzaghi, M. Legendre wavelets method for the nonlinear Volterra-Fredholm integral equations, Math. Comput. Simulat. 70 (1) (2005) 1-8.
[22] Zieniuk, Eugeniusz Bezier curves in the modification of boundary integral equations (BIE) for potential boundary-values problems International Journal of Solids and Structures Volume 40, Issue 9, May 2003, Pages 2301-2320
[23] Zhao, J. and Corless, R.M.,Compact finite difference method for integro-differential equations, Applied Mathematics and Computation, vol 177, 271–288.
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