Numerical solution of the system of Fredholm integro-differential equations by the Tau method

The Tau method, produces approximate polynomial solution of differential, integral and integro-differential equations (see [E.l,Ortiz, The Tau method, SIAM J. Numer. Anal. 6 (3) (1969) 480–492; E.l. Ortiz, H. Samara, An operational approach to the Tau method for the numerical solution of non-linear differential equations, Computing 27 (1981) 15–25; S.M. Hosseini, S. Shahmorad, A matrix formulation of the Tau for Fredholm and Volterra linear integro-differential equations, The Korean J. Comput. Appl. Math. 9 (2) (2002) 497–507; S.M. Hosseini, S. Shahmorad, Numerical solution of a class of integro-differential equations by the Tau method with an error estimation, Appl. Math. Comput. 136 (2003) 559–570]). In this paper, we extend the Tau method for the numerical solution of integro-differential equations systems (IDES). We also give a brief description of the structure of the Tau program by the Maple software. An efficient error estimation of the numerical solution of the method is also introduced. Some examples are given to clarify the efficiency and high accuracy of the method. 2004 Elsevier Inc. All rights reserved.