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Numerical solution of system of Fredholm-Volterra integro-differential equations using Legendre polynomials

In this paper, two collocation methods based on the shifted Legendre
 polynomials are proposed for solving system of nonlinear Fredholm-Volterra
 integro-differential equations. The equation considered in this paper
 involves the derivative of unknown functions in the integral term, which
 makes its numerical solution more complicated. We first introduce a
 single-step Legendre collocation method on the interval [0, 1]. Next, a
 multi-step version of the proposed method is derived on the arbitrary
 interval [0, T] that is based on the domain decomposition strategy and
 specially suited for large domain calculations. The first scheme converts
 the problem to a system of algebraic equations whereas the later solves the
 problem step by step in subintervals and produces a sequence of systems of
 algebraic equations. Some error estimates for the proposed methods are
 investigated. Numerical examples are given and comparisons with other
 methods available in the literature are done to demonstrate the high
 accuracy and efficiency of the proposed methods.