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.
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