Shifted Chebyshev polynomial method for solving systems of linear and nonlinear Fredholm-Volterra integro-differential equations

This paper suggests a novel and efficient method for solving systems of Fredholm-Volterra integro-differential equations (FVIDEs). A Chebyshev matrix approach is implemented for solving linear and nonlinear FVIDEs under initial boundary conditions. The aim of this work is to construct a quick and precise numerical approximation by a simple, tasteful and powerful algorithm based on the Chebyshev se-ries representation for solving such systems. The properties of shifted Chebyshev polynomials are used to transform the system of FVIDEs into a system of algebraic equations. Then, the corresponding matrix equation will be solved by using the Galerkin-like procedure to find the unknown coefficients which are related to the approximate solution. Also, the polynomial convergence rate of our method is discussed by preparing some theorems and lemmas. Finally, some numerical exam-ples are given to illuminate the reliability and high accuracy of this algorithm in comparison with some other well-known methods.