Boubaker Matrix Polynomials and Nonlinear Volterra-Fredholm Integro-differential Equations

In this paper, a new scheme for the formulation of operational matrix form of the integration, differentiation, and approximate functions in any arbitrary interval which will be used in the Galerkin method is presented. There are so many methods to obtain operational matrices for Chebyshev, Legendre, Bernstein, Bessel, etc., polynomials. Our aim is to determine the exact operational matrices for the Boubaker polynomials. This method transforms the nonlinear Volterra-Fredholm integro-differential equations into a system of nonlinear algebraic equations which will be solved easily by the Galerkin procedure. Also, an error estimate and numerical analysis of the proposed method are shown through some theorems. Moreover, the existence and the uniqueness of the solution in this scheme have been proved. To illustrate the efficiency and capability of the method, some examples are presented and their results are compared with the results of some other well-known methods. In addition, an application of this algorithm to the population model is given. The outcome of these examples and their CPU times confirm the capability and validity of this algorithm.