A generalized operational method for solving integro-partial differential equations based on Jacobi polynomials

In this paper, a numerical method is developed for solving linear and nonlinear integro-partial differential equations in terms of the two variables Jacobi polynomials. First, some properties of these polynomials and several theorems are presented then a generalized approach implementing a collocation method in combination with two dimensional operational matrices of Jacobi polynomials is introduced to approximate the solution of some integro-partial differential equations with initial or boundary conditions. Also, it is shown that the resulted approximate solution is the best approximation for the considered problem. The main advantage is to derive the Jacobi operational matrices of integration and product to achieve the best approximation of the two dimensional integro-differential equations. Numerical results are given to confirm the reliability of the proposed method for solving these equations.