The General Two Dimensional Shifted Jacobi Matrix Method for Solving the Second Order Linear Partial Difference-differential Equations with Variable Coefficients

In this paper, a new and efficient approach for numerical approximation of second order linear partial differential-difference equations (PDDEs) with variable coefficients is introduced. Explicit formulae which express the two dimensional Jacobi expansion coefficients for the derivatives and moments of any differentiable function in terms of the original expansion coefficients of the function itself are given in the matrix form. The main importance of this scheme is that using this approach reduces solving the general linear PDDEs to solve a system of linear algebraic equations, wherein greatly simplify the problem. In addition, some experiments are given to demonstrate the validity and applicability of the method.