Jordan Block Based Polynomial Collocation Technique for the Numerical Solution of Two Dimensional PVIDEs

Abstract

The paper present a polynomial method based on the Jordan block operational matrix for the solution of 2-dimensional partial Volterra Integro Differential Equations (VIDEs). The technique transforms the partial VIDEs to integral equation, this is then approximated through polynomial collocation at standard points, leading to a system of algebraic equations. It establishes the uniqueness of the solution by applying Banach fixed-point theory. To demonstrate the accuracy and reliability of the method, linear and nonlinear mathematical problems established in literature with known exact solutions are solved with the aid of a written MATLAB code. The results obtained prove that the technique is capable of reproducing the exact solutions with zero absolute error when the exact solution lies within the chosen polynomial space. The computational results further indicate rapid convergence and low computational cost, highlighting the spectral-type accuracy of the method. Hence, the Jordan block polynomial collocation technique provides a simple, accurate, and computationally efficient numerical technique for solving 2-dimensional partial VIDEs.