NUMERICAL SOLUTION OF 2D PARTIAL VOLTERRA INTEGRO DIFFERENTIAL EQUATIONS USING POLYNOMIAL COLLOCATION WITH MATRIX FORMULATION

Abstract

Partial Volterra integro-differential equations are equations that mix partial derivatives with Volterra-type integral terms, representing process where the current state depends on both local changes and the accumulated history. This study presents a numerical method for solving two-dimensional Partial Volterra Integro Differential Equations (PVIDEs) using a polynomial collocation with matrix formulation. The original integro-differential equation is first reformulated into a continuous time-integrated form through the Fundamental Theorem of Calculus (FTC). This reformulated equation is then discretized on a hybrid space-time collocation grid. A polynomial collocation scheme is constructed using standard basis functions over the grid points to transform the problem into a solvable system of algebraic equations. The method incorporates consistent numerical quadrature for time-integration of the nonlinear kernel ensuring computational efficiency through matrix formulation. Theoretical analysis demonstrates the method's consistency, stability, and convergence using Lax-Richtmyer equivalence theorem and discrete Grönwall inequality. Numerical examples including both linear and nonlinear 2D PVIDEs implemented in MATLAB confirm the validity and accuracy of the method. The approach gives a close form solution, which show its consistency, stability and accuracy. This approach offers a robust and efficient solution of 2D PVIDEs, extending the applicability of polynomial collocation methods to integro differential equations.