Fully Jacobi–Galerkin algorithm for two-dimensional time-dependent PDEs arising in physics

Abstract

Herein, a pure shifted Jacobi–Galerkin (SJG) method is offered for handling linear two-dimensional space-time diffusion and telegraph equations but by considering their integrated forms. The semi-analytic approximate solution is expanded in spatial and temporal variables as polynomials bases built as a linear combination of the shifted Jacobi Polynomials (JPs). Expanding the exact solution regarding these polynomials satisfies the homogeneous initial and boundary conditions. The proposed Jacobi–Galerkin (JG) procedure yields exponential convergence rates when the solution is smooth enough. A careful investigation of the convergence analysis of the suggested approximate triple series expansion examines the offered numerical algorithm. Two test illustrative examples are given to prove the high accuracy of the proposed scheme.