A Legendre tau approach for high-order pantograph Volterra-Fredholm integro-differential equations

High-order pantograph Volterra-Fredholm integro-differential equations (P-VF-IDEs) arise in various scientific and engineering applications, where systems exhibit delay or scaling in their dynamics. This paper investigates a class of high-order P-VF-IDEs characterized by the presence of both Volterra and Fredholm integral terms as well as pantograph delay elements. We propose a spectral tau approach to approximate the solution P-VF-IDEs in both one- and two-dimensions. In particular, we employ the operational differentiation and integration matrices to approximate the integro-differential operator, transforming the continuous problem into a system of algebraic equations, and providing high accuracy with fewer computational modes. Numerical experiments illustrate the accuracy and convergence properties of the spectral Legendre tau method in solving high-order P-VF-IDEs and demonstrate its efficacy compared to other spectral approaches.