An operational approach for one- and two-dimension high-order multi-pantograph Volterra integro-differential equation

High-order Volterra integro-differential equations are of great interest to many authors because of their important applications in physics and engineering, especially if they contain delay or pantograph terms that enable them to describe the memory effect. Providing an efficient numerical scheme for high-order Volterra integro-differential equations helps to explain many problems in mathematical biology and quantum mechanics. In this manuscript, we use shifted Jacobi polynomials as the basis for a spectral collocation approach to solve high-order one- and two-dimensional Volterra integro-differential equations with variable coefficients. A pantograph operational matrix, based on shifted Jacobi polynomials, is used for the first time, together with the Gauss-Jacobi quadrature rule, to reduce the problem to the problem of solving a system of algebraic equations. To ensure the validity of the proposed approach, we compare the numerical results with those of other numerical schemes in the literature.