High accurate pseudo-spectral Galerkin scheme for pantograph type Volterra integro-differential equations with singular kernels

Abstract

Spectral and pseudo-spectral Galerkin techniques, by using the standard Jacobi polynomials, are implemented to calculate numerically the solutions of pantograph type Volterra delay integro-differential equations that have kernels with the property of weak singularity. Because of the complex structure of the considered problems, pseudo-spectral Galerkin approaches are more desirable with respect to the spectral Galerkin approaches, since they have the property of integral approximator by using high order convergent Gauss quadrature formulas. A deep and detailed analysis of convergence of the numerical solutions to the exact solutions are given under some mild conditions such as smoothness of the solutions. Some test problems are illustrated and efficiency of the suggested numerical approach is investigated with respect to a recently proposed Jacobi pseudo-spectral collocation technique via some figures and tables experimentally.