An ultraspherical product integration-spectral collocation method for multidimensional partial Volterra integro-differential equations and its convergence analysis

Abstract

We present a reliable numerical method for solving multidimensional partial Volterra integro-differential equations (PVIDEs). This comprehensive approach integrates techniques from product integration, the Nyström method, and spectral collocation, all founded on ultraspherical polynomials. The primary objective of our methodology is to employ variable and function transformations to reformulate the equations into a novel class of PVIDEs. Subsequently, the ultraspherical product integration-spectral collocation approach is applied to derive equivalent algebraic equations. Newton’s iterative method is then utilized to simultaneously compute the numerical solution and the first-order partial derivative. We rigorously analyze the error bounds of the proposed method in both $L^{\infty }$- and $L^2$-norms. Our results demonstrate that the errors in the numerical solution, as well as in the numerical first-order partial derivative, decay exponentially. Numerical examples are provided to validate reliability and efficiency of the ultraspherical product integration-spectral collocation approach.