Spectral Galerkin Method And Its Convergence Analysis For Volterra Integro-differential Equations

The main work of this paper is to solve the Volterra integro-differential equations based on Jacobi spectral and pseudo-spectral Galerkin methods.Such equations usually appear in mathematical modeling with genetic phenomena,such as material mechanics problems with delayed memory,population dynamics in biology and the spread of epidemics,etc.This paper is mainly divided into three contents.The first part apply Jacobi spectral and pseudo-spectral Galerkin methods to solve the high-order nonlinear Volterra integro-differential equations and prove the convergence of the two methods in the sense of L^?and L²norms.In addition,corresponding numerical examples are given to verify the validity of the methods and the correctness of the theoretical results.For the second part,since the integral term of the considered equation contains a weak singular kernel,it is necessary to use the orthogonal quadrature formula to calculate the relevant integral operator before giving the corresponding discretization scheme.Moreover,the convergence of the discretization scheme is strictly proved and specific numerical examples are given to verify the theoretical results.In the third part,the equations considered are nonlinear weakly singular Volterra integro-differential equations with non-smooth solutions.Since the solution of the equations are not sufficiently smooth,some transformations are used to make the solution of the transformed equation sufficiently smooth.Then,we use the Jacobi pseudo-spectral Galerkin method to numerically solve the transformed equation and prove the convergence of the discrete scheme.Finally,the theoretical results are verified by numerical examples.