Spectral Galerkin Methods And Iterated Methods For Volterra Integro-differential Equations

Our research covers two major parts, one is consider spectral Jacobi-Galerkin iterated approaches for Volterra integral equations of the second kind with weakly singular kernel, the other is discusses spectral and pseudo-spectral Jacobi-Galerkin methods for Volterra integro-differential equations of the second kind with weakly singular kernel.In the first part, the Gauss-Jacobi quadrature formula is used to approximate the integral operator and the inner product based on the Jacobi weight is implemented in the weak formulation in the numerical implementation. We obtain the convergence rates for the approximated solution and iterated solution in weakly singular Volterra integral equations, which show that the iteration methods can improve convergence order of the numberical solution. The numerical examples are given to illustrate the theoretical results.In the second part, we introduce the spectral Jacobi-Galerkin methods and pseudo-spectral Jacobi-Galerkin methods for Volterra integro-differential equations of the second kind with weakly singular kernel. Then we prove the exponential convergence property of this two methods in the sense of L2 and L∞theoretically.Numerical experiments validate the theoretical prediction.