A numerical method on Bakhvalov-Shishkin mesh for Volterra integro-differential equations with a boundary layer

Abstract

We construct a finite difference scheme for a first-order linear singularly perturbed Volterra integro-differential equation (SPVIDE) on Bakhvalov-Shishkin mesh. For the discretization of the problem, we use the integral identities and deal with the emerging integrals terms with interpolating quadrature rules which also yields remaining terms. The stability bound and the error estimates of the approximate solution are established. Further, we demonstrate that the scheme on Bakhvalov-Shishkin mesh is O(N −1 ) uniformly convergent, where N is the mesh parameter. The numerical results are also provided for a couple of examples.