Convergence analysis of the homogeneous second order difference method for a singularly perturbed Volterra delay-integro-differential equation

Abstract

• A new approach for designing the computational method has been considered. • This new approach results in a local truncation error containing much lower derivatives of exact solution compared to classical methods. • The method allows to weaken or get rid of the smoothness of the data functions, a determining factor for convergence analysis. • The previous works related to Volterra integro-differential equation were only concerned with regular cases. • Problems involving boundary layers have a solution with bad behaviours in applications encountered in different fields of engineering. A linear Volterra delay-integro-differential equation with a singular perturbation parameter ε is considered. The problem is discretized using exponentially fitted schemes on the Shishkin type meshes. It is proved that the numerical approximations generated by this method are O ( N − 2 ln N ) convergent in the discrete maximum norm, where N is the mesh parameter. Numerical results show a good agreement with the theoretical analysis.