Uniformly convergent difference schemes for singularly perturbed Volterra delay-integro-differential equations

In this thesis, numerical solutions of first order singularly perturbed Volterra delay-integro-differential equations are considered. This thesis study consists of five main sections. In the first section, singular perturbation theory, the history of integro-differential equations and their application fields are mentioned. Literatures are given in the second section. Some basic definitions and formulas used in this study are given in the third section. In the fourth section of this study, first, initial value problems for the first order linear and nonlinear singularly perturbed Volterra delay-integro-differential equations are discussed. Exponential fitted difference schemes are constructed for these problems on a uniform mesh separately. Second, a finite difference scheme on a piecewise uniform mesh (Shishkin mesh) to solve a singularly perturbed initial value problem for a linear first order Volterra integro-differential equation with delay is presented. Later, a fitted finite difference scheme on the piecewise uniform mesh for the numerical solution of the first order linear singularly perturbed Volterra integro-differential equation is proposed. In the end, a homogeneous (nonhybrid) type difference scheme on the piecewise uniform mesh for numerical solutions of the first order singularly perturbed Volterra delay-integro-differential equation is introduced. For these presented problems, the properties of continuous problems are given and the difference schemes are constructed by the method of integral identities with the use of exponential basis functions and interpolating quadrature rules with the weight and remainder terms in integral form. It has been shown that the approximate solutions are uniformly convergent according to epsilon-parameter to the exact solution in the discrete maximum norm and the rates of convergence are specified. The obtained results are supported by numerical examples.