Homogeneous second order numerical method for singularly perturbed integro-differential equations with integral boundary condition

In this thesis, homogeneous second-order numerical methods are presented for singularly perturbed linear integro-differential equations with integral boundary conditions. This study consists of 8 main chapters. Historical developments and literature declarations are presented in chapters one and two. The third section outlines the materials and methods used for the thesis. Some fundamental definitions and theorem required to solve the problems discussed in the thesis are given in the fourth chapter. In the fifth chapter, after giving asymptotic estimates for the first introduced problem, a finite difference scheme is constructed using quadrature rules with the remainder term in integral form and basis functions with exponential coefficients. Convergence analysis and error estimates of the established scheme are given. Numerical examples and results are shown in tables to demonstrate the effectiveness of the proposed numerical method. In the sixth chapter, the necessary asymptotic estimates are made for the second introduced problem and a finite difference scheme was established on the non-uniform mesh using the basis function with exponential coefficients. Finally, numerical examples supporting the theoretical results are given. In the seventh chapter, asymptotic estimates are made for the third introduced problem, and the finite difference scheme is established using linear basis functions. Finally, the effectiveness of the method is demonstrated with numerical examples. The results acquired for these three problems are assessed and examined in the last chapter's discussion and conclusion section.