Uniform numerical methods for first and second order singularly perturbed integro-differential equations

In this thesis, firstly, the initial value problem for first order singularly perturbed Fredholm integro-differential equation is discussed. The properties of the continuous problem are given, the difference scheme using the quadrature formulas with the remainder term in integral form and containing the basis functions is created in uniform and Shishkin meshes. It has been shown that the approximate solution is uniform convergent according to parameter to the exact solution and the rate of convergence is of the first order in the uniform mesh and the second order in Shishkin mesh. Next, the boundary value problem for the second order singularly perturbed Fredholm integro-differential equation is discussed. The properties of the continuous problem are given, the difference scheme using the quadrature formulas is created in uniform and Shishkin meshes. It has been shown that the approximate solution is uniform convergent according to parameter of the exact solution and the rate of convergence is of the first order and the second order in the uniform mesh and the second order in Shishkin mesh. Finally, boundary value problem for second order singularly perturbed Fredholm integro-differential equation with zeroth order reduced equation and non-self adjoint form is discussed. The properties of the continuous problem are given, the difference scheme using the quadrature formulas is created in Shishkin mesh. It has been shown that the approximate solution is uniform convergent according to parameter to the exact solution and the rate of convergence is of the second order. The presented methods are supported by various numerical examples