Finite difference solution of parameterized differential problems with initial layer

In this thesis, numerical solutions of nonlinear differential equations with linear and integral boundary conditions according to the parameterized singularly perturbed solution function are examined by using finite difference method. Before introducing a numerical method for these problems, the characteristics of singular perturbation problems were investigated. The finite difference scheme for linear problem is established by using the basis functions of exponential coefficients on the uniform mesh and the quadrature formulas with weight functions and integral remainder terms. For nonlinear integral boundary conditional problems, it is established by using interpolation quadrature formulas with piecewise constant basis functions on a Bakhvalov mesh and integral remainder terms. In the discrete norm, the first order uniform convergence of difference schemes of the linear and nonlinear problems, independent of the perturbation parameter, was investigated. Finally, an effective and appropriate solution algorithm for the efficiency of the finite difference method and numerical results supporting the theory are given.