Asimptotic and numerical solutions for first and second order singularly perturbed problems

In this study, and type asymptotic expansions for the first order singularly perturbed initial value problem and and type asymptotic expansions for second order singularly perturbed boundary value problem have been considered. These expansions contain inner and outer layer functions. The convergence rates of the remainder terms of the expansions were evaluated using the maximum principle. In addition, in the study, the -uniform convergent difference scheme and its rate of convergence for the second order self adjoint boundary value problem was analyzed. The asymptotic estimates used in establishing and analyzing the difference scheme were determined before this. At the end of each section, numerical examples supporting the theory are given.