Improvement of numerical solution by boundary value technique for singularly perturbed one dimensional reaction diffusion problem

摘要

A class of singularly perturbed two point boundary value problems (BVPs) for second order ordinary differential equations in self-adjoint form arising in the study of chemical catalysis and Michaelis–Menten process in biology is considered. In order to solve them, a numerical method as in [Appl. Math. Comput. 55 (1993) 31] is proposed. The essential idea in this method is to divide the domain of the differential equation into three non-overlapping subdomains and solve the given equations over these regions separately as three two-point BVPs numerically. The inner region problems are solved using a fitted operator method [Uniform Numerical Methods for Problems with Initial and Boundary Layers, Boole press, Dublin, 1980], whereas the outer region problem is solved using a classical central difference method. The boundary conditions at the transition points are obtained using the zero order asymptotic expansion approximation to the solution of the problem. This method is well suited for parallel computing and an algorithm for the same is given. Error estimates are derived for the numerical solution. Some schemes for self-adjoint equations in conservation form are given. Using Newton’s method of quasilinearization, a class of semilinear problems are also solved. Numerical experiments are conducted. It is found that the present method performs better than the fitted mesh method [Fitted Numerical Methods for Singular Perturbation Problems, World Scientific, Singapore, 1996] and higher order method [Numeriche Mathematik 56 (1990) 675].