Asymptotic Numerical Solutions for Second-order Quasilinear Singularly Perturbed Problems

Abstract

For a second-order quasilinear singularly perturbed problem under the Dirichlet boundary conditions, we propose a new asymptotic numerical method, which involves two problems: a reduced problem with a one-side boundary condition and a novel boundary layer correction problem with a two-sided boundary condition. Through the introduction of two new variables, both problems are transformed to a set of three first-order initial value problems with zero initial conditions. The Runge-Kutta method is then applied to integrate the differential equations and to determine two unknown terminal values of the new variables until they converge. The modified asymptotic numerical solution satisfies the Dirichlet boundary conditions. Some examples confirm that the newly proposed method can achieve a better asymptotic solution to the quasilinear singularly perturbed problem. For most values of the perturbing parameter, the present method not only preserves the inherent asymptotic property within the boundary layer but also improves the accuracy within the entire domain.