Numerical treatment of singular perturbation problems exhibiting dual boundary layers

In this paper, we employed a fitted operator finite difference method on a uniform mesh for solving singularly perturbed two-point boundary value problems exhibiting dual boundary layers. In this method, we have extended the Numerov method to the second order singularly perturbed two-point boundary value problem with first order derivative. By using nonsymmetric finite differences for the first order derivative, we have derived the finite difference scheme. A fitting factor is introduced in this finite difference scheme which takes care of the rapid changes that occur in the boundary layer. This fitting factor is obtained from the asymptotic approximate solution of singular perturbations. Discrete invariant imbedding algorithm is used to solve the tridiagonal system of the fitted finite difference method. We have discussed the convergence analysis of the proposed method. Maximum absolute errors of the several numerical examples are presented to illustrate the proposed method

Computational Method for Singularly Perturbed Boundary Value Problems with Dual Boundary Layer

AbstractIn this paper, we proposed a computational method on a uniform mesh for solving singularly perturbed two-point boundary value problems exhibiting dual boundary layers using exponentially fitting factor. In this method, we extended the Numerov Scheme to the singularly perturbed two-point boundary value problem with first order derivative. By using non symmetric finite differences and mixed finite difference for the first order derivative, the finite difference scheme is derived. An exponential fitting factor is introduced in this finite difference scheme which takes care of the rapid behaviour occurs in the boundary layers. Using the asymptotic approximate solution of singular perturbations, the fitting factor is derived. Discrete invariant imbedding algorithm is used to solve the tridiagonal system of the fitted finite difference method. The method is analyzed for convergence. Numerical experiments are presented to demonstrate the utility and efficiency of the proposed computational method