A brief survey on numerical methods for solving singularly perturbed problems

This paper is concerned with a new algorithm for giving the analytical and approximate solutions of a class of partial differential equations with a singularly perturbed term in a reproducing kernel space. Two numerical examples are studied to demonstrate the accuracy of the present method. Results obtained by the method indicate the method is simple and effective.

Using reproducing kernel for solving a class of singularly perturbed problems

An asymptotic initial value method for boundary value problems for a system of singularly perturbed second order ordinary differential equations

A computational method for solving third-order singularly perturbed boundary-value problems

A boundary value approach for a class of linear singularly perturbed boundary value problems

A computational method is presented to solve a class of nonturning-point singularly-perturbed two-point boundary-value problems for second-order ordinary differential equations with a small parameter multiplying the highest derivative, subject to Dirichlet-type boundary conditions. In this method, first we construct a zeroth order asymptotic expansion for the solution of the given boundary-value problem. Then, this problem is integrated to get an equivalent initial-value problem for first-order ordinary differential equations. This initial-value problem is solved by either a classical method or a fitted operator method after approximating some of the terms in the differential equations by using the zeroth order asymptotic expansion. This method is effective and easy to implement. An error estimate is derived for the numerical solution. Examples are given to illustrate the method.

N. Ramanujan

Papers

Asymptotic Initial-Value Method for Singularly-Perturbed Boundary-Value Problems for Second-Order Ordinary Differential Equations