A solution of the discrepancy occurs due to using the fitted mesh approach rather than to the fitted operator for solving singularly perturbed differential equations

摘要

The solution of the boundary value problems for singularly perturbed differential equations, i.e., where the highest order derivative is multiplied by a small parameter, exhibits layer behavior. The classical numerical schemes to solve such types of the boundary value problems do not give satisfactory result when the perturbation parameter is sufficiently small. To resolve this difficulty, there are mainly two approaches, namely, fitted operator and fitted mesh. Both the scheme are uniformly convergent, i.e., their convergence is independent of the small perturbation parameter.It is justified to adopt the two approach rather than the classical numerical schemes to solve the boundary value problems for singularly perturbed differential equations. Now if we compare the two approaches, one thing is common that both the approaches give the parameter-uniform schemes which is the primary requirement in construction of the numerical scheme to solve such type of problem. Secondly, one desire a higher order numerical scheme to approximate the solution of a problem. As far as order of convergence is concerned, the numerical scheme based on fitted operator approach is better than the numerical scheme constructed using fitted mesh approach.The researchers who adopted the fitted mesh rather than fitted operator approach to solve a singularly perturbed problem faced the question due to the loss of order of convergence, most of them justified it by quoting the simplicity of the method and there are some non-linear problems for which a parameter uniform scheme cannot be constructed based on fitted operator approach while for the same problem, a parameter uniform scheme is constructed based on fitted mesh method. Now question remains unanswered in the case of linear problem. In this article, we replied to this question by giving an example of linear problem for which one cannot construct a parameter uniform scheme based on fitted operator approach while for the same problem a parameter uniform numerical scheme based fitted mesh approach has been constructed [M.K. Kadalbajoo, K.K. Sharma, ε uniform fitted mesh method for singularly perturbed differential difference equations with mixed type of shifts with layer behavior, Int. J. Comput. Math. 81 (2004) 49–62]. A theoretical reason behind the inability in construction of a parameter uniform scheme using fitted operator approach is revealed. In support of the predicted theory, a number of numerical experiments are carried out.