Fitted Numerical Schemes for a Class of Singularly Perturbed Differential-Difference Equations

Abstract

Singularly perturbation problem is a problem involving small positive parameter, which causes a large effect on the problem. A class of delay differential equations that would be obtained by maintaining the small parameter multiplying the highest order derivative is said to be a singularly perturbed differential-difference equation. Due to the effect of the small parameter, narrow regions (boundary layers) frequently adjoin the boundaries of the domain are exhibited in the solution, and the delay term give rise to an interior layer. In the layer regions, the solu tion changes rapidly with changing values of the perturbation parameter. This rapid changing behavior of the layer region makes it difficult to solve the problem analytically. Standard nu merical methods, on the other hand, do not give satisfactory results as they do not consider the layer behaviors. As a result, formulating parameter-uniform numerical schemes for a singularly perturbed problems has become a popular research topic. The main aim of this dissertation is to formulate and analyze fitted numerical schemes that converge independent of the perturba tion parameter for a class of singularly perturbed differential-difference equations in which the diffusion term is dominated by the reaction term. The numerical schemes are either fitted op erator finite difference method or fitted mesh finite difference method. Construction of a fitted operator parameter-uniform numerical scheme involves replacing the standard finite difference operator by an operator which reflects the singularly perturbed nature of the differential op erator on a uniform mesh. The other successful approach of formulating a parameter-uniform numerical scheme involves the use of standard finite difference operators on a special piece wise uniform meshes condensed around the layer regions. Formulating such form of numerical schemes for a class of singularly perturbed differential-difference equation yields a uniformly convergent numerical results, that overcomes the limitations of the standard numerical methods. In this dissertation, six uniformly convergent numerical schemes are formulated. The stability estimate of each scheme is established and the parameter-uniform convergence of the schemes are investigated and proved. To demonstrate the validity and applicability of the formulated numerical schemes, various numerical experiments are performed and confirmed with the theo retical findings. The theoretical and numerical results confirm that the formulated schemes are more accurate and uniformly convergent as compared to other previous research works in the literature.