Abstract
In this paper, a class of linear second-order singularly perturbed differential-difference turning point problems with mixed shifts exhibiting two exponential boundary layers is considered. For the numerical treatment of these problems, first we employ a second-order Taylor’s series approximation on the terms containing shift parameters and obtain a modified singularly perturbed problem which approximates the original problem. Then a hybrid finite difference scheme on an appropriate piecewise-uniform Shishkin mesh is constructed to discretize the modified problem. Further, we proved that the method is almost second-order ɛ-uniformly convergent in the maximum norm. Numerical experiments are considered to illustrate the theoretical results. In addition, the effect of the shift parameters on the layer behavior of the solution is also examined.
Highlights
Many real-life phenomena in different fields of science are modeled mathematically by delay differential or differentialdifference equations (DDEs)
DDEs are prominent in describing several aspects of infectious disease dynamics such as primary infection, drug therapy, and immune response
If we restrict the class of DDEs in which the highest derivative is multiplied by a small parameter, it is said to be a singularly perturbed differential-difference equations (SPDDEs) [3]
Summary
Many real-life phenomena in different fields of science are modeled mathematically by delay differential or differentialdifference equations (DDEs). E papers by Ria and Sharma [22,23,24,25] are the first and the only notable works in the treatment of such problems when the solutions exhibit both interior and boundary layers, where the authors used a fitted mesh and fitted operator methods and obtained an almost first-order uniform convergence. We consider the following second-order linear singularly perturbed differential-difference problem containing mixed shifts and with a turning point at x 0:. Roughout this paper, M (sometimes subscripted) denotes a generic positive constant independent of the singular perturbation parameter ε and in the case of discrete problems, independent of the mesh parameter N. e maximum norm (i.e., ‖f‖ max− 1≤x≤1|f(x)|) is used for studying the convergence of the approximate solution to the exact solution of the problem
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