Abstract
In this paper, we proposed a new class of finite difference schemes for solving singularly perturbed delay differential equation of second order. The proposed schemes are oscillation-free and more accurate than conventional schemes on a uniform mesh. These schemes are easily adaptable on special meshes like Shishkin mesh or Bakhvalov mesh and are uniformly convergent with respect to the perturbation parameter. The error analysis has been carried out and numerical examples are presented to show the accuracy and efficiency of the proposed schemes.
Highlights
Singular perturbation problems (SPPs) arise very frequently in fluid dynamics, elasticity, aerodynamics, plasma dynamics, magneto hydrodynamics, rarefied gas dynamics, oceanography, and other domains of the great world of fluid motion
A more general type of the differential equations, often called functional differential equations, is one in which the unknown function occur with various different arguments
Delay differential equations are similar to ordinary differential equations, but their evolution involves past values of the state variable
Summary
Singular perturbation problems (SPPs) arise very frequently in fluid dynamics, elasticity, aerodynamics, plasma dynamics, magneto hydrodynamics, rarefied gas dynamics, oceanography, and other domains of the great world of fluid motion. In such cases, a more realistic model should include some of the past and the future states of the system; a real system should be modeled by differential equations with delay or advance Such type of equation arises frequently in the mathematical modeling of various practical phenomena, for example, in the modeling of the human pupil-light reflex, model of HIV infection, the study of bistable devices in digital electronics, variational problem in control theory, first exist time problems in modeling of activation of neuronal variability, immune response, mathematical ecology, population dynamics, the modeling of biological oscillators and in a variety of models for physiological process. Subburayan and Ramanujam [16, 17] developed an initial value method for singularly perturbed delay differential equations on Shishkin mesh.
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