Abstract
This paper presents the study of singularly perturbed differential-difference equations of delay and advance parameters. The proposed numerical scheme is a fitted fourth-order finite difference approximation for the singularly perturbed differential equations at the nodal points and obtained a tridiagonal scheme. This is significant because the proposed method is applicable for the perturbation parameter which is less than the mesh size , where most numerical methods fail to give good results. Moreover, the work can also help to introduce the technique of establishing and making analysis for the stability and convergence of the proposed numerical method, which is the crucial part of the numerical analysis. Maximum absolute errors range from 10 − 03 up to 10 − 10 , and computational rate of convergence for different values of perturbation parameter, delay and advance parameters, and mesh sizes are tabulated for the considered numerical examples. Concisely, the present method is stable and convergent and gives more accurate results than some existing numerical methods reported in the literature.
Highlights
Perturbed differential-difference equations (SPDDEs) occur frequently in the mathematical modeling of various physical and biological phenomena, for example, control theory, viscous elasticity, and population dynamics [1]
Reference [2] investigated that the nonlinear thermal radiation and dissipation with the Darcy-Forchheimer equation in the porous medium analysis by using the fifth-order Runge-Kutta method, [3] discussed the cross-fluid flow containing gyrotactic microorganisms and nanoparticles on a horizontal and three-dimensional cylinder by using the Runge-Kutta Fehlberg fifth-order technique, [4] studied the three-dimensional convective heat transfer of magnetohydrodynamics nanofluid flow through a rotating cone by using the fifth-order Runge-Kutta method
Reference [5] discussed that the heat transfer hybrid nanofluid contains 1-butanol as the base fluid and MoS2–Fe3O4 hybrid nanoparticles by using the finite element method
Summary
Perturbed differential-difference equations (SPDDEs) occur frequently in the mathematical modeling of various physical and biological phenomena, for example, control theory, viscous elasticity, and population dynamics [1]. Many researchers have started developing different numerical methods for solving differential equations. Reference [5] discussed that the heat transfer hybrid nanofluid contains 1-butanol as the base fluid and MoS2–Fe3O4 hybrid nanoparticles by using the finite element method. In these papers, the influence of various parameters on velocity profile and temperature has been investigated. We present a stable, convergent, and more accurate exponentially fitted fourth-order numerical scheme for solving SPDDEs and investigate the influence of delay and advance parameters on the solution profile
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