Higher-Order Numerical Method for Singularly Perturbed Delay Reaction-Diffusion Problems

Abstract

In this paper, a higher-order numerical method is presented for solving the singularly perturbed delay differential equations. Such kind of equations have a delay parameter on reaction term and exhibits twin boundary layers or oscillatory behavior. Recently, different numerical methods have been developed to solve the singularly perturbed delay reaction-diffusion problems. However, the obtained accuracy and its rate of convergence are satisfactory. Thus, to solve the considered problem with more satisfactory accuracy and a higher rate of convergence, the higher-order numerical method is presented. First, the given singularly perturbed delay differential equation is transformed to asymptotically equivalent singularly perturbed two-point boundary value convection-diffusion differential equation by using Taylor series approximations. Then, the constructed singularly perturbed boundary value differential equation is replaced by three-term recurrence relation finite difference approximations. The Richardson extrapolation technique is applied to accelerate the fourth-order convergent of the developed method to the sixth-order convergent. The consistency and stability of the formulated method have been investigated very well to guarantee the convergence of the method. The rate of convergence for both the theoretical and numerical have been proven and are observed to be in accord with each other. To demonstrate the efficiency of the method, different model examples have been considered and simulation of numerical results have been presented by using MATLAB software. Numerical experimentation has been done and the results are presented for different values of the parameters. Further, The obtained numerical results described that the finding of the present method is more accurate than the findings of some methods discussed in the literature.