Liouville–Green transformation technique to solve singularly perturbed delay differential equations of reaction diffusion type

Abstract

In this paper, we introduce a numerical method for solving singularly perturbed delay differential equations using Liouville–Green transformation. As an initial step, to approximate the term with negative shift, we use Taylor series, which in turn changes the equation into a singular perturbation problem with the same asymptotic behavior. Then we utilize the Liouville–Green transformation technique to derive a recurrence relation comprising three terms, which can be solved by using the Thomas algorithm. A look into the method's stability and convergence has been done. The MATLAB R2022a mathematical software has been used to obtain the computational results and plots. Almost second‐order accuracy is achieved with the scheme derived. The numerical findings have been compiled into tables and graphs have been provided to visualize the impact of delay on the solution boundary layer and oscillatory behavior. The convergence rate of the present method has been determined theoretically and numerically, and both are shown to agree. In brief, the present method enhances the results of certain previously published numerical approaches.