Numerical Treatment for a Coupled System of Singularly Perturbed Reaction–Diffusion Equations with Robin Boundary Conditions and Having Boundary and Interior Layers

Abstract

AbstractA system of \(k(\ge 2)\) linear singularly perturbed differential equations of reaction–diffusion type coupled through their reactive terms is considered with Robin type boundary conditions, and the system has discontinuous source terms. The highest order derivative term of each equation is multiplied by a small positive parameter and these parameters are assumed to be different in magnitude, due to which the overlapping and interacting interior and boundary layers may appear in the solution of the considered problem. A numerical scheme involving a central difference scheme for the differential equations and a cubic spline technique for the Robin boundary conditions is developed on an appropriate piecewise-uniform Shishkin mesh. Error analysis is done and the constructed scheme is proved to be almost second-order uniformly convergent with respect to each perturbation parameter. Numerical experiments are conducted to verify the theoretical findings.KeywordsCoupled systemSingular perturbationShishkin meshBakhvalov meshDiscontinuous source termRobin boundary conditionsBoundary layerParameter-uniform convergenceFinite difference schemeInterior layerCubicspline