Numerical Method for a Boundary Value Problem for a Linear System of Partially Singularly Perturbed Parabolic Delay Differential Equations of Reaction-Diffusion Type

Abstract

The problem of a partially singularly perturbed boundary value problem for a linear system of reaction-diffusion type parabolic second-order delay differential equations is explored. The boundary value problem is partially perturbed when \(\varepsilon _{m+1}=\dots =\varepsilon _{n}=1\) for some \(m<n\) and the system is partial with respect to delay when the co-efficient function of delay terms \(b_i(x,t)=0\) for some \(i, i=1,2,\dots ,n.\) The components of the solution of this system exhibit boundary layers at \((x,t)=(0,t)\) and \((x,t)=(2,t)\) and interior layers at \((x,t)=(1,t).\) To approximate the solution, a numerical technique is presented that uses a classical finite difference scheme on a Shishkin piecewise-uniform mesh. For all values of singular perturbation parameters, the approach is shown to be uniformly first-order convergent. To corroborate the theoretical results, numerical illustrations are provided.