A uniformly convergent numerical scheme for solving singularly perturbed differential equations with large spatial delay

Abstract

In this study, a parameter-uniform numerical scheme is built and analyzed to treat a singularly perturbed parabolic differential equation involving large spatial delay. The solution of the considered problem has two strong boundary layers due to the effect of the perturbation parameter, and the large delay causes a strong interior layer. The behavior of the layers makes it difficult to solve such problem analytically. To treat the problem, we developed a numerical scheme using the weighted average ($$\theta$$ θ -method) difference approximation on a uniform time mesh and the central difference method on a piece-wise uniform spatial mesh. We established the Stability and convergence analysis for the proposed scheme and obtained that the method is uniformly convergent of order two in the temporal direction and almost second order in the spatial direction. To validate the applicability of the proposed numerical scheme, two model examples are treated and confirmed with the theoretical findings.

Solving singularly perturbed fredholm integro-differential equation using exact finite difference method

“…In general, the solutions to such equations exhibit multi-scale phenomena [ 20 ]. Within certain thin sub-regions of the domain, the scale of some derivatives is significantly larger than other derivatives [ 10 , 27 ]. These thin regions of rapid change are referred to as boundary layers [ 1 ].…”

A Parameter‐Uniform Numerical Scheme for Solving Singularly Perturbed Parabolic Reaction‐Diffusion Problems with Delay in the Spatial Variable

“…Bansal and Sharma [19] and Swaminathan et al [20] solved singularly perturbed parabolic problem involving signifcant delay in the spatial variable by constructing a scheme applying implicit Euler techniques on uniform time grids and central diference method on piecewise uniform spatial grids. In [21], a uniformly convergent numerical scheme has been developed using the ftted mesh method for a time-dependent singularly perturbed delay diferential equation.…”

A robust numerical scheme for singularly perturbed differential equations with spatio-temporal delays

“…Bansal and Sharma [29] formulated a scheme for a singularly perturbed with delay using the implicit Euler on the temporal meshes and standard central difference on nonuniform spatial meshes. Ejere et al [30] developed a numerical scheme for a time-dependent singularly perturbed differential equation with large spatial delays using the weighted-average method in the temporal direction and the central difference method in the spatial direction on piece-wise uniform Shishkin meshes. Alam and Khan [31] proposed a new numerical algorithm for singularly perturbed differential equations involving the shift and the advance parameters.…”