In this study, we focus on the formulation and analysis of an exponentially fitted numerical scheme by decomposing the domain into subdomains to solve singularly perturbed differential equations with large negative shift. The solution of problem exhibits twin boundary layers due to the presence of the perturbation parameter and strong interior layer due to the large negative shift. The original domain is divided into six subdomains, such as two boundary layer regions, two interior (interfacing) layer regions, and two regular regions. Constructing an exponentially fitted numerical scheme on each boundary and interior layer subdomains and combining with the solutions on the regular subdomains, we obtain a second order ε -uniformly convergent numerical scheme. To demonstrate the theoretical results, numerical examples are provided and analyzed.

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.

The objective of this research work is to develop and analyse a numerical scheme for solving singularly perturbed parabolic reaction-diffusion problems with large spatial delay. The presence of the small positive parameter on the term with the highest order of derivative exhibits two strong boundary layers in the solution of the problem, and the large delay term gives rise to a strong interior layer. The layers’ behavior makes it difficult to solve the problem analytically. To treat such a problem, we developed a numerical scheme using the Crank–Nicolson method in the time direction and the central difference method in the spatial direction via nonstandard finite difference methods on uniform meshes. Stability and convergence analyses for the obtained scheme have been established, which show that the developed numerical scheme is uniformly convergent. To confirm the theoretical analysis, model numerical examples are considered and demonstrated.

Objective: A numerical scheme is developed and analyzed for a singularly perturbed reaction-diffusion problem with a negative shift. The influence of the perturbation parameter exhibits boundary layers at the two ends of the domain, and the negative shift causes a strong interior layer. The rapidly changing behavior of the solution in the layers brings significant difficulties in solving the problem analytically. The problem is treated by proposing a numerical scheme using the implicit Euler method in the temporal direction and the spline tension method in the spatial direction with uniform meshes. Result: Error estimate is investigated for the developed numerical scheme. The scheme is demonstrated by numerical examples. The theoretical and numerical results show that the method is uniformly convergent. Mathematics Subject Classification: Primary 65M06; secondary 65M12, 65M15

No abstract