An Exponentially Fitted Numerical Scheme via Domain Decomposition for Solving Singularly Perturbed Differential Equations with Large Negative Shift

et al. 2023

Preprint

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

Section: Introductionmentioning

confidence: 99%

A tension spline fitted numerical scheme for singularly perturbed reaction-diffusion problem with negative shift

et al. 2023

Preprint

“…In reference [15], a singularly perturbed delay diferential equation is treated by decomposing the domain into subdomains and obtaining a convergent numerical result. In reference [16], a singularly perturbed Fredholm integrodiferential equation is considered.…”

Section: Introductionmentioning

confidence: 99%

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

International Journal of Mathematics and Mathematical Sciences

et al. 2023

Self Cite

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.

“…Appadu and Tijani [26] treated a one-dimensional generalized Burgers-Huxley equation by proposing two solutions using the classical finite difference scheme and nonstandard finite difference scheme and obtained that one of the proposed solutions contains a minor error. In [27], a singularly perturbed ordinary differential equation with a large negative shift is treated by developing a numerical scheme using the fitted operator method via domain decomposition.…”

Section: Introductionmentioning

confidence: 99%

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

Front. Appl. Math. Stat.

et al. 2023

Self Cite

In this article, we proposed and analyzed a numerical scheme for singularly perturbed differential equations with both spatial and temporal delays. The presence of the perturbation parameter exhibits strong boundary layers, and the large negative shift gives rise to a strong interior layer in the solution. The abruptly changing behaviors of the solution in the layers make it difficult to solve the problem analytically. Standard numerical methods do not give satisfactory results, unless a large mesh number is considered, which needs a massive computational cost. We treated such problem by proposing a numerical scheme using the implicit Euler method in the temporal variable and the nonstandard finite difference method in the spatial variable on uniform meshes. The stability and uniform convergence of the proposed scheme have been investigated and proved. To demonstrate the theoretical results, numerical experiments are carried out. From the theoretical and numerical results, we observed that the method is uniformly convergent of order one in time and of order two in space.