Fitted Finite Difference Method for Third Order Singularly Perturbed Delay Differential Equations of Convection Diffusion Type

In this paper, an asymptotic numerical method based on a fitted finite difference scheme and the fourth-order Runge-Kutta method with piecewise cubic Hermite interpolation on Shishkin mesh is suggested to solve singularly perturbed boundary value problems for thirdorder ordinary differential equations of convection diffusion type with a delay. An error estimate is derived using the supremum norm and it is of almost first-order convergence. A nonlinear problem is also solved using the Newton's quasi linearization technique and the present asymptotic numerical method. Numerical results are provided to illustrate the theoretical results. Keywords Third-order differential equations • Convection diffusion equation • Boundary value problem • Singularly perturbed problem • Shishkin mesh • Delay differential equations • Asymptotic numerical methods Mathematics Subject Classification 34K10 • 34K26 • 34K28

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“…Proof Refer [Mahendran and Subburayan (2018), Theorem 3.1] An immediate consequence of the above theorem is the following stability result.…”

Section: Stability Resultsmentioning

confidence: 96%

Asymptotic numerical method for third-order singularly perturbed convection diffusion delay differential equations

Subburayan

Mahendran

2020

Comp. Appl. Math.

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“…A subclass of differential equations in which the term with the highest order derivative is multiplied by a small positive parameter (ε) and involves one or more shift arguments is said to be a singularly perturbed differential equation with delay [9]. Such problems frequently arise in the modeling of various physical systems, such as the human pupil-light reflex [10], the study of bistable devices in digital electronics [11], variational problem in control theory [12], immune response modeling [13], mathematical modeling in ecology [14], models to stabilize rotating and frozen waves [15], models for the physiological process [16].…”

Section: Introductionmentioning

confidence: 99%

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

Ejere

Duressa

Woldaregay

et al. 2023

Front. Appl. Math. Stat.

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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.

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“…Chen and Xu 9 proved the stability and accuracy of FEM and SDFEM for singularly perturbed problems. On the other side, the research on the higher‐order finite difference methods is considerably increased compared to FEM and SDFEM 10‐12 . Many authors suggested various types of finite difference schemes for solving the same type of equations.…”

Section: Introductionmentioning

confidence: 99%

An asymptotic streamline diffusion finite element method for singularly perturbed convection‐diffusion delay differential equations with point source

Sethurathinam

Subburayan

Arasamudi

et al. 2021

Comp and Math Methods

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In this article, we presented an asymptotic SDFEM for singularly perturbed convection diffusion type differential difference equations with point source term. First, the solution is decomposed into two functions, among them one is the solution of delay differential equation and other one is the solution of differential equation with point source. Furthermore, using the asymptotic expansion approximation, the delay differential equation is modified as a nondelay differential equations. Streamline diffusion finite element methods are applied to approximate the solutions of the two problems. We prove that the present method gives an almost second‐order convergence in maximum norm and square integrable norm, whereas first‐order convergence in H1 norm. Numerical results are presented to validate the theoretical results.

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Fitted Finite Difference Method for Third Order Singularly Perturbed Delay Differential Equations of Convection Diffusion Type

Cited by 11 publications

References 8 publications

Asymptotic numerical method for third-order singularly perturbed convection diffusion delay differential equations

Asymptotic numerical method for third-order singularly perturbed convection diffusion delay differential equations

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

An asymptotic streamline diffusion finite element method for singularly perturbed convection‐diffusion delay differential equations with point source

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