Solving singularly perturbed delay differential equations via fitted mesh and exact difference method

Woldaregay, Mesfin Mekuria

doi:10.1080/27684830.2022.2109301

Cited by 11 publications

(6 citation statements)

References 23 publications

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“…Due to the appearance of the boundary layer in the solution of a singularly perturbed differential equation, classical numerical methods on equidistant grids are inadequate and fail to provide a reliable approximation, when the perturbation parameter tends to zero, unless otherwise, if one uses an unacceptably large number of grid points. Several articles have been written on the solution method for singularly perturbed delay differential equations, to cite a few [9][10][11][12][13]. Among the recently conducted studies on SPTDDE of the convection-diffusion type, having a right end boundary layer, in [14], the authors used an implicit-trapezoidal scheme on uniform mesh for temporal discretization, and for spatial discretization, a hybrid scheme, which is a combination of the midpoint upwind scheme and the central difference scheme on Shishkin type meshes, is applied.…”

Section: Introductionmentioning

confidence: 99%

Hybrid Fitted Numerical Scheme for Singularly Perturbed Convection-Diffusion Problem with a Small Time Lag

Ayele

Tiruneh

Derese

2023

Abstract and Applied Analysis

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In this article, a singularly perturbed convection-diffusion problem with a small time lag is examined. Because of the appearance of a small perturbation parameter, a boundary layer is observed in the solution of the problem. A hybrid scheme has been constructed, which is a combination of the cubic spline method in the boundary layer region and the midpoint upwind scheme in the outer layer region on a piecewise Shishkin mesh in the spatial direction. For the discretization of the time derivative, the Crank-Nicolson method is used. Error analysis of the proposed method has been performed. We found that the proposed scheme is second-order convergent. Numerical examples are given, and the numerical results are in agreement with the theoretical results. Comparisons are made, and the results of the proposed scheme give more accurate solutions and a higher rate of convergence as compared to some previous findings available in the literature.

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Section: Introductionmentioning

confidence: 99%

Hybrid Fitted Numerical Scheme for Singularly Perturbed Convection-Diffusion Problem with a Small Time Lag

Ayele

Tiruneh

Derese

2023

Abstract and Applied Analysis

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“…The authors accomplished derivation of the method by including a fitted factor in the constructed numerical scheme which is based on a special type of grids, and the fitted factor is evaluated applying the theory of singular perturbation. Woldaregay [8] solved singularly perturbed differential equations with delays by developing three different schemes. The author used nonstandard mid-point upwind finite difference method on uniform grids, a standard mid-point upwind finite difference method on Shiskin's meshes and a nonstandard mid-point upwind finite difference method on Shiskin's meshes.…”

Section: Introductionmentioning

confidence: 99%

Parameter-Robust Numerical Scheme for Computing Singularly Perturbed Differential Equations with Mixed Large Shifts

Ejere

2023

Preprint

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This study focuses on the formulation and analysis of a numerical method to compute singularly perturbed functional differential equations (SP-FDEs) involving large mixed shifts. SP-FDEs are a class of differential equations with both shifts and small perturbation parameter. Such equations arise in various scientific fields to model phenomena with memory effects and perturbed dynamics. In this particular study, the considered SP-FDEs involve large negative shift (delay) and large positive shift (advance). Dealing with large mixed shifts in SP-FDEs gives rise to challenges in terms of stability analysis, numerical methods, and convergence of a solution. In this paper, the influence of the large mixed shifts is treated by choosing a uniform mesh discretized in such a way that the term with the shifts lie on the nodal points. Then, using the Numerov’s finite difference method, the continuous problem is transformed into a finite difference scheme and the influence of the perturbation parameter is handled by introducing a fitting factor in the difference scheme. Stability estimate and convergence analysis of the developed scheme are investigated and proved. Numerical experiments are carried out to demonstrate the validity and applicability of the developed scheme. It is shown that the numerical scheme obtained in this work is parameter-robust of second order convergence rate.

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“…Te nonstandard FDM on Shishkin mesh has better accuracy and order of convergence than the equivalent standard FDM on Shishkin mesh [20]. In addition, the uniform convergence analysis of the nonstandard FDMs were restricted to uniform mesh [7].…”

Section: Introductionmentioning

confidence: 99%

“…In [21], He and Wang developed a new form of the nonstandard FDM for solving a convection dominated difusion equation. In the study of [20], Woldaregay developed numerical schemes using the nonstandard FDM on uniform mesh and Shishkin mesh for solving singularly perturbed boundary value problem. Motivated by the results in [20,21], in this paper we proposed a uniformly convergent numerical scheme for solving the problem in (1).…”

Section: Introductionmentioning

confidence: 99%

“…In the study of [20], Woldaregay developed numerical schemes using the nonstandard FDM on uniform mesh and Shishkin mesh for solving singularly perturbed boundary value problem. Motivated by the results in [20,21], in this paper we proposed a uniformly convergent numerical scheme for solving the problem in (1). Te proposed scheme uses the Crank-Nicolson method for time derivative discretization and the midpoint upwind nonstandard FDM on the uniform mesh and the Shishkin mesh for the spatial derivative discretization.…”

Section: Introductionmentioning

confidence: 99%

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Boundary Layer Resolving Exact Difference Scheme for Solving Singularly Perturbed Convection-Diffusion-Reaction Equation

Woldaregay

Duressa

2022

Mathematical Problems in Engineering

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This paper considers the numerical treatment of singularly perturbed time-dependent convection-diffusion-reaction equation. The diffusion term of the equation is multiplied by a small perturbation parameter (ε), which takes an arbitrary value in the interval (0, 1]. For small values of ε, the solution of the equation exhibits an exponential boundary layer which makes it difficult to solve it analytically or using classical numerical methods. We proposed numerical schemes using the Crank–Nicolson method in time derivative discretization and the nonstandard finite difference method (exact finite difference method) in space derivative discretization on a uniform and piecewise uniform Shishkin mesh. The existence of unique discrete solutions and the stability of the schemes are discussed and proved. Uniform convergence of the schemes is proved. The formulated schemes converge uniformly with linear order of convergence. The method on Shishkin mesh possesses boundary layer resolving property. We validated the methods by considering two numerical examples for different values of ε and mesh length.

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Solving singularly perturbed delay differential equations via fitted mesh and exact difference method

Cited by 11 publications

References 23 publications

Hybrid Fitted Numerical Scheme for Singularly Perturbed Convection-Diffusion Problem with a Small Time Lag

Hybrid Fitted Numerical Scheme for Singularly Perturbed Convection-Diffusion Problem with a Small Time Lag

Parameter-Robust Numerical Scheme for Computing Singularly Perturbed Differential Equations with Mixed Large Shifts

Boundary Layer Resolving Exact Difference Scheme for Solving Singularly Perturbed Convection-Diffusion-Reaction Equation

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