A robust method of lines solution for singularly perturbed delay parabolic problem

Mbroh, Nana Adjoah; Noutchie, Suares Clovis Oukouomi; Massoukou, Rodrigue Yves M’pika

doi:10.1016/j.aej.2020.03.042

Cited by 14 publications

(4 citation statements)

References 27 publications

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“…At (p + 1) st time step, we substitute (22) for time derivative and average values between (x, t p ) and (x, t P+1 ) for spatial derivatives into (20) and we obtain…”

Section: Semidiscretizationmentioning

confidence: 99%

“…In other models of physical problems like heat and mass transfer, control theory, and chemical processes, the system may depend on the present and past history of the state which yields a delay term in the diferential equation. Such singularly perturbed time delay parabolic problems with uniperturbation parameter are addressed by some literature [17][18][19][20][21][22][23]. Besides solving singularly perturbed diferential equations, obtaining higher-order and robust numerical solutions is the main devotion of researchers [24][25][26][27][28][29][30].…”

Section: Introductionmentioning

confidence: 99%

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A Numerical Approach for Diffusion‐Dominant Two‐Parameter Singularly Perturbed Delay Parabolic Differential Equations

Worku,

Duressa

2023

International Journal of Mathematics and Mathematical Sciences

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A numerical scheme is developed to solve a large time delay two-parameter singularly perturbed one-dimensional parabolic problem in a rectangular domain. Two small parameters multiply the convective and diffusive terms, which determine the width of the left and right lateral surface boundary layers. Uniform mesh and piece-wise uniform Shishkin mesh discretization are applied in time and spatial dimensions, respectively. The numerical scheme is formulated by using the Crank–Nicolson method on two consecutive time steps and the average central finite difference approximates in spatial derivatives. It is proved that the method is uniformly convergent, independent of the perturbation parameters, where the convection term is dominated by the diffusion term. The resulting scheme is almost second-order convergent in the spatial direction and second-order convergent in the temporal direction. Numerical experiments illustrate theoretical findings, and the method provides more accurate numerical solutions than prior literature.

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“…At (p + 1) st time step, we substitute (22) for time derivative and average values between (x, t p ) and (x, t P+1 ) for spatial derivatives into (20) and we obtain…”

Section: Semidiscretizationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Numerical Approach for Diffusion‐Dominant Two‐Parameter Singularly Perturbed Delay Parabolic Differential Equations

Worku,

Duressa

2023

International Journal of Mathematics and Mathematical Sciences

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“…N. A. Mbroh et. al [41] have designed a numerical discretization using fitted operator finite difference method on spatially direction and Crank Nicolson finite difference approach on time direction. S. Yadav and P. Rai [62] have constructed a higher-order difference method consisting of hybrid scheme on Shishkin mesh and implicit Euler method on a uniform mesh to examine singularly perturbed delay parabolic turning point problems of convection-diffusion type.…”

Section: Introductionmentioning

confidence: 99%

A second-order numerical method for pseudo-parabolic equations having both layer behavior and delay argument

Güneş,

Duru

2024

Commun. Fac. Sci. Univ. Ank. Ser. A1 Math. Stat.

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In this paper, singularly perturbed pseudo-parabolic initial-boundary value problems with time-delay parameter are considered by numerically. Initially, the asymptotic properties of the analytical solution are investigated. Then, a discretization with exponential coefficient is suggested on a uniform mesh. The error approximations and uniform convergence of the presented method are estimated in the discrete energy norm. Finally, some numerical experiments are given to clarify the theory.

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“…Most of the scholars studied on time delayed singularly perturbed problems. Mbroh et al [3] proposed parameter uniform method for solving a time delay nonautonomous singularly perturbed parabolic differential equation. Clavero and Gracia [4] studied singularly perturbed time-dependent problem of reaction-diffusion type using Richardson extrapolation technique.…”

Section: Introductionmentioning

confidence: 99%

Numerical Treatment on Parabolic Singularly Perturbed Differential Difference Equation via Fitted Operator Scheme

Tefera¹,

Tiruneh²,

Derese³

2021

Abstract and Applied Analysis

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This paper proposes a new fitted operator strategy for solving singularly perturbed parabolic partial differential equation with delay on the spatial variable. We decomposed the problem into three piecewise equations. The delay term in the equation is expanded by Taylor series, the time variable is discretized by implicit Euler method, and the space variable is discretized by central difference methods. After developing the fitting operator method, we accelerate the order of convergence of the time direction using Richardson extrapolation scheme and obtained O h 2 + k 2 uniform order of convergence. Finally, three examples are given to illustrate the effectiveness of the method. The result shows the proposed method is more accurate than some of the methods that exist in the literature.

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A robust method of lines solution for singularly perturbed delay parabolic problem

Cited by 14 publications

References 27 publications

A Numerical Approach for Diffusion‐Dominant Two‐Parameter Singularly Perturbed Delay Parabolic Differential Equations

A Numerical Approach for Diffusion‐Dominant Two‐Parameter Singularly Perturbed Delay Parabolic Differential Equations

A second-order numerical method for pseudo-parabolic equations having both layer behavior and delay argument

Numerical Treatment on Parabolic Singularly Perturbed Differential Difference Equation via Fitted Operator Scheme

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