Numerical investigation of time delay parabolic differential equation involving two small parameters

Sahu, Subal Ranjan; Mohapatra, Jugal

doi:10.1108/ec-07-2020-0369

Cited by 13 publications

(2 citation statements)

References 18 publications

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“…For SPPDEs with two small parameters, in (Govindarao et al ., 2019), the authors used a combination of implicit Euler in time and upwind scheme in space, on a uniform mesh and S-mesh, respectively. For the same class of problems, Sahu and Mohapatra have used the hybrid scheme in space on both the S-mesh and Bakhvalov-Shishkin mesh (B-S-mesh) along with the implicit Euler scheme in the time direction on a uniform mesh (Sahu and Mohapatra, 2021b). Recently, Priyadarshana et al., have used the Richardson extrapolation technique to obtain a global second-order accuracy over three different layer resolving meshes (Priyadarshana et al ., 2022).…”

Section: Introductionmentioning

confidence: 99%

Uniformly convergent computational method for singularly perturbed time delayed parabolic differential-difference equations

Mohapatra

Priyadarshana

Reddy

2023

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PurposeThe purpose of this work is to introduce an efficient, global second-order accurate and parameter-uniform numerical approximation for singularly perturbed parabolic differential-difference equations having a large lag in time.Design/methodology/approachThe small delay and advance terms in spatial direction are handled with Taylor's series approximation. The Crank–Nicholson scheme on a uniform mesh is applied in the temporal direction. The derivative terms in space are treated with a hybrid scheme comprising the midpoint upwind and the central difference scheme at appropriate domains, on two layer-resolving meshes namely, the Shishkin mesh and the Bakhvalov–Shishkin mesh. The computational effectiveness of the scheme is enhanced by the use of the Thomas algorithm which takes less computational time compared to the usual Gauss elimination.FindingsThe proposed scheme is proved to be second-order accurate in time and to be almost second-order (up to a logarithmic factor) uniformly convergent in space, using the Shishkin mesh. Again, by the use of the Bakhvalov–Shishkin mesh, the presence of a logarithmic effect in the spatial-order accuracy is prevented. The detailed analysis of the convergence of the fully discrete scheme is thoroughly discussed.Research limitations/implicationsThe use of second-order approximations in both space and time directions makes the complete finite difference scheme a robust approximation for the considered class of model problems.Originality/valueTo validate the theoretical findings, numerical simulations on two different examples are provided. The advantage of using the proposed scheme over some existing schemes in the literature is proved by the comparison of the corresponding maximum absolute errors and rates of convergence.

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Section: Introductionmentioning

confidence: 99%

Uniformly convergent computational method for singularly perturbed time delayed parabolic differential-difference equations

Mohapatra

Priyadarshana

Reddy

2023

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“…Kumar and Kumari [22] treated singularly perturbed reaction-diffusion problems with large delay by developing a parameter-uniform numerical scheme using the Crank-Nicolson method in time direction and the central finite difference method in the spatial direction. Singularly perturbed differential equations involving temporal delay are well solved in various research works, some of which can be referred in [23][24][25][26] for the detail discussions.…”

Section: Introductionmentioning

confidence: 99%

A uniformly convergent numerical scheme for solving singularly perturbed differential equations with large spatial delay

et al. 2022

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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.

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An Improved Numerical Scheme for Semilinear Singularly Perturbed Parabolic Delay Differential Equations

Mohapatra

Priyadarshana

2023

Lecture Notes in Networks and Systems

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Numerical investigation of time delay parabolic differential equation involving two small parameters

Cited by 13 publications

References 18 publications

Uniformly convergent computational method for singularly perturbed time delayed parabolic differential-difference equations

Uniformly convergent computational method for singularly perturbed time delayed parabolic differential-difference equations

A uniformly convergent numerical scheme for solving singularly perturbed differential equations with large spatial delay

An Improved Numerical Scheme for Semilinear Singularly Perturbed Parabolic Delay Differential Equations

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