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Computational study for a class of time-dependent singularly perturbed parabolic partial differential equation through tension spline

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Abstract

This article contributes a numerical technique for a class of singularly perturbed time delayed parabolic partial differential equation. A priori results of maximum principle, stability and bounds are discussed. The continuous problem is semi-discretized by the Crank–Nicolson based scheme in the temporal direction and then discretized by the tension spline scheme on non-uniform Shishkin mesh. Error estimation for the discretized problem is derived. To validate the theoretical findings, the numerical outcomes for linear and nonlinear problems are tested.

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References

  • Adomian G, Rach R (1983) Nonlinear stochastic differential delay equations. J Math Anal Appl 91:94–101

    Article  MathSciNet  Google Scholar 

  • Ansari AR, Bakr SA, Shishkin GI (2007) A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations. Appl Math Comput 205:552–566

    Article  MathSciNet  Google Scholar 

  • Aziz T, Khan A, Khan I, Stojanovic M (2004) A variable-mesh approximation method for singularly perturbed boundary-value problems using cubic spline in tension. Int J Comput Math 81:1513–1518

    Article  MathSciNet  Google Scholar 

  • Bashier EBM, Patidar KC (2011) A novel fitted operator finite difference method for a singularly perturbed delay parabolic partial differential equation. Appl Math Comput 217:4728–4739

    MathSciNet  MATH  Google Scholar 

  • Epstein IR (1992) Delay effects and differential delay equations in chemical kinetics. Int Rev Phys Chem 11:135–160

    Article  Google Scholar 

  • Gowrisankar S, Natesan S (2014) A robust numerical scheme for singularly perturbed delay parabolic initial-boundary-value prolems on equidistributed grids. Electron Trans Numer Anal 41:376–395

    MathSciNet  MATH  Google Scholar 

  • Kumar S, Kumar M (2014) High order parameter-uniform discretization for singularly perturbed parabolic partial differential equations with time delay. Comput Math Appl 68:1355–1367

    Article  MathSciNet  Google Scholar 

  • McCartin BJ (1985) Discretization of the semiconductor device equations. In: New problems and new solutions for device and process modelling. Boole Press, Dublin

  • Protter MH, Weinberger HF (1967) Maximum principles in differential equations. Prentice-Hall, Englewood Cliffs

    MATH  Google Scholar 

  • Rao RN, Chakravarthy PP (2014) A fitted numerov method for singularly perturbed parabolic partial differential equation with a small negative shift arising in control theory. Numer Math Theory Methods Appl 7:23–40

    Article  MathSciNet  Google Scholar 

  • Samarskii AA, Vabishchevich PN (1995) Computational heat transfer. Wiley, New York

    Google Scholar 

  • Selvi PA, Ramanujam N (2017) A parameter uniform difference scheme for singularly perturbed parabolic delay differential equation with Robin type boundary condition. Appl Math Comput 296:101–115

    MathSciNet  MATH  Google Scholar 

  • Singh J, Kumar S, Kumar M (2018) A domain decomposition method for solving singularly perturbed parabolic reaction-diffusion problems with time delay. Numer Methods Partial Differ Equ 34:1849–1866

    Article  MathSciNet  Google Scholar 

  • Tikhonov AN, Samarskii AA (1972) Equations of mathematical physics. Nauka, Moscow

    Google Scholar 

  • Van Harten A, Schumacher JM (1978) On a class of partial functional differential equations arising in feedback control theory. North Holland, Amsterdam

    MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Department of Basic Sciences and Humanities, GMR Institute of Technology, Rajam, Srikakulam, Andhra Pradesh, 532 127, India

    P. Murali Mohan Kumar

  2. Department of Mathematics, National Institute of Technology Kurukshetra, Kurukshetra, Haryana, 136 119, India

    P. Murali Mohan Kumar & A. S. V. Ravi Kanth

Authors
  1. P. Murali Mohan Kumar
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  2. A. S. V. Ravi Kanth
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Correspondence to A. S. V. Ravi Kanth.

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Communicated by Frederic Valentin.

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Kumar, P.M.M., Ravi Kanth, A.S.V. Computational study for a class of time-dependent singularly perturbed parabolic partial differential equation through tension spline. Comp. Appl. Math. 39, 233 (2020). https://doi.org/10.1007/s40314-020-01278-5

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  • DOI: https://doi.org/10.1007/s40314-020-01278-5

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