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Numerical Treatment for the Class of Time Dependent Singularly Perturbed Parabolic Problems with General Shift Arguments

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Abstract

In this paper we design two numerical schemes for solving a class of time dependent singularly perturbed parabolic convection–diffusion problems with general shift arguments in the reaction term. The discretization in both the directions is based on finite difference scheme. Special type of mesh and interpolation is used to tackle the terms containing shifts. The earlier numerical schemes for the considered problem are restricted to the case of small delay and advance arguments while in practical situations these shift arguments can be of arbitrary size (i.e., may be big or small enough in size). In this paper we propose two numerical schemes which work in both the situations i.e., when shifts are big or small enough in size. An extensive amount of analysis is presented to show the linear convergence in space and time of both the schemes. Some numerical results are given to confirm the predicted theory and to show the effect of shifts on the solution.

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Acknowledgments

The work of the first author was supported by U.G.C. (Letter No. F.17-7(J)/08(SA-1) dated 01-Feb-2012) New Delhi, India. The work of the second author was supported by the Research and Development grant scheme 2014–2015 of University of Delhi, Delhi under Grant No. RC/2014/6820.

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Authors and Affiliations

  1. Department of Mathematics, Panjab University, Chandigarh, India

    Komal Bansal

  2. Department of Mathematics, University of Delhi, New Delhi, India

    Pratima Rai

  3. Department of Mathematics, South Asian University, New Delhi, India

    Kapil K. Sharma

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  1. Komal Bansal
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  2. Pratima Rai
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  3. Kapil K. Sharma
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Correspondence to Komal Bansal.

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Bansal, K., Rai, P. & Sharma, K.K. Numerical Treatment for the Class of Time Dependent Singularly Perturbed Parabolic Problems with General Shift Arguments. Differ Equ Dyn Syst 25, 327–346 (2017). https://doi.org/10.1007/s12591-015-0265-7

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  • DOI: https://doi.org/10.1007/s12591-015-0265-7

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