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Parameter uniform numerical scheme for time dependent singularly perturbed convection-diffusion-reaction problems with general shift arguments

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Abstract

In this work, a parameter uniform numerical method is developed to find the approximate solution of time-dependent singularly perturbed convection-diffusion-reaction problems with general shift arguments in the space variable. The earlier work on such type of initial-interval boundary value problems is restricted to the case of small delay and advance arguments while in practical situations the shift arguments can be of arbitrary size (i.e. it may be bigger or small enough in size). The fitted mesh technique to establish parameter uniform error estimates is not extendable for the class of singularly perturbed parabolic partial differential-difference equations (SPPPDDEs) with general shift arguments in the space variable. To observe the dispersive behaviour of the solution of the problem considered in this paper, we use systematically constructed suitable denominator function for the discrete second order derivative. The motivation behind the construction of the scheme is modelling rules for non-standard finite difference methods (NSFDMs), developed by Mickens. The proposed numerical scheme is analysed for consistency and stability. It is proved that the scheme is unconditionally stable and parameter uniform convergent. The scheme is convergent for bigger shift arguments as well as for small shift arguments. The performance of the method is corroborated by numerical examples.

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Authors and Affiliations

  1. Department of Mathematics, (Center for Advance Study in Mathematics), Panjab University, Chandigarh, 160014, India

    Komal Bansal & Kapil K. Sharma

  2. South Asian University, New Delhi, 110021, India

    Kapil K. Sharma

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  1. Komal Bansal
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Correspondence to Kapil K. Sharma.

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Bansal, K., Sharma, K.K. Parameter uniform numerical scheme for time dependent singularly perturbed convection-diffusion-reaction problems with general shift arguments. Numer Algor 75, 113–145 (2017). https://doi.org/10.1007/s11075-016-0199-3

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  • DOI: https://doi.org/10.1007/s11075-016-0199-3

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