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Uniformly convergent finite difference methods for singularly perturbed parabolic partial differential equations with mixed shifts

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Abstract

In this paper, exponentially fitted finite difference methods are presented for singularly perturbed one-dimensional parabolic partial differential equation with mixed shifts in the spatial variable. Such boundary value problems are ubiquitous in control theory, namely, the processing of metal sheets and the first exist time problems in modelling of activation of neuronal variability. The methods presented here are based upon the size of the shift parameter. When the shift parameters are smaller to the perturbation parameter, the shift terms are approximated by their Taylor series expansions, while for the shift parameters larger to the perturbation parameter, a special type of mesh is constructed so that the shifts lie on the nodal points after discretization. The proposed methods are checked for convergence. Also two test problems are solved with the proposed methods to demonstrate the applicability. Graphs are plotted to illustrate the effect of the shifts on the solution of the problems.

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Acknowledgements

Authors are grateful to the anonymous referees for their valuable suggestions and comments that improved the quality of this paper.

Funding

The authors wish to thank the National Board for Higher Mathematics, Department of Atomic Energy, Government of India, for their financial support under the project No. 02011/8/2021 NBHM(R.P)/R &D II/7224, dated 24.06.2021.

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Authors and Affiliations

  1. Department of Mathematics, School of Advanced Sciences, Vellore Institute of Technology, Vellore, Tamil Nadu, 632 014, India

    T. Prathap & R. Nageshwar Rao

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  1. T. Prathap
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Correspondence to R. Nageshwar Rao.

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Prathap, T., Rao, R.N. Uniformly convergent finite difference methods for singularly perturbed parabolic partial differential equations with mixed shifts. J. Appl. Math. Comput. 69, 1679–1704 (2023). https://doi.org/10.1007/s12190-022-01802-2

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  • DOI: https://doi.org/10.1007/s12190-022-01802-2

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