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Wavelet-based approximation with nonstandard finite difference scheme for singularly perturbed partial integrodifferential equations

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Abstract

A non-standard finite difference scheme with Haar wavelet basis functions is constructed for the convection–diffusion type singularly perturbed partial integrodifferential equations. The scheme comprises the Crank–Nicolson time semi-discretization followed by the Haar wavelet approximation in the spatial direction. The presence of the perturbation parameter leads to a boundary layer in the solution’s vicinity of \(x = 1.\) The Shishkin mesh is constructed to resolve the boundary layer. The method is proved to be parameter-uniform convergent of order two in the \(L^2\)-norm through meticulous error analysis. Compared to the recent methods developed to solve such problems, the present method is a boundary layer resolving, fast, and elegant.

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Acknowledgements

The authors are grateful to the unknown reviewers for their insightful observations leading to the improvement of the manuscript. The first author expresses his sincere thanks to DST-SERB, New Delhi, for providing financial support (award letter No: MTR/2018/000784) under the MATRICS scheme, and the second author is thankful to CSIR, New Delhi, India (award letter No. 09/719(0096)/2019-EMR-I). The third author thanks UGC, New Delhi, India (award letter No. 1078/(CSIR-UGC NET JUNE 2019)) for providing financial support.

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  1. Department of Mathematics, Birla Institute of Technology and Science, Pilani, Rajasthan, 333031, India

    Devendra Kumar, Komal Deswal & Satpal Singh

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  1. Devendra Kumar
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  2. Komal Deswal
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  3. Satpal Singh
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Correspondence to Devendra Kumar.

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Communicated by Hui Liang.

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Kumar, D., Deswal, K. & Singh, S. Wavelet-based approximation with nonstandard finite difference scheme for singularly perturbed partial integrodifferential equations. Comp. Appl. Math. 41, 341 (2022). https://doi.org/10.1007/s40314-022-02053-4

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