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A new numerical algorithm for time-dependent singularly perturbed differential-difference convection–diffusion equation arising in computational neuroscience

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Abstract

In this paper, we study and construct a higher order numerical algorithm for singularly perturbed differential-difference convection–diffusion equation with retarded term appearing in computational biological science. The solution of the considered class of problems may exhibit a boundary layer due to the presence of the perturbation parameter and retarded term. We have discussed the analytical behaviour of the exact solution and its partial derivatives. We have used Taylor’s series expansion to approximate the advance and delay terms of the model problem and then the problem is discretized using Crank–Nicolson’s method on equidistant mesh in the time direction and modified cubic B-spline basis functions on generalized Shishkin mesh in the spatial direction. We have also shown that the developed algorithm is unconditionally stable. The proposed numerical algorithm is proved to be \(\varepsilon \)-uniformly convergent of order four in the spatial direction up to a logarithmic factor and second-order convergent in the time direction. To demonstrate the accuracy and to validate the theoretical results of the proposed numerical algorithm, we have presented three numerical experiments. We have also compared our method with existing schemes in the literature to prove the accuracy of the proposed numerical algorithm.

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Acknowledgements

The authors are thankful to the anonymous reviewers for their careful reading and valuable comments/suggestions which improved the organization and quality of the manuscript. First author is also thankful to CSIR for providing Senior Research Fellowship with file number (09/466(0193)/2018 EMR-I).

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  1. Department of Mathematics, Jamia Millia Islamia, New Delhi, 110025, India

    Mohammad Prawesh Alam & Arshad Khan

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  1. Mohammad Prawesh Alam
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Alam, M.P., Khan, A. A new numerical algorithm for time-dependent singularly perturbed differential-difference convection–diffusion equation arising in computational neuroscience. Comp. Appl. Math. 41, 402 (2022). https://doi.org/10.1007/s40314-022-02102-y

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