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A numerical approach for nonlinear time-fractional diffusion equation with generalized memory kernel

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Abstract

In this manuscript, a nonlinear time-fractional diffusion equation with a generalized memory kernel is studied. Initially, the original model problem is linearized by implementing the Newton’s quasilinearization technique. In the time-fractional term, a generalized Caputo derivative is considered and approximated using the non-uniform L1-scheme as the solution has a singularity at \(t=0\). The main contribution of this work is to develop a generalized discrete fractional Grönwall inequality. Thereafter, permitting its use to establish the stability and analyze the error estimate, under a proper regularity condition in the \(L^2\)-norm, and an optimal convergence order \(\mathcal {O}\left( N^{-(2-\zeta )}\right) \) is obtained for the L1-scheme with respect to the graded mesh. Numerical results are inserted to corroborate the effectiveness of the theoretical analysis.

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Acknowledgements

The first author would like to express the thanks to Indian Institute of Technology Guwahati, India for funding this project. The authors wish to acknowledge the referees for their valuable comments and suggestions, which helped to improve the presentation.

Funding

This work was supported by the Indian Institute of Technology Guwahati, India.

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Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati, Assam, 781039, India

    Aniruddha Seal & Srinivasan Natesan

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  1. Aniruddha Seal
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  2. Srinivasan Natesan
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Contributions

A.S. implemented the method, prepared the first draft of the manuscript, and implemented the computer codes. S.N. provided the concept, methodology, correcting the manuscript, etc.

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Correspondence to Srinivasan Natesan.

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Seal, A., Natesan, S. A numerical approach for nonlinear time-fractional diffusion equation with generalized memory kernel. Numer Algor (2023). https://doi.org/10.1007/s11075-023-01714-7

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