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A spectral approach to non-linear weakly singular fractional integro-differential equations

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Abstract

In this work, a class of non-linear weakly singular fractional integro-differential equations is considered, and we first prove existence, uniqueness, and smoothness properties of the solution under certain assumptions on the given data. We propose a numerical method based on spectral Petrov-Galerkin method that handling to the non-smooth behavior of the solution. The most outstanding feature of our approach is to evaluate the approximate solution by means of recurrence relations despite solving complex non-linear algebraic system. Furthermore, the well-known exponential accuracy is established in \(L^{2}\)-norm, and we provide some examples to illustrate the theoretical results and the performance of the proposed method.

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Acknowledgements

This work is funded by national funds through the FCT - Fundação para a Ciência e a Tecnologia, I.P., under the scope of the projects UIDB/00297/2020 and UIDP/00297/2020 (Center for Mathematics and Applications).

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Basic Sciences, Sahand University of Technology, Tabriz, Iran

    Amin Faghih

  2. Center for Mathematics and Applications (NovaMath) and Department of Mathematics, FCT NOVA, Quinta da Torre, 2829-516, Caparica, Portugal

    Magda Rebelo

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  1. Amin Faghih
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  2. Magda Rebelo
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Correspondence to Magda Rebelo.

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Faghih, A., Rebelo, M. A spectral approach to non-linear weakly singular fractional integro-differential equations. Fract Calc Appl Anal 26, 370–398 (2023). https://doi.org/10.1007/s13540-022-00113-4

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  • DOI: https://doi.org/10.1007/s13540-022-00113-4

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