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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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
Keywords
- Weakly singular fractional integro-differential equation
- Caputo derivative operator
- Generalized Jacobi polynomials
- Spectral Petrov-Galerkin method
- Convergence