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

2022/07/13 by Faghih, Amin, Rebelo, Magda
#33C45 #34K37 #45E10 #45J05 #65D15 #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.2207.06113

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 L²-norm, and we provide some examples to illustrate the theoretical results and the performance of the proposed method.

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