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Journal of Scientific Computing

Erschienen in:

12.06.2019

A Fractional Spectral Collocation for Solving Second Kind Nonlinear Volterra Integral Equations with Weakly Singular Kernels

verfasst von: Haotao Cai

Erschienen in: Journal of Scientific Computing | Ausgabe 3/2019

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Abstract

The classical integer-order Jacobi spectral methods for solving second kind nonlinear Volterra integral equations with weakly singular kernels may cause a low-order accuracy in numerically approximating the exact solution. To overcome the shortcomings, we in this paper present a fractional spectral collocation method for solving weakly singular nonlinear Volterra integral equations. Based on the behavior of the original solution near the initial point of integration, we construct the fractional interpolation basis in the collocation method, and then develop an easily implementing technique to approximate the entry with one-fold integral in the resulting nonlinear system produced by the fractional spectral method. Consequently, we establish that both the semi-discrete and the fully discrete nonlinear systems have a unique solution for sufficiently large n, respectively, where \(n+1\) denotes the dimension of the approximate space. We also ensure that two approximate solutions produced by both the semi-discrete and the fully discrete method arrive at the quasi-optimal convergence order in the infinite norm. At last, numerical examples are given to confirm the theoretical results.

Vorheriger Artikel A Multi-modes Monte Carlo Interior Penalty Discontinuous Galerkin Method for the Time-Harmonic Maxwell’s Equations with Random Coefficients

Nächster Artikel Diagnosing Forward Operator Error Using Optimal Transport

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Titel: A Fractional Spectral Collocation for Solving Second Kind Nonlinear Volterra Integral Equations with Weakly Singular Kernels
verfasst von: Haotao Cai
Publikationsdatum: 12.06.2019
Verlag: Springer US
Erschienen in: Journal of Scientific Computing / Ausgabe 3/2019
Print ISSN: 0885-7474
Elektronische ISSN: 1573-7691
DOI: https://doi.org/10.1007/s10915-019-00987-2

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