Time-local discretization of fractional and related diffusive operators using Gaussian quadrature with applications

This paper investigates the time-local discretization, using Gaussian quadrature, of a class of diffusive operators that includes fractional operators, for application in fractional differential equations and related eigenvalue problems. A discretization based on the Gauss–Legendre quadrature rule i...

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Bibliographic Details
Published in:Applied numerical mathematics 2020-09, Vol.155, p.73-92
Main Authors: Monteghetti, Florian, Matignon, Denis, Piot, Estelle
Format: Article
Language:English
Subjects:
Completely monotone kernel
Diffusive representation
Eigenvalue problems
Fractional calculus
Fractional derivative
Mathematics
Non-classical method
Numerical Analysis
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Summary:This paper investigates the time-local discretization, using Gaussian quadrature, of a class of diffusive operators that includes fractional operators, for application in fractional differential equations and related eigenvalue problems. A discretization based on the Gauss–Legendre quadrature rule is analyzed both theoretically and numerically. Numerical comparisons with both optimization-based and quadrature-based methods highlight its applicability. In addition, it is shown, on the example of a fractional delay differential equation, that quadrature-based discretization methods are spectrally correct, i.e. that they yield an unpolluted and convergent approximation of the essential spectrum linked to the fractional derivative, by contrast with optimization-based methods that can yield polluted spectra whose convergence is difficult to assess.
ISSN:0168-9274
1873-5460
DOI:10.1016/j.apnum.2018.12.003