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

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Title: Time-local discretization of fractional and related diffusive operators using Gaussian quadrature with applications.
Authors: Monteghetti, Florian1 (AUTHOR) florian.monteghetti@onera.fr, Matignon, Denis2 (AUTHOR) denis.matignon@isae.fr, Piot, Estelle1 (AUTHOR) estelle.piot@onera.fr
Source: Applied Numerical Mathematics. Sep2020, Vol. 155, p73-92. 20p.
Subjects: Fractional differential equations, Laplacian operator, Delay differential equations, Quadrature domains, Discretization methods, Eigenvalue equations, Fractional calculus
Abstract: 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. [ABSTRACT FROM AUTHOR]
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Database: Engineering Source
Description
Abstract: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. [ABSTRACT FROM AUTHOR]
ISSN:01689274
DOI:10.1016/j.apnum.2018.12.003