Convergence analysis of an L1-continuous Galerkin method for nonlinear time-space fractional Schrödinger equations

This paper develops and analyses a finite difference/spectral-Galerkin scheme for the nonlinear fractional Schrödinger equations with Riesz space- and Caputo time-fractional derivatives. The finite difference approximation is used for the discretization of the Caputo fractional derivative and the Le...

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Bibliographic Details
Published inInternational journal of computer mathematics Vol. 98; no. 7; pp. 1420 - 1437
Main Authors Zaky, Mahmoud A., Hendy, Ahmed S.
Format Journal Article
LanguageEnglish
Published Abingdon Taylor & Francis 03.07.2021
Taylor & Francis Ltd
Subjects
Approximation
Convergence
discrete Grönwall inequality
Finite difference method
Galerkin method
Schrodinger equation
Spectra
Spectral approximation of Galerkin form
Spectral methods
theoretical analysis
time-space fractional Schrödinger equation
type scheme
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ISSN0020-7160
1029-0265
DOI10.1080/00207160.2020.1822994

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Summary:This paper develops and analyses a finite difference/spectral-Galerkin scheme for the nonlinear fractional Schrödinger equations with Riesz space- and Caputo time-fractional derivatives. The finite difference approximation is used for the discretization of the Caputo fractional derivative and the Legendre-Galerkin spectral method is used for the spatial approximation. Additionally, by using a proper form of discrete Grönwall inequality, the scheme is proved to be unconditionally stable and convergent with accuracy in time and spectral accuracy in space in case of smooth solutions. Finally, some numerical tests are preformed to distinguish the validity of our theoretical results.
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ISSN:0020-7160
1029-0265
DOI:10.1080/00207160.2020.1822994