ETNA - Electronic Transactions on Numerical Analysis
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |
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DATUM, UNTERSCHRIFT / DATE, SIGNATURE
BANK AUSTRIA CREDITANSTALT, WIEN (IBAN AT04 1100 0006 2280 0100, BIC BKAUATWW), DEUTSCHE BANK MÜNCHEN (IBAN DE16 7007 0024 0238 8270 00, BIC DEUTDEDBMUC)
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ETNA - Electronic Transactions on Numerical Analysis ISBN 978-3-7001-8258-0 Online Edition Research Article
Shervan Erfani,
Esmail Babolian,
Shahnam Javadi
S. 150 - 175 doi:10.1553/etna_vol54s150 Verlag der Österreichischen Akademie der Wissenschaften doi:10.1553/etna_vol54s150
Abstract: The main purpose of this paper is to introduce generalized fractional pseudospectral integration and differentiation matrices using a family of fractional interpolants, called fractional Lagrange interpolants. We develop novel approaches to the numerical solution of fractional differential equations with a singular behavior at an end-point. To achieve this goal, we present efficient and stable methods based on three-term recurrence relations, generalized barycentric representations, and Jacobi-Gauss quadrature rules to evaluate the corresponding matrices. In a special case, we prove the equivalence of the proposed fractional pseudospectral methods using a suitable fractional Birkhoff interpolation problem. In fact, the fractional integration matrix yields the stable inverse of the fractional differentiation matrix, and the resulting system is well-conditioned. We develop efficient implementation procedures for providing optimal error estimates with accurate convergence rates for the interpolation operators and the proposed schemes in the $L^{2}$-norm. Some numerical results are given to illustrate the accuracy and performance of the algorithms and the convergence rates. Keywords: convergence rate, fractional differential equations, fractional Birkhoff interpolation, fractional pseudospectral methods, fractional Lagrange interpolants, singularity Published Online: 2021/01/21 12:47:05 Object Identifier: 0xc1aa5576 0x003c2afe Rights: . Electronic Transactions on Numerical Analysis (ETNA) is an electronic journal for the publication of significant new developments in numerical analysis and scientific computing. Papers of the highest quality that deal with the analysis of algorithms for the solution of continuous models and numerical linear algebra are appropriate for ETNA, as are papers of similar quality that discuss implementation and performance of such algorithms. New algorithms for current or new computer architectures are appropriate provided that they are numerically sound. However, the focus of the publication should be on the algorithm rather than on the architecture. The journal is published by the Kent State University Library in conjunction with the Institute of Computational Mathematics at Kent State University, and in cooperation with the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences (RICAM). Reviews of all ETNA papers appear in Mathematical Reviews and Zentralblatt für Mathematik. Reference information for ETNA papers also appears in the expanded Science Citation Index. ETNA is registered with the Library of Congress and has ISSN 1068-9613. …
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Verlag der Österreichischen Akademie der Wissenschaften Austrian Academy of Sciences Press
A-1011 Wien, Dr. Ignaz Seipel-Platz 2
Tel. +43-1-515 81/DW 3420, Fax +43-1-515 81/DW 3400 https://verlag.oeaw.ac.at, e-mail: verlag@oeaw.ac.at |