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Fractional Calculus and Applied Analysis

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New spectral techniques for systems of fractional differential equations using fractional-order generalized Laguerre orthogonal functions

Ali Bhrawy
  • Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia
  • Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt
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/ Yahia Alhamed
  • Chemical and Materials Engineering Department, Faculty of Engineering, King Abdulaziz University, Jeddah, 21589, Saudi Arabia
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/ Dumitru Baleanu
  • Chemical and Materials Engineering Department, Faculty of Engineering, King Abdulaziz University, Jeddah, 21589, Saudi Arabia
  • Department of Mathematics and Computer Sciences, Cankaya University, Ankara, Turkey
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/ Abdulrahim Al-Zahrani
  • Chemical and Materials Engineering Department, Faculty of Engineering, King Abdulaziz University, Jeddah, 21589, Saudi Arabia
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Published Online: 2014-09-03 | DOI: https://doi.org/10.2478/s13540-014-0218-9

Abstract

Fractional-order generalized Laguerre functions (FGLFs) are proposed depends on the definition of generalized Laguerre polynomials. In addition, we derive a new formula expressing explicitly any Caputo fractional-order derivatives of FGLFs in terms of FGLFs themselves. We also propose a fractional-order generalized Laguerre tau technique in conjunction with the derived fractional-order derivative formula of FGLFs for solving Caputo type fractional differential equations (FDEs) of order ν (0 < ν < 1). The fractional-order generalized Laguerre pseudo-spectral approximation is investigated for solving nonlinear initial value problem of fractional order ν. The extension of the fractional-order generalized Laguerre pseudo-spectral method is given to solve systems of FDEs. We present the advantages of using the spectral schemes based on FGLFs and compare them with other methods. Several numerical example are implemented for FDEs and systems of FDEs including linear and nonlinear terms. We demonstrate the high accuracy and the efficiency of the proposed techniques.

MSC: 26A33; 34A08; 33C47; 65M70

Keywords: multi-term fractional differential equations; fractional-order generalized Laguerre orthogonal functions; generalized Laguerre polynomials; Tau method; pseudo-spectral methods

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About the article

Published Online: 2014-09-03

Published in Print: 2014-12-01


Citation Information: Fractional Calculus and Applied Analysis, Volume 17, Issue 4, Pages 1137–1157, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.2478/s13540-014-0218-9.

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© 2014 Diogenes Co., Sofia. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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