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

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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.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Ali H. Bhrawy

  2. Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

    Ali H. Bhrawy

  3. Chemical and Materials Engineering Department, Faculty of Engineering, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Yahia A. Alhamed, Dumitru Baleanu & Abdulrahim A. Al-Zahrani

  4. Department of Mathematics and Computer Sciences, Cankaya University, Ankara, Turkey

    Dumitru Baleanu

Authors
  1. Ali H. Bhrawy
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  2. Yahia A. Alhamed
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  3. Dumitru Baleanu
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  4. Abdulrahim A. Al-Zahrani
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Correspondence to Ali H. Bhrawy.

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Bhrawy, A.H., Alhamed, Y.A., Baleanu, D. et al. New spectral techniques for systems of fractional differential equations using fractional-order generalized Laguerre orthogonal functions. Fract Calc Appl Anal 17, 1137–1157 (2014). https://doi.org/10.2478/s13540-014-0218-9

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  • DOI: https://doi.org/10.2478/s13540-014-0218-9

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