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A New Legendre Spectral Galerkin and Pseudo‐Spectral Approximations for Fractional Initial Value Problems

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  • A. H. Bhrawy
  • M. A. Alghamdi

Abstract

We extend the application of the Galerkin method for treating the multiterm fractional differential equations (FDEs) subject to initial conditions. A new shifted Legendre‐Galerkin basis is constructed which satisfies exactly the homogeneous initial conditions by expanding the unknown variable using a new polynomial basis of functions which is built upon the shifted Legendre polynomials. A new spectral collocation approximation based on the Gauss‐Lobatto quadrature nodes of shifted Legendre polynomials is investigated for solving the nonlinear multiterm FDEs. The main advantage of this approximation is that the solution is expanding by a truncated series of Legendre‐Galerkin basis functions. Illustrative examples are presented to ensure the high accuracy and effectiveness of the proposed algorithms are discussed.

Suggested Citation

  • A. H. Bhrawy & M. A. Alghamdi, 2013. "A New Legendre Spectral Galerkin and Pseudo‐Spectral Approximations for Fractional Initial Value Problems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    • Handle: RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:306746
    DOI: 10.1155/2013/306746
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    References listed on IDEAS

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    1. D. Baleanu & A. H. Bhrawy & T. M. Taha, 2013. "Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    2. B. Bandrowski & A. Karczewska & P. Rozmej, 2012. "Numerical Solutions to Fractional Perturbed Volterra Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    3. Adel Al-Rabtah & Shaher Momani & Mohamed A. Ramadan, 2012. "Solving Linear and Nonlinear Fractional Differential Equations Using Spline Functions," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    4. D. Baleanu & A. H. Bhrawy & T. M. Taha, 2013. "Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-10, June.
    5. Mohammad Maleki & Ishak Hashim & Majid Tavassoli Kajani & Saeid Abbasbandy, 2012. "An Adaptive Pseudospectral Method for Fractional Order Boundary Value Problems," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
    6. B. Bandrowski & A. Karczewska & P. Rozmej, 2012. "Numerical Solutions to Fractional Perturbed Volterra Equations," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-19, December.
    7. Adel Al-Rabtah & Shaher Momani & Mohamed A. Ramadan, 2012. "Solving Linear and Nonlinear Fractional Differential Equations Using Spline Functions," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-9, March.
    8. Mohammad Maleki & Ishak Hashim & Majid Tavassoli Kajani & Saeid Abbasbandy, 2012. "An Adaptive Pseudospectral Method for Fractional Order Boundary Value Problems," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-19, December.
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    Cited by:

    1. Fukang Yin & Junqiang Song & Yongwen Wu & Lilun Zhang, 2013. "Numerical Solution of the Fractional Partial Differential Equations by the Two‐Dimensional Fractional‐Order Legendre Functions," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).

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