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An Adaptive Pseudospectral Method for Fractional Order Boundary Value Problems

Author

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  • Mohammad Maleki
  • Ishak Hashim
  • Majid Tavassoli Kajani
  • Saeid Abbasbandy

Abstract

An adaptive pseudospectral method is presented for solving a class of multiterm fractional boundary value problems (FBVP) which involve Caputo‐type fractional derivatives. The multiterm FBVP is first converted into a singular Volterra integrodifferential equation (SVIDE). By dividing the interval of the problem to subintervals, the unknown function is approximated using a piecewise interpolation polynomial with unknown coefficients which is based on shifted Legendre‐Gauss (ShLG) collocation points. Then the problem is reduced to a system of algebraic equations, thus greatly simplifying the problem. Further, some additional conditions are considered to maintain the continuity of the approximate solution and its derivatives at the interface of subintervals. In order to convert the singular integrals of SVIDE into nonsingular ones, integration by parts is utilized. In the method developed in this paper, the accuracy can be improved either by increasing the number of subintervals or by increasing the degree of the polynomial on each subinterval. Using several examples including Bagley‐Torvik equation the proposed method is shown to be efficient and accurate.

Suggested Citation

  • 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).
    • Handle: RePEc:wly:jnlaaa:v:2012:y:2012:i:1:n:381708
    DOI: 10.1155/2012/381708
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    References listed on IDEAS

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    1. D. Baleanu & H. Mohammadi & Sh. Rezapour, 2012. "Positive Solutions of an Initial Value Problem for Nonlinear Fractional Differential Equations," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-7, June.
    2. D. Baleanu & H. Mohammadi & Sh. Rezapour, 2012. "Positive Solutions of an Initial Value Problem for Nonlinear Fractional Differential Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
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    4. 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).
    5. 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.
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    Cited by:

    1. 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).
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    3. A. H. Bhrawy & M. A. Alghamdi & D. Baleanu, 2013. "Numerical Solution of a Class of Functional‐Differential Equations Using Jacobi Pseudospectral Method," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    4. A. H. Bhrawy & L. M. Assas & M. A. Alghamdi, 2013. "Fast Spectral Collocation Method for Solving Nonlinear Time‐Delayed Burgers‐Type Equations with Positive Power Terms," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).

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