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Numerical Solution of the Fractional Partial Differential Equations by the Two‐Dimensional Fractional‐Order Legendre Functions

Author

Listed:
  • Fukang Yin
  • Junqiang Song
  • Yongwen Wu
  • Lilun Zhang

Abstract

A numerical method is presented to obtain the approximate solutions of the fractional partial differential equations (FPDEs). The basic idea of this method is to achieve the approximate solutions in a generalized expansion form of two‐dimensional fractional‐order Legendre functions (2D‐FLFs). The operational matrices of integration and derivative for 2D‐FLFs are first derived. Then, by these matrices, a system of algebraic equations is obtained from FPDEs. Hence, by solving this system, the unknown 2D‐FLFs coefficients can be computed. Three examples are discussed to demonstrate the validity and applicability of the proposed method.

Suggested Citation

  • 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).
    • Handle: RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:562140
    DOI: 10.1155/2013/562140
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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. Odibat, Zaid & Momani, Shaher, 2008. "Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order," Chaos, Solitons & Fractals, Elsevier, vol. 36(1), pages 167-174.
    3. Ali Ahmadian & Mohamed Suleiman & Soheil Salahshour, 2013. "An Operational Matrix Based on Legendre Polynomials for Solving Fuzzy Fractional-Order Differential Equations," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-29, June.
    4. 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, Hindawi, vol. 2013, pages 1-10, August.
    5. Junqiang Song & Fukang Yin & Xiaoqun Cao & Fengshun Lu, 2013. "Fractional Variational Iteration Method versus Adomian’s Decomposition Method in Some Fractional Partial Differential Equations," Journal of Applied Mathematics, John Wiley & Sons, vol. 2013(1).
    6. A. M. A. El-Sayed & A. Elsaid & D. Hammad, 2012. "A Reliable Treatment of Homotopy Perturbation Method for Solving the Nonlinear Klein-Gordon Equation of Arbitrary (Fractional) Orders," Journal of Applied Mathematics, Hindawi, vol. 2012, pages 1-13, February.
    7. Fukang Yin & Junqiang Song & Xiaoqun Cao & Fengshun Lu, 2013. "Couple of the Variational Iteration Method and Legendre Wavelets for Nonlinear Partial Differential Equations," Journal of Applied Mathematics, John Wiley & Sons, vol. 2013(1).
    8. 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.
    9. Ali Ahmadian & Mohamed Suleiman & Soheil Salahshour, 2013. "An Operational Matrix Based on Legendre Polynomials for Solving Fuzzy Fractional‐Order Differential Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    10. 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).
    11. A. M. A. El-Sayed & A. Elsaid & D. Hammad, 2012. "A Reliable Treatment of Homotopy Perturbation Method for Solving the Nonlinear Klein‐Gordon Equation of Arbitrary (Fractional) Orders," Journal of Applied Mathematics, John Wiley & Sons, vol. 2012(1).
    12. Fukang Yin & Junqiang Song & Xiaoqun Cao & Fengshun Lu, 2013. "Couple of the Variational Iteration Method and Legendre Wavelets for Nonlinear Partial Differential Equations," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-11, February.
    13. Junqiang Song & Fukang Yin & Xiaoqun Cao & Fengshun Lu, 2013. "Fractional Variational Iteration Method versus Adomian’s Decomposition Method in Some Fractional Partial Differential Equations," Journal of Applied Mathematics, Hindawi, vol. 2013, pages 1-10, February.
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    Cited by:

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    3. A. H. Bhrawy & A. S. Alofi & R. A. Van Gorder, 2014. "An Efficient Collocation Method for a Class of Boundary Value Problems Arising in Mathematical Physics and Geometry," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).

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