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Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems

Author

Listed:
  • D. Baleanu
  • A. H. Bhrawy
  • T. M. Taha

Abstract

We present a direct solution technique for approximating linear multiterm fractional differential equations (FDEs) on semi‐infinite interval, using generalized Laguerre polynomials. We derive the operational matrix of Caputo fractional derivative of the generalized Laguerre polynomials which is applied together with generalized Laguerre tau approximation for implementing a spectral solution of linear multiterm FDEs on semi‐infinite interval subject to initial conditions. The generalized Laguerre pseudo‐spectral approximation based on the generalized Laguerre operational matrix is investigated to reduce the nonlinear multiterm FDEs and its initial conditions to nonlinear algebraic system, thus greatly simplifying the problem. Through several numerical examples, we confirm the accuracy and performance of the proposed spectral algorithms. Indeed, the methods yield accurate results, and the exact solutions are achieved for some tested problems.

Suggested Citation

  • 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).
    • Handle: RePEc:wly:jnlaaa:v:2013:y:2013:i:1:n:546502
    DOI: 10.1155/2013/546502
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    References listed on IDEAS

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    1. A. A. Hemeda, 2013. "New Iterative Method: An Application for Solving Fractional Physical Differential Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    2. 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).
    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. 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).
    5. A. A. Hemeda, 2013. "New Iterative Method: An Application for Solving Fractional Physical Differential Equations," Abstract and Applied Analysis, Hindawi, vol. 2013, pages 1-9, May.
    6. 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.
    7. 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).
    8. 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.
    9. 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.
    10. 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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    1. B. A. Jacobs & C. Harley, 2014. "Two Hybrid Methods for Solving Two‐Dimensional Linear Time‐Fractional Partial Differential Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
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    3. 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).
    4. A. H. Bhrawy, 2014. "A New Legendre Collocation Method for Solving a Two‐Dimensional Fractional Diffusion Equation," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    5. Yong-Ju Yang & Liu-Qing Hua, 2014. "Variational Iteration Transform Method for Fractional Differential Equations with Local Fractional Derivative," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    6. Ali H. Bhrawy & Abdulrahim AlZahrani & Dumitru Baleanu & Yahia Alhamed, 2014. "A Modified Generalized Laguerre‐Gauss Collocation Method for Fractional Neutral Functional‐Differential Equations on the Half‐Line," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    7. A. H. Bhrawy & M. A. Alghamdi & Eman S. Alaidarous, 2014. "An Efficient Numerical Approach for Solving Nonlinear Coupled Hyperbolic Partial Differential Equations with Nonlocal Conditions," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    8. 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).
    9. M. M. Tharwat & S. M. Al-Harbi, 2013. "Approximation of Eigenvalues of Sturm‐Liouville Problems by Using Hermite Interpolation," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    10. 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).
    11. Shao-Hong Yan & Xiao-Hong Chen & Gong-Nan Xie & Carlo Cattani & Xiao-Jun Yang, 2014. "Solving Fokker‐Planck Equations on Cantor Sets Using Local Fractional Decomposition Method," Abstract and Applied Analysis, John Wiley & Sons, vol. 2014(1).
    12. Ali Ahmadian & Norazak Senu & Farhad Larki & Soheil Salahshour & Mohamed Suleiman & Md. Shabiul Islam, 2013. "Numerical Solution of Fuzzy Fractional Pharmacokinetics Model Arising from Drug Assimilation into the Bloodstream," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    13. A. H. Bhrawy, 2013. "A New Numerical Algorithm for Solving a Class of Fractional Advection‐Dispersion Equation with Variable Coefficients Using Jacobi Polynomials," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).
    14. Yin Yang & Yunqing Huang, 2013. "Spectral‐Collocation Methods for Fractional Pantograph Delay‐Integrodifferential Equations," Advances in Mathematical Physics, John Wiley & Sons, vol. 2013(1).
    15. A. H. Bhrawy & M. A. Alghamdi, 2013. "Approximate Solutions of Fisher′s Type Equations with Variable Coefficients," Abstract and Applied Analysis, John Wiley & Sons, vol. 2013(1).

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