Research Papers

The Use of Generalized Laguerre Polynomials in Spectral Methods for Solving Fractional Delay Differential Equations

[+] Author and Article Information
M. M. Khader

Department of Mathematics and Statistics,
College of Science,
Al-Imam Mohammed Ibn Saud,
Islamic University (IMSIU),
Riyadh 11566, Saudi Arabia
e-mail: mohamedmbd@yahoo.com

1Permanent address: Department of Mathematics, Faculty of Science, Benha University, Benha, 13518 Egypt.

Contributed by Design Engineering Division of ASME for publication in the Journal of Computational and Nonlinear Dynamics. Manuscript received August 23, 2012; final manuscript received April 2, 2013; published online July 18, 2013. Assoc. Editor: J. A. Tenreiro Machado.

J. Comput. Nonlinear Dynam 8(4), 041018 (Jul 18, 2013) (5 pages) Paper No: CND-12-1130; doi: 10.1115/1.4024852 History: Received August 23, 2012; Revised April 02, 2013

In this paper, an efficient numerical method for solving the fractional delay differential equations (FDDEs) is considered. The fractional derivative is described in the Caputo sense. The proposed method is based on the derived approximate formula of the Laguerre polynomials. The properties of Laguerre polynomials are utilized to reduce FDDEs to a linear or nonlinear system of algebraic equations. Special attention is given to study the error and the convergence analysis of the proposed method. Several numerical examples are provided to confirm that the proposed method is in excellent agreement with the exact solution.

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Bagley, R. L., and Torvik, P. J., 1984, “On the Appearance of the Fractional Derivative in the Behavior of Real Materials,” ASME J. Appl. Mech., 51(2), pp. 294–298. [CrossRef]
Enelund, M., and Josefson, B. L., 1997, “Time-Domain Finite Element Analysis of Viscoelastic Structures With Fractional Derivatives Constitutive Relations,” AIAA J., 35(10), pp. 1630–1637. [CrossRef]
Diethelm, K., 1997, “An Algorithm for the Numerical Solution of Differential Equations of Fractional Order,” Electron. Trans. Numer. Anal., 5, pp. 1–6.
He, J. H., 2011, “A Short Remark on Fractional Variational Iteration Method,” Phys. Lett. A, 375(38), pp. 3362–3364. [CrossRef]
He, J. H., 1999, “Homotopy Perturbation Technique,” Comput. Methods Appl. Mech. Eng., 178(3–4), pp. 257–262. [CrossRef]
Khabibrakhmanov, I. Z., and Summers, D., 1998, “The Use of Generalized Laguerre Polynomials in Spectral Methods for Non-Linear Differential Equations,” Comput. Math. Appl., 36, pp. 65–70. [CrossRef]
Khader, M. M., 2011, “On the Numerical Solutions for the Fractional Diffusion Equation,” Commun. Nonlinear Sci. Numer. Simul., 16, pp. 2535–2542. [CrossRef]
Khader, M. M., 2012, “Introducing an Efficient Modification of the VIM by Using Chebyshev Polynomials,” Appl. Appl. Math., 7(1), pp. 283–299.
Ramadan, M. A., and Sharif, M. N., 2006, “Numerical Solution of System of First Order Delay Differential Equations Using Spline Functions,” Int. J. Comput. Math., 83(12), pp. 925–937. [CrossRef]
Samko, S., Kilbas, A., and Marichev, O., 1993, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, London.
Sweilam, N. H., Khader, M. M., and Al-Bar, R. F., 2007, “Numerical Studies for a Multi-Order Fractional Differential Equation,” Phys. Lett. A, 371, pp. 26–33. [CrossRef]
Sweilam, N. H., Khader, M. M., and Nagy, A. M., 2011, “Numerical Solution of Two Sided Space Fractional Wave Equation Using Finite Difference Method,” Comput. Appl. Math., 235, pp. 2832–2841. [CrossRef]
Sweilam, N. H., Khader, M. M., and Mahdy, A. M. S., 2012, “Numerical Studies for Fractional-Order Logistic Differential Equation With Two Different Delays,” J. Appl. Math., 2012, pp. 1–14. [CrossRef]
Bell, W. W., 1968, Special Functions for Scientists and Engineers, Butler and Tanner Ltd, Frome, London
Canuto, C., Hnssalni, M. Y., Quarteroni, A., and Zang, T. A., 1988, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York.
Funaro, D., 1992, Polynomial Approximation of Differential Equations, Springer-Verlag, New York.
Ramadan, M. A., 2005, “Spline Solution of First Order Delay Differential Equation,” J. Egypt. Math. Soc., 1, pp. 7–18.
Khader, M. M., and Hendy, A. S., 2012, “The Approximate and Exact Solutions of the Fractional-Order Delay Differential Equations Using Legendre Pseudospectral Method,” Int. J. Pure Appl. Math., 74(3), pp. 287–297.
Podlubny, I., 1999, Fractional Differential Equations, Academic, New York, 1999.
Michalska, M., and Szynal, J., 2001, “A New Bound for the Laguerre Polynomials,” J. Comput. Appl. Math., 133, pp. 489–493. [CrossRef]
Wang, L., and Guo, B., 2006, “Stair Laguerre Pseudospectral Method for Differential Equations on the Half Line,” Adv. Comput. Math., 25, pp. 305–322. [CrossRef]
Askey, R., and Gasper, G., 1977, “Convolution Structures for Laguerre Polynomials,” J. Anal. Math., 31, pp. 48–68. [CrossRef]
Khader, M. M., El Danaf, T. S., and Hendy, A. S., 2012, “Efficient Spectral Collocation Method for Solving Multi-Term Fractional Differential Equations Based on the Generalized Laguerre Polynomials,” Fractional Calculus Appl., 3(13), pp. 1–14. [CrossRef]
Lewandowski, Z., and Szynal, J., 1998, “An Upper Bound for the Laguerre Polynomials,” J. Comput. Appl. Math., 99, pp. 529–533. [CrossRef]


Grahic Jump Location
Fig. 1

The behavior of the exact solution and the approximate solution at m = 3 and m = 5



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