Solving a Nonlinear Multi-Order Fractional Differential Equation Using Legendre Pseudo-Spectral Method

ABSTRACT

In this paper, we apply the Legendre spectral-collocation method to obtain approximate solutions of nonlinear multi-order fractional differential equations (M-FDEs). The fractional derivative is described in the Caputo sense. The study is conducted through illustrative example to demonstrate the validity and applicability of the presented method. The results reveal that the proposed method is very effective and simple. Moreover, only a small number of shifted Legendre polynomials are needed to obtain a satisfactory result.

KEYWORDS

Legendre Pseudo-Spectral Method; Multi-Order Fractional Differential Equations; Caputo Derivative

Legendre Pseudo-Spectral Method; Multi-Order Fractional Differential Equations; Caputo Derivative

Cite this paper

Y. Yang, "Solving a Nonlinear Multi-Order Fractional Differential Equation Using Legendre Pseudo-Spectral Method,"*Applied Mathematics*, Vol. 4 No. 1, 2013, pp. 113-118. doi: 10.4236/am.2013.41020.

Y. Yang, "Solving a Nonlinear Multi-Order Fractional Differential Equation Using Legendre Pseudo-Spectral Method,"

References

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[8] X. Li and C. Xu, “A Space-Time Spectral Method for the Time Fractional Diffusion Equation,” SIAM Journal on Numerical Analysis, Vol. 47, No. 3, 2009, pp. 2108-2131. doi:10.1137/080718942

[9] Y. Wei and Y. Chen, “Convergence Analysis of the Legendre Spectral Collocation Methods for Second Order Volterra Integro-Differential Equations,” Numerical Mathematics: Theory, Methods and Applications, Vol. 4, No. 4, 2011, pp. 339-358.

[10] Y. Li, “Solving a Nonlinear Fractional Differential Equation Using Chebyshev Wavelets,” Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 9, 2010, pp. 2284-2292. doi:10.1016/j.cnsns.2009.09.020

[11] V. Gejji and H. Jafari, “Solving a Multi-Order Fractional Differential Equation Using Adomian Decomposition,” Applied Mathematics and Computation, Vol. 189, No. 1, 2007, pp. 541-548.

[12] S. Yang, A. Xiao and H. Su, “Convergence of the Variational Iteration Method for Solving Multi-Order Fractional Differential Equations,” Computers and Mathematics with Applications, Vol. 60, No. 10, 2010, pp. 2871-2879. doi:10.1016/j.camwa.2010.09.044

[13] A. El-Mesiry, A. El-Sayed and H. El-Saka, “Numerical Methods for Multi-Term Fractional (Arbitrary) Orders Differential Equations,” Computational and Applied Mathematics, Vol. 160, No. 3, 2005, pp. 683-699. doi:10.1016/j.amc.2003.11.026

[14] B. Baeumer, M. Kovcs and M. M. Meerschaer, “Numerical Solutions for Fractional Reaction-Diffusion Equations,” Computers and Mathematics with Applications, Vol. 55, No. 10, 2008, pp. 2212-2226. doi:10.1016/j.camwa.2007.11.012

[15] K. Diethelm and N. Ford, “Multi-Order Fractional Differential Equations and Their Numerical Solution,” Applied Mathematics and Computation, Vol. 154, No. 3, 2004, pp. 621-640. doi:10.1016/S0096-3003(03)00739-2

[1] H. H. Sun, B. Onaral and Y. Tsao, “Application of Positive Reality Principle to Metal Electrode Linear Polarization Phenomena,” IEEE Transactions on Biomedical Engineering, Vol. 31, No. 10, 1984, pp. 664-674. doi:10.1109/TBME.1984.325317

[2] H. H. Sun, A. A. Abdelwahab and B. Onaral, “Linear Approximation of Transfer Function with a Pole of Fractional Order,” IEEE Transactions on Automatic Control, Vol. 29, No. 5, 1984, pp. 441-444. doi:10.1109/TAC.1984.1103551

[3] C. Li and G. Chen, “Chaos and Hyperchaos in the Fractional-Order Rossle Equations,” Physica A, Vol. 341, No. 1, 2004, pp. 55-61. doi:10.1016/j.physa.2004.04.113

[4] N. H. Sweilam, M. M. Khader and R. F. AlffBar, “Numerical Studies for a Multi-Order Fractional Differential Equation,” Physics Letter A, Vol. 371, No. 1-2, 2007, pp. 26-33. doi:10.1016/j.physleta.2007.06.016

[5] V. S. Erturk, S. Momani and Z. Odibat, “Application of Generalized Differential Transform Method to Multi-Order Fractional Differential Equations,” Communications in Nonlinear Science and Numerical Simulation, Vol. 13, No. 8, 2008, pp. 1642-1654. doi:10.1016/j.cnsns.2007.02.006

[6] Z. Odibat and S. Momani, “Modified Homotopy Perturbation Method: Application to Quadratic Riccati Differential Equation of Fractional Order,” Chaos, Solitons and Fractals, Vol. 36, No. 1, 2008, pp. 167-174. doi:10.1016/j.chaos.2006.06.041

[7] Y. Chen and T. Tang, “Convergence Analysis for the Jacobi Collocation Methods to Volterra Integral Equations with a Weakly Singular Kernel,” Mathematics of Computation, Vol. 79, No. 269, 2010, pp. 147-167. doi:10.1090/S0025-5718-09-02269-8

[8] X. Li and C. Xu, “A Space-Time Spectral Method for the Time Fractional Diffusion Equation,” SIAM Journal on Numerical Analysis, Vol. 47, No. 3, 2009, pp. 2108-2131. doi:10.1137/080718942

[9] Y. Wei and Y. Chen, “Convergence Analysis of the Legendre Spectral Collocation Methods for Second Order Volterra Integro-Differential Equations,” Numerical Mathematics: Theory, Methods and Applications, Vol. 4, No. 4, 2011, pp. 339-358.

[10] Y. Li, “Solving a Nonlinear Fractional Differential Equation Using Chebyshev Wavelets,” Communications in Nonlinear Science and Numerical Simulation, Vol. 15, No. 9, 2010, pp. 2284-2292. doi:10.1016/j.cnsns.2009.09.020

[11] V. Gejji and H. Jafari, “Solving a Multi-Order Fractional Differential Equation Using Adomian Decomposition,” Applied Mathematics and Computation, Vol. 189, No. 1, 2007, pp. 541-548.

[12] S. Yang, A. Xiao and H. Su, “Convergence of the Variational Iteration Method for Solving Multi-Order Fractional Differential Equations,” Computers and Mathematics with Applications, Vol. 60, No. 10, 2010, pp. 2871-2879. doi:10.1016/j.camwa.2010.09.044

[13] A. El-Mesiry, A. El-Sayed and H. El-Saka, “Numerical Methods for Multi-Term Fractional (Arbitrary) Orders Differential Equations,” Computational and Applied Mathematics, Vol. 160, No. 3, 2005, pp. 683-699. doi:10.1016/j.amc.2003.11.026

[14] B. Baeumer, M. Kovcs and M. M. Meerschaer, “Numerical Solutions for Fractional Reaction-Diffusion Equations,” Computers and Mathematics with Applications, Vol. 55, No. 10, 2008, pp. 2212-2226. doi:10.1016/j.camwa.2007.11.012

[15] K. Diethelm and N. Ford, “Multi-Order Fractional Differential Equations and Their Numerical Solution,” Applied Mathematics and Computation, Vol. 154, No. 3, 2004, pp. 621-640. doi:10.1016/S0096-3003(03)00739-2