Numerical Solution for the Fractional Wave Equation Using Pseudo-Spectral Method Based on the Generalized Laguerre Polynomials

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References

[1] Miller, K.S. and Ross, B. (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. John Wiley and Sons, New York.

[2] Podlubny, I. (1999) Fractional Differential Equations. Academic Press, New York.

[3] Sweilam, N.H., Khader, M.M. and Al-Bar, R.F. (2007) Numerical Studies for a Multi-Order Fractional Differential Equation. Physics Letters A, 371, 26-33.

http://dx.doi.org/10.1016/j.physleta.2007.06.016

[4] Hashim, I., Abdulaziz, O. and Momani, S. (2009) Homotopy Analysis Method for Fractional IVPs. Communications in Nonlinear Science and Numerical Simulations, 14, 674-684.

http://dx.doi.org/10.1016/j.cnsns.2007.09.014

[5] Funaro, D. (1992) Polynomial Approximation of Differential Equations, Springer Verlag, New York.

[6] Khader, M.M. (2011) On the Numerical Solutions for the Fractional Diffusion Equation. Communications in Nonlinear Science and Numerical Simulations, 16, 2535-2542.

http://dx.doi.org/10.1016/j.cnsns.2010.09.007

[7] Sweilam, N.H., Khader, M.M. and Adel, M. (2014) Chebyshev Pseudo-Spectral Method for Solving Fractional Advection-Dispersion Equation. Applied Mathematics, 5, 3240-3248.

http://dx.doi.org/10.4236/am.2014.519301

[8] Sweilam, N.H., Khader, M.M. and Mahdy, A.M.S. (2012) Numerical Studies for Fractional-Order Logistic Differential Equation with Two Different Delays. Journal of Applied Mathematics, 2012, Article ID: 764894, 14 p.

[9] Sweilam, N.H. and Khader, M.M. (2010) A Chebyshev Pseudo-Spectral Method for Solving Fractional Integro-Differential Equations. ANZIAM, 51, 464-475.

http://dx.doi.org/10.1017/S1446181110000830

[10] Sweilam, N.H., Khader, M.M. and Adel, M. (2014) Numerical Simulation of Fractional Cable Equation of Spiny Neuronal Dendrites. Journal of Advanced Research (JAR), 5, 253-259.

http://dx.doi.org/10.1016/j.jare.2013.03.006

[11] Smith, G.D. (1965) Numerical Solution of Partial Differential Equations. Oxford University Press, New York.

[12] Jafari, H. and Daftardar-Gejji, V. (2006) Solving Linear and Nonlinear Fractional Diffusion and Wave Equations by Adomian Decomposition Method. Applied Mathematics and Computation, 180, 488-497.

http://dx.doi.org/10.1016/j.amc.2005.12.031

[13] Sweilam, N.H., Khader, M.M. and Nagy, A.M. (2011) Numerical Solution of Two-Sided Space-Fractional Wave Equation Using Finite Difference Method. Journal of Computational and Applied Mathematics, 235, 2832-2841.

http://dx.doi.org/10.1016/j.cam.2010.12.002

[14] Sweilam, N.H., Khader, M.M. and Adel, M. (2012) On the Stability Analysis of Weighted Average Finite Difference Methods for Fractional Wave Equation. Fractional Differential Calculus, 2, 17-29.

http://dx.doi.org/10.7153/fdc-02-02

[15] Chen, S., Liu, F., Zhuang, P. and Anh, V. (2009) Finite Difference Approximations for the Fractional Fokker-Planck Equation. Applied Mathematical Modelling, 33, 256-273.

http://dx.doi.org/10.1016/j.apm.2007.11.005

[16] Khader, M.M. (2013) Numerical Treatment for Solving the Perturbed Fractional PDEs Using Hybrid Techniques. Journal of Computational Physics, 250, 565-573.

http://dx.doi.org/10.1016/j.jcp.2013.05.032

[17] Lubich, Ch. (1986) Discretized Fractional Calculus. SIAM Journal on Mathematical Analysis, 17, 704-719.

http://dx.doi.org/10.1137/0517050

[18] Meerschaert, M.M. and Tadjeran, C. (2006) Finite Difference Approximations for Two-Sided Space-Fractional Partial Differential Equations. Applied Numerical Mathematics, 56, 80-90.

http://dx.doi.org/10.1016/j.apnum.2005.02.008

[19] Liu, F., Zhuang, P. and Burrage, K. (2012) Numerical Methods and Analysis for a Class of Fractional Advection-Dispersion Models. Computer and Mathematics with Application, 64, 2990-3007.

http://dx.doi.org/10.1016/j.camwa.2012.01.020

[20] Canuto, C., Hussaini, M.Y., Quarteroni, A. and Zang, T.A. (2006) Spectral Methods. Springer-Verlag, New York.

[21] Xu, C.-L. and Guo, B.-Y. (2002) Laguerre Pseudo-Spectral Method for Non-Linear Partial Differential Equations. Journal of Computational Mathematics, 20, 413-428.

[22] Wang, L. and Guo, B.Y. (2006) Stair Laguerre Pseudo-Spectral Method for Differential Equations on the Half Line. Advances in Computational Mathematics, 25, 305-322.

http://dx.doi.org/10.1007/s10444-003-7608-6

[23] Khader, M.M. (2013) The Use of Generalized Laguerre Polynomials in Spectral Methods for Fractional-Order Delay Differential Equations. Journal of Computational and Nonlinear Dynamics, 8, Article ID: 041018.

[24] Doha, E.H., Bhrawy, A.H. and Ezz-Eldien, S.S. (2011) Efficient Chebyshev Spectral Methods for Solving Multi-Term Fractional Orders Differential Equations. Applied Mathematical Modelling, 35, 5662-5672.

http://dx.doi.org/10.1016/j.apm.2011.05.011

[25] Khader, M.M. and Babatin, M.M. (2013) On Approximate Solutions for Fractional Logistic Differential Equation. Mathematical Problems in Engineering, 2013, Article ID: 391901.

http://dx.doi.org/10.1155/2013/391901