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

Affiliation(s)

^{1}
Department of Mathematics, Faculty of Science, Cairo University, Giza, Egypt.

^{2}
Department of Mathematics and Statistics, College of Science, Al-Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh, Kingdom of Saudi Arabia.

^{3}
Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt.

ABSTRACT

In this paper, an efficient numerical method is considered for solving the fractional wave equation (FWE). The fractional derivative is described in the Caputo sense. The method is based on Laguerre approximations. The properties of Laguerre polynomials are utilized to reduce FWE to a system of ordinary differential equations, which is solved by the finite difference method. An approximate formula of the fractional derivative is given. Special attention is given to study the convergence analysis and estimate an error upper bound of the presented formula. Numerical solutions of FWE are given and the results are compared with the exact solution.

In this paper, an efficient numerical method is considered for solving the fractional wave equation (FWE). The fractional derivative is described in the Caputo sense. The method is based on Laguerre approximations. The properties of Laguerre polynomials are utilized to reduce FWE to a system of ordinary differential equations, which is solved by the finite difference method. An approximate formula of the fractional derivative is given. Special attention is given to study the convergence analysis and estimate an error upper bound of the presented formula. Numerical solutions of FWE are given and the results are compared with the exact solution.

KEYWORDS

Fractional Wave Equation, Caputo Derivative, Finite Difference Method, Laguerre Polynomials, Convergence Analysis

Fractional Wave Equation, Caputo Derivative, Finite Difference Method, Laguerre Polynomials, Convergence Analysis

Cite this paper

Sweilam, N. , Khader, M. and Adel, M. (2015) Numerical Solution for the Fractional Wave Equation Using Pseudo-Spectral Method Based on the Generalized Laguerre Polynomials.*Applied Mathematics*, **6**, 647-654. doi: 10.4236/am.2015.64058.

Sweilam, N. , Khader, M. and Adel, M. (2015) Numerical Solution for the Fractional Wave Equation Using Pseudo-Spectral Method Based on the Generalized Laguerre Polynomials.

References

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[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

[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