Pseudo-Spectral Method for Space Fractional Diffusion Equation

ABSTRACT

This paper presents a numerical scheme for space
fractional diffusion equations (SFDEs) based on pseudo-spectral method. In this
approach, using the Guass-Lobatto nodes, the unknown function is approximated
by orthogonal polynomials or interpolation polynomials. Then, by using
pseudo-spectral method, the SFDE is reduced to a system of ordinary
differential equations for time variable *t*.
The high order Runge-Kutta scheme can be used to solve the system. So, a high
order numerical scheme is derived. Numerical examples illustrate that the
results obtained by this method agree well with the analytical solutions.

Cite this paper

Huang, Y. and Zheng, M. (2013) Pseudo-Spectral Method for Space Fractional Diffusion Equation.*Applied Mathematics*, **4**, 1495-1502. doi: 10.4236/am.2013.411202.

Huang, Y. and Zheng, M. (2013) Pseudo-Spectral Method for Space Fractional Diffusion Equation.

References

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http://dx.doi.org/10.1103/PhysRevE.58.1621

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http://dx.doi.org/10.1016/j.jcp.2004.11.025

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http://dx.doi.org/10.1137/030602666

[9] S. B. Yuste, “Weighted Average Finite Difference Methods for Fractional Diffusion Equations,” Journal of Computational Physics, Vol. 216, No. 1, 2006, pp. 264-274.

http://dx.doi.org/10.1016/j.jcp.2005.12.006

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http://dx.doi.org/10.1016/j.jcp.2005.08.008

[11] M. M. Meerschaert and C. Tadjeran, “Finite Difference Approximations for Fractional Advection-Dispersion Flow Equations,” Journal of Computational and Applied Mathematics, Vol. 172, No. 1, 2004, pp. 65-77.

http://dx.doi.org/10.1016/j.cam.2004.01.033

[12] L. Blank, “Numerical Treatment of Differential Equations of Fractional Order,” Numerical Analysis Report 287, Manchester Centre for Computational Mathematics, Manchester, 1996.

[13] K. Diethelm and G. Walz, “Numerical Solution of Fractional Order Differential Equations by Extroplation,” Numerical Algorithms, Vol. 16, 1997, pp. 231-253.

http://dx.doi.org/10.1023/A:1019147432240

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[15] N. Ford and A. Simpson, “The Numerical Solution of Fractional Differential Equations: Speed versus Accuracy,” Numerical Analysis Report 385, Manchester Centre for Computational Mathematics, Manchester, 2001.

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http://dx.doi.org/10.1023/A:1016592219341

[17] C.-M. Chen, F. Liu and K. Burrage, “Finite Difference Methods and a Fourier Analysis for the Fractional Reaction-Subdiffusion Equation,” Applied Mathematics and Computation, Vol. 198, No. 2, 2008, pp. 754-769.

http://dx.doi.org/10.1016/j.amc.2007.09.020

[18] B. Baeumer, M. Kovacs and M. M. Meerschaert, “Numerical Solutions for Fractional Reaction-Diffusion Equations,” Computers & Mathematics with Applications, Vol. 55, No. 10, 2008, pp. 2212-2226.

http://dx.doi.org/10.1016/j.camwa.2007.11.012

[19] S. Shen, F. Liu and V. Anh, “Numerical Approximations and Solution Techniques for the Space-Time Riesz-Caputo Fractional Advection-Diffusion Equation,” Numerical Algorithms, Vol. 56, No. 3, 2011, pp. 383-403.

http://dx.doi.org/10.1007/s11075-010-9393-x

[20] C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, “Spectral Methods. Fundamentals in Single Domains,” Springer-Verlag, Berlin, 2006.

[21] D. Funaro and D. Gottlieb, “A New Method of Imposing Boundary Conditions in Pseudospectral Approximations of Hyperbolic Equations,” Mathematical and Computer, Vol. 51, No. 184, 1988, pp. 599-613.

http://dx.doi.org/10.1090/S0025-5718-1988-0958637-X

[22] G.-Q. Chen, Q. Du and E. Tadmor, “Spectral Viscosity Approximations to Multidimensional Scalar Conservation Laws,” Mathematical and Computer, Vol. 61, No. 204, 1993, pp. 629-643.

http://dx.doi.org/10.1090/S0025-5718-1993-1185240-3

[23] T. Y. Hou and R. Li, “Computing Nearly Singular Solutions Using Pseudo-Spectral Methods,” Journal of Computational Physics, Vol. 226, No. 1, 2007, pp. 379-397.

http://dx.doi.org/10.1016/j.jcp.2007.04.014

[24] S. Esmaeili and M. Shamsi, “A Pseudo-Spectral Scheme for the Approximate Solution of a Family of Fractional Differential Equations,” Communications in Nonlinear Science & Numerical Simulation, Vol. 16, No. 9, 2011, pp. 3646-3654.

http://dx.doi.org/10.1016/j.cnsns.2010.12.008

[25] C. Li, F. Zeng and F. Liu, “Spectral Approximations to the Fractional Integral and Derivative,” Fractional Calculus & Applied Analysis, Vol. 15, No. 3, 2012, pp. 383-406.

http://dx.doi.org/10.2478/s13540-012-0028-x

[26] J. Shen and T. Tang, “Spectral and High-Order Methods with Applications,” Science Press, Beijing, 2007.

[27] A. Quarteroni and A. Valli, “Numerical Approximation of Partial Differential Equations,” Springer-Verlag, Berlin, 1997.

[1] M. Giona and H. E. Roman, “Fractional Diffusion Equation for Transport Phenomena in Random Media,” Journal of Physics A, Vol. 185, No. 1-4, 1992, pp. 87-97.

[2] R. Metzler, J. Klafter and I. M. Sokolov, “Anomalous Transport in External Fields: Continuous Time Random Walks and Fractional Diffusion Equations Extends,” Physical Review E, Vol. 58, No. 3, 1998, pp. 1621-1633.

http://dx.doi.org/10.1103/PhysRevE.58.1621

[3] R. Metzler and J. Klafter, “Boundary Value Problems for Fractional Diffusion Equations,” Journal of Physics A, Vol. 278, No. 1-2, 2000, pp. 107-125.

[4] B. I. Henry and S. L. Wearne, “Fractional Reaction-Diffusion,” Journal of Physics A, Vol. 276, No. 3-4, 2000, pp. 448-445.

[5] Y. Zhang, M. Meerschaert and B. Baeumer, “Particle Tracking for Time-Fractional Diffusion,” Physical Review E, Vol. 78, No. 3, 2008, Article ID: 036705.

[6] H. G. Sun, W. Chen and Y. Q. Chen, “Variable-Order Fractional Differential Operators in Anomalous Diffusion Modeling,” Journal of Physics A, Vol. 338, No. 21, 2009, pp. 4586-4592.

[7] A. T. M. Langlands and B. I. Henry, “The Accuracy and Stability of an Implicit Solution Method for the Fractional Diffusion Equation,” Journal of Computational Physics, Vol. 205, No. 2, 2005, pp. 719-736.

http://dx.doi.org/10.1016/j.jcp.2004.11.025

[8] S. B. Yuste and L. Acedo, “On an Explicit Finite Difference Method for Fractional Diffusion Equations,” SIAM Journal on Numerical Analysis, Vol. 42, No. 5, 2005, pp. 1862-1874.

http://dx.doi.org/10.1137/030602666

[9] S. B. Yuste, “Weighted Average Finite Difference Methods for Fractional Diffusion Equations,” Journal of Computational Physics, Vol. 216, No. 1, 2006, pp. 264-274.

http://dx.doi.org/10.1016/j.jcp.2005.12.006

[10] C. Tadjeran, M. M. Meerschaert and H.-P. Scheffler, “A Second Order Accurate Numerical Approximation for the Fractional Diffusion Equation,” Journal of Computational Physics, Vol. 213, No. 1, 2006, pp. 205-213.

http://dx.doi.org/10.1016/j.jcp.2005.08.008

[11] M. M. Meerschaert and C. Tadjeran, “Finite Difference Approximations for Fractional Advection-Dispersion Flow Equations,” Journal of Computational and Applied Mathematics, Vol. 172, No. 1, 2004, pp. 65-77.

http://dx.doi.org/10.1016/j.cam.2004.01.033

[12] L. Blank, “Numerical Treatment of Differential Equations of Fractional Order,” Numerical Analysis Report 287, Manchester Centre for Computational Mathematics, Manchester, 1996.

[13] K. Diethelm and G. Walz, “Numerical Solution of Fractional Order Differential Equations by Extroplation,” Numerical Algorithms, Vol. 16, 1997, pp. 231-253.

http://dx.doi.org/10.1023/A:1019147432240

[14] K. Diethelm, “An Algorithm for the Numerical Solution of Differential Equations of Fractional Order,” Electronic Transactions on Numerical Analysis, Vol. 5, 1997, pp. 1-6.

[15] N. Ford and A. Simpson, “The Numerical Solution of Fractional Differential Equations: Speed versus Accuracy,” Numerical Analysis Report 385, Manchester Centre for Computational Mathematics, Manchester, 2001.

[16] K. Diethelm, N. Ford and A. Freed, “A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations,” Nonlinear Dynamics, Vol. 29, No. 1-4, 2002, pp. 3-22.

http://dx.doi.org/10.1023/A:1016592219341

[17] C.-M. Chen, F. Liu and K. Burrage, “Finite Difference Methods and a Fourier Analysis for the Fractional Reaction-Subdiffusion Equation,” Applied Mathematics and Computation, Vol. 198, No. 2, 2008, pp. 754-769.

http://dx.doi.org/10.1016/j.amc.2007.09.020

[18] B. Baeumer, M. Kovacs and M. M. Meerschaert, “Numerical Solutions for Fractional Reaction-Diffusion Equations,” Computers & Mathematics with Applications, Vol. 55, No. 10, 2008, pp. 2212-2226.

http://dx.doi.org/10.1016/j.camwa.2007.11.012

[19] S. Shen, F. Liu and V. Anh, “Numerical Approximations and Solution Techniques for the Space-Time Riesz-Caputo Fractional Advection-Diffusion Equation,” Numerical Algorithms, Vol. 56, No. 3, 2011, pp. 383-403.

http://dx.doi.org/10.1007/s11075-010-9393-x

[20] C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, “Spectral Methods. Fundamentals in Single Domains,” Springer-Verlag, Berlin, 2006.

[21] D. Funaro and D. Gottlieb, “A New Method of Imposing Boundary Conditions in Pseudospectral Approximations of Hyperbolic Equations,” Mathematical and Computer, Vol. 51, No. 184, 1988, pp. 599-613.

http://dx.doi.org/10.1090/S0025-5718-1988-0958637-X

[22] G.-Q. Chen, Q. Du and E. Tadmor, “Spectral Viscosity Approximations to Multidimensional Scalar Conservation Laws,” Mathematical and Computer, Vol. 61, No. 204, 1993, pp. 629-643.

http://dx.doi.org/10.1090/S0025-5718-1993-1185240-3

[23] T. Y. Hou and R. Li, “Computing Nearly Singular Solutions Using Pseudo-Spectral Methods,” Journal of Computational Physics, Vol. 226, No. 1, 2007, pp. 379-397.

http://dx.doi.org/10.1016/j.jcp.2007.04.014

[24] S. Esmaeili and M. Shamsi, “A Pseudo-Spectral Scheme for the Approximate Solution of a Family of Fractional Differential Equations,” Communications in Nonlinear Science & Numerical Simulation, Vol. 16, No. 9, 2011, pp. 3646-3654.

http://dx.doi.org/10.1016/j.cnsns.2010.12.008

[25] C. Li, F. Zeng and F. Liu, “Spectral Approximations to the Fractional Integral and Derivative,” Fractional Calculus & Applied Analysis, Vol. 15, No. 3, 2012, pp. 383-406.

http://dx.doi.org/10.2478/s13540-012-0028-x

[26] J. Shen and T. Tang, “Spectral and High-Order Methods with Applications,” Science Press, Beijing, 2007.

[27] A. Quarteroni and A. Valli, “Numerical Approximation of Partial Differential Equations,” Springer-Verlag, Berlin, 1997.