Abstract and Applied Analysis

Numerical Treatment of the Modified Time Fractional Fokker-Planck Equation

Yuxin Zhang

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Abstract

A numerical method for the modified time fractional Fokker-Planck equation is proposed. Stability and convergence of the method are rigorously discussed by means of the Fourier method. We prove that the difference scheme is unconditionally stable, and convergence order is O ( τ + h 4 ) , where τ and h are the temporal and spatial step sizes, respectively. Finally, numerical results are given to confirm the theoretical analysis.

Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 282190, 10 pages.

Dates
First available in Project Euclid: 7 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1412687046

Digital Object Identifier
doi:10.1155/2014/282190

Mathematical Reviews number (MathSciNet)
MR3191030

Zentralblatt MATH identifier
07022083

Citation

Zhang, Yuxin. Numerical Treatment of the Modified Time Fractional Fokker-Planck Equation. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 282190, 10 pages. doi:10.1155/2014/282190. https://projecteuclid.org/euclid.aaa/1412687046


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References

  • K. S. Miller and B. Ross, An Introduction To the Fractional Calculus and Fractional Differential Equations, John Wiley, New York, NY, USA, 1993.
  • I. Podlubny, Fractional Differential Equations, Academic Press, 1999.
  • A. Barari, M. Omidvar, A. R. Ghotbi, and D. D. Ganji, “Application of homotopy perturbation method and variational iteration method to nonlinear oscillator differential equations,” Acta Applicandae Mathematicae, vol. 104, no. 2, pp. 161–171, 2008.
  • S. Das, “Analytical solution of a fractional diffusion equation by variational iteration method,” Computers and Mathematics with Applications, vol. 57, no. 3, pp. 483–487, 2009.
  • D. D. Ganji and A. Sadighi, “Application of homotopy-perturbation and variational iteration methods to nonlinear heat transfer and porous media equations,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 24–34, 2007.
  • E. Hesameddini and F. Fotros, “Solution for time-fractional coupled Klein-Gordon Schrodinger equation using decomposition method,” International Mathematics Olympiad, vol. 7, pp. 1047–1056, 2012.
  • G.-C. Wu and E. W. M. Lee, “Fractional variational iteration method and its application,” Physics Letters A, vol. 374, no. 25, pp. 2506–2509, 2010.
  • 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, pp. 754–769, 2008.
  • C. Çelik and M. Duman, “Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative,” Journal of Computational Physics, vol. 231, no. 4, pp. 1743–1750, 2012.
  • X. Hu and L. Zhang, “Implicit compact difference schemes for the fractional cable equation,” Applied Mathematical Modelling, vol. 36, pp. 4027–4043, 2012.
  • M. M. Meerschaert and C. Tadjeran, “Finite difference approximations for two-sided space-fractional partial differential equations,” Applied Numerical Mathematics, vol. 56, no. 1, pp. 80–90, 2006.
  • D. K. Salkuyeh, “On the finite difference approximation to the convection-diffusion equation,” Applied Mathematics and Computation, vol. 179, no. 1, pp. 79–86, 2006.
  • E. Sousa, “Numerical approximations for fractional diffusion equations via splines,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 938–944, 2011.
  • E. Sousa, “Finite difference approximations for a fractional advection diffusion problem,” Journal of Computational Physics, vol. 228, no. 11, pp. 4038–4054, 2009.
  • C. Tadjeran and M. M. Meerschaert, “A second-order accurate numerical method for the two-dimensional fractional diffusion equation,” Journal of Computational Physics, vol. 220, no. 2, pp. 813–823, 2007.
  • S. B. Yuste and L. Acedo, “An explicit finite difference method and a new von Neumann-type stability analysis for fractional diffusion equations,” SIAM Journal on Numerical Analysis, vol. 42, no. 5, pp. 1862–1874, 2005.
  • Q. Yang, F. Liu, and I. Turner, “Numerical methods for fractional partial differential equations with Riesz space fractional derivatives,” Applied Mathematical Modelling, vol. 34, no. 1, pp. 200–218, 2010.
  • B. I. Henry, T. A. M. Langlands, and P. Straka, “Fractional Fokker-Planck equations for subdiffusion with space- and time-dependent forces,” Physical Review Letters, vol. 105, no. 17, Article ID 170602, 2010. \endinput