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Numerical solution of fractional advection-diffusion equation with a nonlinear source term

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Abstract

In this paper we use the Jacobi collocation method for solving a special kind of the fractional advection-diffusion equation with a nonlinear source term. This equation is the classical advection-diffusion equation in which the space derivatives are replaced by the Riemann-Liouville derivatives of order 0 < σ ≤ 1 and 1 < μ ≤ 2. Also the stability and convergence of the presented method are shown for this equation. Finally some numerical examples are solved using the presented method.

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References

  1. Al-Khaled, K., Momani, S.: An approximate solution for a fractional diffusion-wave equation using the decomposition method. Appl. Math. Comput. 165, 473–483 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. Baeumer, B., Kovacs, M., Meerschaert, M. M.: Numerical solutions for fractional reaction-diffusion equations. Comput. Math. Applic. 55, 2212–2226 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bueno-Orovio, A., Kay, D., Burrage, K.: Fourier spectral methods for fractional–in–space reaction-diffusion equations, to appear in BIT (2014)

  4. Butcher, J.C.: The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods. Wiley, Chichester (1987)

    MATH  Google Scholar 

  5. Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral methods fundamentals in single domains. Springer, Berlin (2006)

    MATH  Google Scholar 

  6. Celik, C., Duman, M.: Crank-Nicolson method for the fractional diffusion equation with the Riesz fractional derivative. J. Comput. Phys. 231, 1743–1750 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  7. Chen, C., Liu, F., Burrage, K.: Finite difference methods and a Fourier analysis for the fractional reaction-subdiffusion equation. Appl. Math. Comput. 198, 754–769 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dehghan, M., Manafian, J., Saadatmandi, A.: Solving nonlinear fractional partial diffrential equations using the homotopy analysis method. Numer. Methods Partial Differ. Equ. 26, 448–479 (2010)

    MATH  MathSciNet  Google Scholar 

  9. Diethelm, K.: The analysis of fractional differential equations. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  10. Ervin, V.J., Heuer, N., Roop, J.P.: Numerical approximation of a time-dependent, nonlinear, space-fractional diffusion equation. SIAM J. Numer. Anal. 45, 572–591 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  11. Ervin, V.J., Roop, J.P.: Variational formulation for the stationary fractional advection dispersion equation. Numer. Methods Partial Differ. Equ. 22, 558–576 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  12. Ervin, V.J., Roop, J.P.: Variational solution of fractional advection-dispersion equations on bounded domains in \(\mathbb {R}^{d}\). Numer. Methods Partial Differ. Equ. 23, 256–281 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. Eslahchi, M.R., Dehghan, M.: Application of Taylor series in obtaining the orthogonal operational matrix. Comput. Math. Applic. 61, 2596–2604 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  14. Gautschi, W.: Construction of GaussChristoffel quadrature formulas. Math. Comput. 22, 251–270 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  15. Gautschi, W.: Orthogonal polynomials–construction theory and applications. J. Comput. Appl. Math. 1213, 61–67 (1985)

    Article  MathSciNet  Google Scholar 

  16. Golbabai, A., Sayevand, K.: Analytical modelling of fractional advection–dispersion equation defined in a bounded space domain. Math. Comput. Modelling 53, 1708–1718 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  17. Golub, G.H., Welsch, J.H.: Calculation of Gauss quadrature rule. Math. Comput. 23, 221–230 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  18. Guo, B.Y.: Spectral methods and their applications. World Scientific, Singapore (1998)

    Book  MATH  Google Scholar 

  19. Gu, Y.T., Zhuang, P.H.: Anomalous subdiffusion equations by the meshless collocation method. Aust. J. Mech. Eng. 10(1), 1–8 (2012)

    Article  MathSciNet  Google Scholar 

  20. Gu, Y.T., Zhuang, P.H., Liu, Q.X.: An advanced meshless method for time fractional diffusion equation. Int. J. Comput. Methods 8(4), 653–665 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  21. Huang, Q., Huang, G., Zhan, H.: A finite element solution for the fractional advection-dispersion equation. Adv. Water Resour. 31, 1578–1589 (2008)

    Article  Google Scholar 

  22. Huang, F., Liu, F., et al.: The fundamental solution of the space-time fractional advection-dispersion equation. J. Appl. Math. Comput. 18, 339–350 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. Jiang, W., Lin, Y.: Approximate solution of the fractional advection–dispersion equation. Comput. Phy. Commun. 181, 557–561 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  24. Jiang, H., Liu, F., Turner, I., Burrage, K.: Analytical solutions for the multi-term time-space Caputo-Riesz fractional advection-diffusion equations on a finite domain. J. Math. Anal. Applic. 389, 1117–1127 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  25. Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric Orthogonal Polynomials Theirq-Analogues. Springer, Berlin (2010)

    Book  Google Scholar 

  26. Liu, F., Anh, V., Turner, I., Zhuang, P.: Time fractional advection-dispersion equation. J. Appl. Math. Comput. 13, 233–245 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  27. Liu, F., Turner, I., Anh, V., Yang, Q., Burrage, K.: A numerical method for the fractional Fitzhugh–Nagumo monodomain model. ANZIAM 54, C608–C629 (2012)

    MathSciNet  Google Scholar 

  28. Liu, F., Chen, S., Turner, I., Burrage, K., Anh, V., et al.: Numerical simulation for two–dimensional Riesz space fractional diffusion equations with a nonlinear reaction term. Cent. Eur. J. Phys. in press (2013)

  29. Liu, F., Zhuang, P., Burrage, K.: Numerical methods and analysis for a class of fractional advection–dispersion models. Comput. Math. Appl. 64, 2990–3007 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  30. Liu, F., Zhuang, P., Anh, V., Turner, I., Burrage, K.: Stability and convergence of the difference methods for the space–time fractional advection–diffusion equation. App. Math. Comput. 191, 12–20 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  31. Liu, F., Anh, V., Turner, I.: Numerical solution of the space fractional Fokker–Planck Equation. J. Comput. Appl. Math. 166, 209–219 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  32. Liu, Q.X., Gu, Y.T., Zhuang, P.H., Liu, F., Nie, Y.F.: An implicit RBF meshless approach for time fractional diffusion equations. Comput. Mech. 48(1), 1–12 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  33. Li, X., Xu, C.: A space time spectral methods for the time fractional diffusion equation. SIAM J. Numer. Anal. 47, 2108–2131 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  34. Meerschaert, M.M., Tadjeran, C.: Finite difference approximations for fractional advection-dispersion equations. J. Comput. Appl. Math. 172, 65–77 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  35. Meerschaert, M.M., Tadjeran, C.: Finite difference approximation for two-sided space-fractional partial differential equations. Appl. Numer. Math. 56, 80–90 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  36. Mohebbi, A., Abbaszadeh, M., Dehghan, M.: A high order and unconditionally stable scheme for the modified anomalous fractional subdifusion equation with a nonlinear source term. J. Comput. Phys. 240, 36–48 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  37. Ozdemir, N., Avci, D., Billur Iskender, B.: The numerical solutions of a two–dimensional space-time Riesz–Caputo fractional diffusion equation. Int. J. Optim. Control Theor. Appl. 1, 17–26 (2011)

    Article  Google Scholar 

  38. Pandey, R.K., Singh, O.P., Baranwal, V.K.: An analytic algorithm for the space–time fractional advection–dispersion equation. Comput. Phy. Commun. 182, 1134–1144 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  39. Podlubny, I.: Fractional differential equations. Academic (1999)

  40. Roop, J.P.: Numerical approximation of a one-dimensional space fractional advection–dispersion equation with boundary layer. Comput. Math. Appl. 56, 1808–1819 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  41. Roop, J.P.: Variational solution of the fractional advection dispersion equation, PhD thesis (2004)

  42. Saadatmandi, A., Dehghan, M., Azizi, M.R.: The sinc–Legendre collocation method for a class of fractional convection–diffusion equations with variable coefficients. Commun. Nonlinear Sci. Numer. Simulat. 17, 593–601 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  43. Saadatmandi, A., Dehghan, M.: A tau approach for solution of the space fractional diffusion equation. Comput. Math. Appl. 62, 1135–1142 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  44. El–Sayed, A.M.A., Behiry, S.H., Raslan, W.E.: Adomian’s decomposition method for solving an intermediate fractional advection–dispersion equation. Comput. Math. Appl. 59, 1759–1765 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  45. Shen, J., Tang, T., Wang, L–L.: Spectral methods, algorithms, analysis and applications. Springer, Berlin (2011)

    MATH  Google Scholar 

  46. Shen, S., Anh, V., Turner, I.: The fundamental solution and numerical solution of the Riesz fractional advection–dispersion equation. IMA J. Appl. Math. 73, 850–872 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  47. Shen, S., Liu, F., Anh, V.: Numerical approximations and solution techniques for the space–time Riesz–Caputo fractional advection–diffusion equation. Numer. Algor. 56, 383–403 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  48. Sousa, E.: Finite difference approximations for a fractional advection-diffusion problem. J. Comput. Phys. 228, 4038–4054 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  49. Su, L., Wang, W., Xu, Q.: Finite difference methods for fractional dispersion equations. Appl. Math. Comput. 216, 3329–3334 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  50. Wang, K., Wang, H.: A fast characteristic finite difference method for fractional advection–diffusion equations. Adv. Water Resour. 34, 810–816 (2011)

    Article  Google Scholar 

  51. Yang, Q., Moroney, T., Burrage, K., Turner, I., Liu, F.: Novel numerical methods for time-space fractional reaction–diffusion equations in two domains. J. ANZIAM 52, 461–473 (2004)

    Google Scholar 

  52. Yang, Q., Turner, I., Liu, F.: Analytical and numerical solutions for the time and space-symmetric fractional diffusion equation. J. ANZIAM 50, 800–814 (2009)

    MathSciNet  Google Scholar 

  53. Yang, Q., Liu, F., Turner, I.: Numerical methods for fractional partial differential equations with Riesz space fractional derivatives. Appl. Math. Model. 34, 200–218 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  54. Zheng, Y., Li, C., Zhao, Z.: A note on the finite element method for the space–fractional advection diffusion equation. Comput. Math. Appl. 59, 1718–1726 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  55. Zhuang, P., Liu, F., Anh, V., Turner, I.: Numerical methods for the variable–order fractional advection–diffusion with a nonlinear source term. SIAM J. Numer. Anal. 47, 1760–1781 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  56. Zhuang, P., Gu, Y.T., Liu, F., Turner, I., Yarlagadda, P.K.: Time–dependent fractionaladvection–diffusion equations by an implicit MLS meshless method. Submitted to InternationalJournal for Numerical Methods in Engineering, (2011). (Also see the Technical reportr http://eprints.qut.edu.au/45619).

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Authors and Affiliations

  1. Department of Applied Mathematics, Faculty of Mathematical Sciences, Tarbiat Modares University, P.O. Box 14115-134, Tehran, Iran

    M. Parvizi & M. R. Eslahchi

  2. Department of Applied Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., 15914, Tehran, Iran

    Mehdi Dehghan

Authors
  1. M. Parvizi
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  2. M. R. Eslahchi
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  3. Mehdi Dehghan
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Correspondence to M. R. Eslahchi or Mehdi Dehghan.

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Parvizi, M., Eslahchi, M.R. & Dehghan, M. Numerical solution of fractional advection-diffusion equation with a nonlinear source term. Numer Algor 68, 601–629 (2015). https://doi.org/10.1007/s11075-014-9863-7

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  • DOI: https://doi.org/10.1007/s11075-014-9863-7

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