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A second-order compact difference scheme for the fourth-order fractional sub-diffusion equation

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Abstract

In the present work, a compact difference scheme with convergence order O(τ 2 + h 4) is proposed for the fourth-order fractional sub-diffusion equation, where h and τ are space and temporal step length, respectively. The method is based on applying the L2−1 σ formula to approximate the time Caputo fractional derivative and employing compact operator to approximate the spatial fourth-order derivative. Using the special properties of L2−1 σ formula and mathematical induction method, we obtain the unconditional stability and convergence for our scheme by discrete energy method. Furthermore, the extension to the two-dimensional case is also considered. Numerical examples are given to verify the theoretical analysis and efficiency of the new developed scheme.

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References

  1. Podlubny, I.: Fractional Differential Equations. Academic (1999)

  2. Hilfer, R. (ed.): Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

  3. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equation. Elsevier, Amsterdam (2006)

    MATH  Google Scholar 

  4. Solomon, T.H., Weeks, E.R., Swinney, H.L.: Observations of anomalous diffusion and Lvy flights in a 2-dimensional rotating flow. Phys. Rev. Lett. 71, 3975–3979 (1993)

    Article  Google Scholar 

  5. Klages, R., Radons, G., Sokolov, I.M.: Anomalous Transport: Foundations and Applications. Wiley, Weinheim (2008)

    Book  Google Scholar 

  6. Balakrishnan, V.: Anomalous diffusion in one dimension. Phys. A 132, 569–580 (1985)

    Article  MATH  Google Scholar 

  7. Schneider, W.R., Wyss, W.: Fractional diffusion and wave equations. J. Math. Phys. 30, 134–144 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  8. Mainardi, F.: Fractional diffusive waves in viscoelastic solids. In: Wegner, J.I., Norwood, F.R (eds.) Nonlinear Waves in Solids. ASME/AMR, pp. 93–97. Fairfield, NJ (1995)

  9. Mainardi, F.: Some basic problems in continuum and statistical mechanics. In: rpinteri, A., Mainardi, F (eds.) Fractals and Fractional Calculus in Continuum Mechanics. Springer, Wien (1997)

  10. Das, S.: Functional Fractional Calculus. Springer, Berlin (2011)

  11. Tomovski, ž., Sandev, T.: Exact solutions for fractional diffusion equation in a bounded domain with different boundary conditions. Nonlinear Dynam. 71, 671–683 (2013)

    Article  MathSciNet  Google Scholar 

  12. Langlands, T.A.M., Henry, B.I.: The accuracy and stability of an implicit solution method for the fractional diffusion equation. J. Comput. Phys. 205, 719–736 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Yuste, S.B., Acedo, L.: An explicit finite difference method and a new Von-Neumann type stability analysis for fractional diffusion equations. SIAM J. Numer. Anal. 42, 1862–1874 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. Yuste, S.: Weighted average finite difference methods for fractional diffusion equations. J. Comput. Phys. 216, 264–274 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  15. Tadjeran, C., Meerschaert, M.M., Scheffler, H.P.: A second-order accurate numerical approximation for the fractional diffusion equation. J. Comput. Phys. 213, 205–213 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lin, X., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Chen, C.M., Liu, F., Turner, I., Anh, V.: A Fourier method for the fractional diffusion equation describing subdiffusion. J. Comput. Phys. 227, 886–897 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Zhuang, P., Liu, F., Anh, V., Turner, I.: New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation. SIAM J. Numer. Anal. 46, 1079–1095 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Liu, F., Yang, C., Burrage, K.: Numerical method and analytical technique of the modified anomalous subdiffusion equation with a nonlinear source term. J. Comput. Appl. Math. 231, 160–176 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sun, Z.Z., Wu, X.N.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  21. Cui, M.: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228, 7792–7804 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Chen, C.M., Liu, F., Anh, V., Turner, I.: Numerical schemes with high spatial accuracy for a variable-order anomalous subdiffusion equations. SIAM J. Sci. Comput 32, 1740–1760 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gao, G.H., Sun, Z.Z.: A compact difference scheme for the fractional subdiffusion equations. J. Comput. Phys. 230, 586–595 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zhang, Y.N., Sun, Z.Z., Wu, H.W.: Error estimates of Crank-Nicolson type difference schemes for the subdiffusion equation. SIAM J. Numer. Anal. 49, 2302–2322 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Zhao, X., Xu, Q.: Efficient numerical schemes for fractional sub-diffusion equation with the spatially variable coefficient. Appl. Math. Model 38(15-16), 3848–3859 (2014)

    Article  MathSciNet  Google Scholar 

  26. Vong, S., Lyu, P., Wang, Z.: A compact difference scheme for fractional sub-diffusion equations with the spatially variable coefficient under Neumann boundary conditions. J. Sci. Comput. 66(2), 725–739 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  27. Zhao, X., Sun, Z.Z., Karniadakis, G.E.: Second-order approximations for variable order fractional derivatives: algorithms and applications. J. Comput. Phys. 293, 184–200 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gao, G.H., Sun, H.W., Sun, Z.Z.: Stability and convergence of finite difference schemes for a class of time-fractional sub-diffusion equations based on certain superconvergence. J. Comput. Phys. 280, 510–528 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  29. Alikhanov, A.A.: A new difference scheme for the time fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  30. Sneddon, I.N.: Fourier Transforms. McGraw Hill, New York (1951)

    MATH  Google Scholar 

  31. Oldhan, K.B., Spainer, J.: The Fractional Calculus. Academic Press, New York (1974)

    Google Scholar 

  32. Karpman, V.I.: Stabilization of soliton instabilities by higher-order dispersion: fourth order nonlinear Schrödinger-type equations. Phys. Rev. E 53(2), 1336–1339 (1996)

    Article  Google Scholar 

  33. Karpman, V.I., Shagalov, A.G.: Stability of soliton described by nonlinear Schrödinger-type equations with higher order dispersion. Phys. D 144, 194–210 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  34. Agrawal, Om P.: A general solution for a fourth-order fractional diffusion-wave equation in a bounded domain. Comput. Struct. 79, 1497–1501 (2001)

    Article  Google Scholar 

  35. Jafari, H., Dehghan, M., Sayevand, K.: Solving a fourth-order fractional diffusion-wave equation in a bounded domain by decomposition method. Wiley InterScience, New York (2007)

    MATH  Google Scholar 

  36. Golbabai, A., Sayevand, K.: Fractional calculus—a new approach to the analysis of generalized fourthorder diffusion-wave equations. Appl. Math. Comput. 61, 2227–2231 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  37. Liu, Y., Fang, Z.C., Li, H., He, S.: A mixed finite element method for a time-fractional fourth-order partial differential equation. Appl. Math. Comput. 243, 703–717 (2014)

    MathSciNet  MATH  Google Scholar 

  38. Liu, Y., Du, Y., Li, H., He, S., Gao, W.: Finite difference/finite element method for a nonlinear time-fractional fourth-order reaction-diffusion problem. Comput. Math. Appl. 70(4), 573–591 (2015)

    Article  MathSciNet  Google Scholar 

  39. Hu, X.L., Zhang, L.M.: On finite difference methods for fourth-order fractional diffusion-wave and subdiffusion systems. Appl. Math. Comput. 218, 5019–5034 (2012)

    MathSciNet  MATH  Google Scholar 

  40. Guo, J., Li, C.P., Ding, H.F.: Finite difference methods for time subdiffusion equation with space fourth order. Commun. Appl. Math. Comput. 28, 96–108 (2014). in Chinese

    MathSciNet  MATH  Google Scholar 

  41. Siddiqi, S.S., Arshed, S.: Numerical solution of time-fractional fourth-order partial differential equations. Int. J. Comput. Math. 92(7), 1496–1518 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  42. Ji, C.C., Sun, Z.Z., Hao, Z.P.: Numerical algorithms with high spatial accuracy for the fourth-order fractional sub-diffusion equations with the first Dirichlet boundary conditions. J. Sci. Comput. 66(3), 1148–1174 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  43. Liao, H.L., Sun, Z.Z.: Maximum norm error bounds of ADI and compact ADI methods for solving parabolic equations. Numer. Methods Partial Differ. Equ. 26, 37–60 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  44. Samarskii, A.A., Andreev, V.B.: Finite Difference Methods for Elliptic Equation. Moscow, Nauka (1976). in Russian

    MATH  Google Scholar 

  45. Sun, Z.Z.: Numerical Methods of Partial Differential Equations, 2D edn. Science Press, Beijing (2012). in Chinese

    Google Scholar 

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Authors and Affiliations

  1. State Key Laboratory for Geomechanics and Deep Underground Engineering, China University of Mining and Technology, Xuzhou, Jiangsu, 221116, People’s Republic of China

    Pu Zhang & Hai Pu

  2. School of Basic Education Sciences, Xuzhou Medical University, Xuzhou, Jiangsu, 221004, People’s Republic of China

    Pu Zhang

  3. School of Mechanics and civil engineering, China University of Mining and Technology, Xuzhou, Jiangsu, 221116, People’s Republic of China

    Hai Pu

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  1. Pu Zhang
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  2. Hai Pu
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Correspondence to Hai Pu.

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Zhang, P., Pu, H. A second-order compact difference scheme for the fourth-order fractional sub-diffusion equation. Numer Algor 76, 573–598 (2017). https://doi.org/10.1007/s11075-017-0271-7

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  • DOI: https://doi.org/10.1007/s11075-017-0271-7

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