A high-order compact alternating direction implicit method for solving the 3D time-fractional diffusion equation with the Caputo–Fabrizio operator

Abstract

In this paper, a high-order compact finite difference method (CFDM) with an operator-splitting technique for solving the 3D time-fractional diffusion equation is considered. The Caputo–Fabrizio time operator is evaluated by the \(L_1\) approximation, and the second-order space derivatives are approximated by the compact CFDM to obtain a discrete scheme. Alternating direction implicit method (ADI) is used to split the problem into three separate one-dimensional problems. The local truncation error analysis is discussed. Moreover, the convergence and stability of the numerical method are investigated. Finally, some numerical examples are presented to demonstrate the accuracy of the compact ADI method.

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References

  1. 1.

    Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)

    Google Scholar 

  2. 2.

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

    Google Scholar 

  3. 3.

    Samko, S.G., Kilbas, A.A., Marichev, O.I.: Fractional Integrals and Derivatives Theory and Applications. Gordon and Breach, Switzerland (1993)

    Google Scholar 

  4. 4.

    Ray, S.S., Sahoo, S.: Generalized Fractional Order Differential Equations Arising in Physical Models. CRC Press, Boca Raton (2018)

    Google Scholar 

  5. 5.

    Daftardar-Gejji, V.: Fractional Calculus and Fractional Differential Equations. Springer, Berlin (2018)

    Google Scholar 

  6. 6.

    Fallahgoul, H., Focardi, S., Fabozzi, F.: Fractional Calculus and Fractional Processes with Applications to Financial Economics: Theory and Application. Academic Press, Cambridge (2016)

    Google Scholar 

  7. 7.

    Baleanu, D., Lopes, A.M.: Handbook of Fractional Calculus with Applications, Part A, vol. 7. Walter de Gruyter, Berlin (2019)

    Google Scholar 

  8. 8.

    Rudolf, H.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)

    Google Scholar 

  9. 9.

    Agrawal, O.P., Baleanu, D.: A Hamiltonian formulation and a direct numerical scheme for fractional optimal control problems. J. Vib. Control 13(9–10), 1269–1281 (2017)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Su, A., Kanoria, M.: Fractional heat conduction with finite wave speed in a thermo-visco-elastic spherical shell. Lat. Am. J. Solids Struct. 11(7), 1132–1162 (2014)

    Google Scholar 

  11. 11.

    Kumar, A., Bhardwaj, A., Kumar, B.R.: A meshless local collocation method for time fractional diffusion wave equation. Comput. Math. Appl. 78(6), 1851–1861 (2019)

    MathSciNet  Google Scholar 

  12. 12.

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

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Podlubny, I., Chechkin, A., Skovranek, T., Chen, Y., Jara, B.M.V.: Matrix approach to discrete fractional calculus II: partial fractional differential equations. J. Comput. Phys. 228(8), 3137–3153 (2009)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Deng, W.: Finite element method for the space and time fractional Fokker–Planck equation. SIAM J. Numer. Anal. 47(1), 204–226 (2008)

    MathSciNet  MATH  Google Scholar 

  15. 15.

    Ammi, M.S.S., Jamiai, I., Torres, D.F.: A finite element approximation for a class of Caputo time-fractional diffusion equations. Comput. Math. Appl. 78(5), 1334–1344 (2019)

    MathSciNet  Google Scholar 

  16. 16.

    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(5), 1862–1874 (2005)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Duo, S., van Wyk, H.W., Zhang, Y.: A novel and accurate finite difference method for the fractional Laplacian and the fractional Poisson problem. J. Comput. Phys. 355(15), 233–252 (2018)

    MathSciNet  MATH  Google Scholar 

  18. 18.

    Guo, B., Xu, Q., Yin, Z.: Implicit finite difference method for fractional percolation equation with Dirichlet and fractional boundary conditions. Appl. Math. Mech. 37(3), 403–416 (2016)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Zahra, W.K., Hikal, M.M.: Non standard finite difference method for solving variable order fractional optimal control problems. J. Vib. Control 23(6), 948–958 (2017)

    MathSciNet  MATH  Google Scholar 

  20. 20.

    Yuan, Y., Li, C., Sun, T., Liu, Y.: Characteristic fractional step finite difference method for nonlinear section coupled system. Appl. Math. Mech. 35(10), 1311–1330 (2014)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Ansari, R., Hosseini, K., Darvizeh, A., Daneshian, B.: A sixth-order compact finite difference method for non-classical vibration analysis of nanobeams including surface stress effects. Appl. Math. Comput. 219(10), 4977–4991 (2013)

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Ameen, R., Jarad, F., Abdeljawad, T.: Ulam stability for delay fractional differential equations with a generalized Caputo derivative. Filomat 32(15), 5265–5274 (2018)

    MathSciNet  Google Scholar 

  23. 23.

    Kratou, M.: Ground state solutions of p-Laplacian singular Kirchhoff problem involving a Riemann–Liouville fractional derivative. Filomat 33(7), 2073–2088 (2019)

    MathSciNet  Google Scholar 

  24. 24.

    Ragusa, M.A.: Necessary and sufficient condition for a VMO function. Appl. Math. Comput. 218(24), 11952–11958 (2012)

    MathSciNet  MATH  Google Scholar 

  25. 25.

    Li, L., Jiang, Z., Yin, Z.: Fourth-order compact finite difference method for solving two-dimensional convection–diffusion equation. Adv. Differ. Equ. 2018, 234 (2018)

    MathSciNet  MATH  Google Scholar 

  26. 26.

    Usman, M., Badshah, N., Ghaffa, F.: Higher order compact finite difference method for the solution of 2-D time fractional diffusion equation. Matrix Sci. Math. 2, 4–8 (2018)

    Google Scholar 

  27. 27.

    Cui, M.: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228(20), 7792–7804 (2009)

    MathSciNet  MATH  Google Scholar 

  28. 28.

    Cui, M.: Compact difference scheme for time-fractional fourth-order equation with first Dirichlet boundary condition. East Asian J. Appl. Math. 9, 45–66 (2019)

    MathSciNet  Google Scholar 

  29. 29.

    Wu, F., Li, D., Wen, J., Duan, J.: Stability and convergence of compact finite difference method for parabolic problems with delay. Appl. Math. Comput. 322, 129–139 (2018)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Özişik, M.N.: Finite difference methods in heat transfer. CRC Press, Boca Raton (1994)

    Google Scholar 

  31. 31.

    Wu, F., Cheng, X., Li, D., Duan, J.: A two-level linearized compact ADI scheme for two-dimensional nonlinear reaction diffusion equations. Comput. Math. Appl. 75(8), 2835–2850 (2018)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Cheng, X., Duan, J., Li, D.: A novel compact ADI scheme for two-dimensional Riesz space fractional nonlinear reaction-diffusion equations. Appl. Math. Comput. 346, 452–464 (2019)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Hu, D., Cao, X.: A fourth-order compact ADI scheme for two-dimensional Riesz space fractional nonlinear reaction-diffusion equation. Int. J. Comput. Math. 97, 1–21 (2019)

    MathSciNet  Google Scholar 

  34. 34.

    Ishii, A.L., Healy, R.W., Striegl, R.G.: A Numerical Solution for the Diffusion Equation in Hydrogeologic Systems. Department of the Interior, US Geological Survey, Reston (1989)

    Google Scholar 

  35. 35.

    Roisin, B.C.: Environmental Transport and Fate. Thayer School of Engineering Dartmouth College, University Lecture, Hanover (2012)

    Google Scholar 

  36. 36.

    Cui, M.: Convergence analysis of high-order compact alternating direction implicit schemes for the two-dimensional time fractional diffusion equation. Numer. Algorithms 62, 383–409 (2013)

    MathSciNet  MATH  Google Scholar 

  37. 37.

    Liu, Z., Cheng, A., Li, X.: A second order Crank–Nicolson scheme for fractional Cattaneo equation based on new fractional derivative. Appl. Math. Comput. 311, 361–374 (2017)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Caputo, M., Fabrizio, M.: A new definition of fractional derivative without singular kernel. Prog. Fract. Differ. Appl. 1(2), 1–13 (2015)

    Google Scholar 

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Correspondence to Hossein Aminikhah.

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Abdi, N., Aminikhah, H., Refahi Sheikhani, A. et al. A high-order compact alternating direction implicit method for solving the 3D time-fractional diffusion equation with the Caputo–Fabrizio operator. Math Sci (2020). https://doi.org/10.1007/s40096-020-00346-5

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Keywords

  • 3D time-fractional diffusion equation
  • Compact scheme
  • Finite difference method
  • Alternating direction implicit method
  • Caputo–Fabrizio operator