The Temporal Second Order Difference Schemes Based on the Interpolation Approximation for the Time Multi-term Fractional Wave Equation (original) (raw)

References

  1. Compte, A., Metzler, R.: The generalized Cattaneo equation for the description of anomalous transport processes. J. Phys. A Math. Gen. 30, 7277–7289 (1997)
    Article MathSciNet MATH Google Scholar
  2. Godoy, S., Garcia-Colin, L.S.: From the quantum random walk to classical mesoscopic diffusion in crystalline solids. Phys. Rev. E 53, 5779–5785 (1996)
    Article Google Scholar
  3. Srivastava, V., Rai, K.N.: A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues. Math. Comput. Model. 51, 616–624 (2010)
    Article MathSciNet MATH Google Scholar
  4. Povstenko, Y.Z.: Fractional Cattaneo-type equations and generalized thermoelasticity. J. Therm. Stress. 34, 97–114 (2011)
    Article Google Scholar
  5. Sun, H.G., Li, Z.P., Zhang, Y., Chen, W.: Fractional and fractal derivative models for transient anomalous diffusion: Model comparison. Chaos Solitons Fractals (2017). https://doi.org/10.1016/j.chaos.2017.03.060
  6. Sun, H.G., Chen, W., Chen, Y.Q.: Variable-order fractional differential operators in anomalous diffusion modeling. Phys. A 388, 4586–4592 (2009)
    Article Google Scholar
  7. Sun, H.G., Zhang, Y., Chen, W., Reeves, D.M.: Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media. J. Contam. Hydrol. 157, 47–58 (2014)
    Article Google Scholar
  8. Luchko, Y.: Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation. J. Math. Anal. Appl. 374, 538–548 (2011)
    Article MathSciNet MATH Google Scholar
  9. Shiralashetti, S.C., Deshi, A.B.: An efficient Haar wavelet collocation method for the numerical solution of multi-term fractional differential equations. Nonlinear Dyn. 83, 293–303 (2016)
    Article MathSciNet Google Scholar
  10. Srivastava, V., Rai, K.N.: A multi-term fractional diffusion equation for oxygen delivery through a capillary to tissues. Math. Comput. Model. 51, 616–624 (2010)
    Article MathSciNet MATH Google Scholar
  11. Jin, B., Lazarov, R., Liu, Y., Zhou, Z.: The Galerkin finite element method for a multi-term time-fractional diffusion equation. J. Comput. Phys. 281, 825–843 (2015)
    Article MathSciNet MATH Google Scholar
  12. Liu, F., Meerschaert, M.M., McGough, R.J., Zhuang, P., Liu, Q.: Numerical methods for solving the multi-term time-fractional wave-diffusion equation. Fract. Calc. Appl. Anal. 16, 9–25 (2013)
    Article MathSciNet MATH Google Scholar
  13. Dehghan, M., Abbaszadeh, M.: Analysis of the element free Galerkin (EFG) method for solving fractional cable equation with Dirichlet boundary condition. Appl. Numer. Math. 109, 208–234 (2016)
    Article MathSciNet MATH Google Scholar
  14. Abbaszadeh, M., Dehghan, M.: An improved meshless method for solving two-dimensional distributed order time-fractional diffusion-wave equation with error estimate. Numer. Algorithm 75, 173–211 (2017)
    Article MathSciNet MATH Google Scholar
  15. Dehghan, M., Abbaszadeh, M.: Element free Galerkin approach based on the reproducing kernel particle method for solving 2D fractional Tricomi-type equation with Robin boundary condition. Comput. Math. Appl. 73, 1270–1285 (2017)
    Article MathSciNet Google Scholar
  16. Oldham, K.B., Spanier, J.: The Fractional Calculus. Academic Press, San Diego (1974)
    MATH Google Scholar
  17. Sun, Z.Z., Wu, X.: A fully discrete difference scheme for a diffusion-wave system. Appl. Numer. Math. 56, 193–209 (2006)
    Article MathSciNet MATH Google Scholar
  18. Du, R., Cao, W., Sun, Z.Z.: A compact difference scheme for the fractional diffusion-wave equation. Appl. Math. Model. 34, 2998–3009 (2010)
    Article MathSciNet MATH Google Scholar
  19. Zhang, Y.N., Sun, Z.Z., Zhao, X.: Compact alternating direction implicit schemes for the two-dimensional fractional diffusion-wave equation. SIAM J. Numer. Anal. 50, 1535–1555 (2012)
    Article MathSciNet MATH Google Scholar
  20. Zhao, X., Sun, Z.Z.: Compact Crank-Nicolson schemes for a class of fractional Cattaneo equation in inhomogeneous medium. J. Sci. Comput. 62, 747–771 (2015)
    Article MathSciNet MATH Google Scholar
  21. 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
  22. Sun, H., Sun, Z.Z., Gao, G.H.: Some temporal second order difference schemes for fractional wave equations. Numer. Methods Part. Differ. Equ. 32, 970–1001 (2016)
    Article MathSciNet MATH Google Scholar
  23. Dehghan, M., Abbaszadeh, M., Deng, W.H.: Fourth-order numerical method for the space-time tempered fractional diffusion-wave equation. Appl. Math. Lett. 73, 120–127 (2017)
    Article MathSciNet MATH Google Scholar
  24. Ghazizadeh, H.R., Maerefat, M., Azimi, A.: Explicit and implicitt finite difference schemes for fractional Cattaneo equation. J. Comput. Phys. 229, 7042–7057 (2010)
    Article MathSciNet MATH Google Scholar
  25. Li, C.P., Cao, J.X.: A finite difference method for time-fractional telegraph equation. In: 2012 IEEE/ASME International Conference, pp. 314–318 (2012)
  26. Vong, S.W., Pang, H.K., Jin, X.Q.: A high-order difference scheme for the generalized Cattaneo equation. East Asian J. Appl. Math. 2, 170–184 (2012)
    Article MathSciNet MATH Google Scholar
  27. Zhou, J., Xu, D., Chen, H.B.: A weak Galerkin finite element method for multi-term time-fractional diffusion equations. East Asian J. Appl. Math. 8, 181–193 (2018)
    MathSciNet Google Scholar
  28. Li, G.S., Sun, C.L., Jia, X.Z., Du, D.H.: Numerical solution to the multi-term time fractional diffusion equation in a finite domain. Numer. Math. Theor. Meth. Appl. 9, 337–357 (2016)
    Article MathSciNet MATH Google Scholar
  29. Zheng, M., Liu, F., Anh, V., Turner, I.: A high-order spectral method for the multi-term time-fractional diffusion equations. Appl. Math. Model. 40, 4970–4985 (2016)
    Article MathSciNet Google Scholar
  30. Salehi, R.: A meshless point collocation method for 2-D multi-term time fractional diffusion-wave equation. Numer. Algorithm 74, 1145–1168 (2017)
    Article MathSciNet MATH Google Scholar
  31. Abdel-Rehim, E.A., El-Sayed, A.M.A., Hashem, A.S.: Simulation of the approximate solutions of the time-fractional multi-term wave equations. Comput. Math. Appl. 73, 1134–1154 (2017)
    Article MathSciNet Google Scholar
  32. Liu, Y.: Strong maximum principle for multi-term time-fractional diffusion equations and its application to an inverse source problem. Comput. Math. Appl. 73, 96–108 (2017)
    Article MathSciNet MATH Google Scholar
  33. Bhrawy, A.H., Zaky, M.A.: A method based on the Jacobi tau approximation for solving multi-term time-space fractional partial differential equations. J. Comput. Phys. 281, 876–895 (2015)
    Article MathSciNet MATH Google Scholar
  34. Dehghan, M., Safarpoor, M., Abbaszadeh, M.: Two high-order numerical algorithms for solving the multi-term time fractional diffusion-wave equations. J. Comput. Appl. Math. 290, 174–195 (2015)
    Article MathSciNet MATH Google Scholar
  35. Ren, J.C., Sun, Z.Z.: Efficient Numerical solution of the multi-term time fractional diffusion-dave equation. East Asian J. Appl. Math. 5, 1–28 (2015)
    Article MathSciNet MATH Google Scholar
  36. Liu, F., Meerschaert, M.M., McGough, R.J., Zhuang, P., Liu, Q.: Numerical methods for solving the multi-term time-fractional wave-diffusion equation. Fract. Calc. Appl. Anal. 16, 1–17 (2013)
    Article MathSciNet MATH Google Scholar
  37. Brunner, H., Han, H., Yin, D.: Artificial boundary conditions and finite difference approximations for a time-fractional diffusion-wave equation on a two-dimensional unbounded spatial domain. J. Comput. Phys. 276, 541–562 (2014)
    Article MathSciNet MATH Google Scholar
  38. Gao, G.H., Alikhanov, A.A., Sun, Z.Z.: The temporal second order difference schemes based on the interpolation approximation for solving the time multi-term and distributed-order fractional sub-diffusion equations. J. Sci. Comput. 73, 93–121 (2017)
    Article MathSciNet MATH Google Scholar
  39. Jiang, S.D., Zhang, J.W., Zhang, Q., Zhang, Z.M.: Fast evaluation of the Caputo fractional derivative and its applications to fractional diffusion equations. Commun. Comput. Phys. 21, 650–678 (2017)
    Article MathSciNet Google Scholar
  40. Yan, Y.G., Sun, Z.Z., Zhang, J.W.: Fast evalution of the Caputo fractional derivative and its applications to fractional diffusion equations: A second-order scheme. Commun. Comput. Phys. 22, 1028–1048 (2017)
    Article MathSciNet Google Scholar
  41. Alikhanov, A.A.: A new difference scheme for the fractional diffusion equation. J. Comput. Phys. 280, 424–438 (2015)
    Article MathSciNet MATH Google Scholar
  42. Quarteroni, A., Valli, A.: Numerical Approximation of Partial Differential Equations. Springer, New York (1997)
    MATH Google Scholar
  43. Sun, Z.Z.: Numerical Methods of Partial Differential Equations, 2nd edn. Science Press, Beijing (2012). in Chinese
    Google Scholar

Download references