409
Views
16
CrossRef citations to date
0
Altmetric
Original Articles

Galerkin–Legendre spectral method for the distributed-order time fractional fourth-order partial differential equation

&
Pages 1183-1196 | Received 24 Dec 2018, Accepted 13 Apr 2019, Published online: 27 Apr 2019

References

  • A.A. Alikhanov, A new difference scheme for the time fractional diffusion equation, J. Comput. Phys. 280 (2015), pp. 424–438. doi: 10.1016/j.jcp.2014.09.031
  • A.H. Bhrawy, E.H. Doha, S.S. Ezz-Eldien, and M.A. Abdelkawy, A numerical technique based on the shifted Legendre polynomials for solving the time-fractional coupled KdV equations, Calcolo 53 (2016), pp. 1–17. doi: 10.1007/s10092-014-0132-x
  • T.A. Biala and A.Q.M. Khaliq, Parallel algorithms for nonlinear time-space fractional parabolic PDEs, J. Comput. Phys. 375 (2018), pp. 135–154. doi: 10.1016/j.jcp.2018.08.034
  • W. Bu, A. Xiao, and W. Zeng, Finite difference/finite element methods for distributed-order time fractional diffusion equations, J. Sci. Comput. 72 (2017), pp. 422–441. doi: 10.1007/s10915-017-0360-8
  • B.L. Buzbee, E.W. Dorr, J.A. George, and G.H. Golub, The direct solution of the discrete poisson's equation on irregular domains, SIAM J. Numer. Anal. 8 (1971), pp. 722–736. doi: 10.1137/0708066
  • A.V. Chechkin, R. Gorenflo, and I.M. Sokolov, Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations, Phys. Rev. E 66 (2002), pp. 046129. doi: 10.1103/PhysRevE.66.046129
  • H. Chen and S. Lü, Finite difference/spectral approximations for the distributed order time fractional reaction–diffusion equation on an unbounded domain, J. Comput. Phys. 315 (2016), pp. 84–97. doi: 10.1016/j.jcp.2016.03.044
  • M. Dehghan, M. Abbaszadeh, and A. Mohebbi, Legendre spectral element method for solving time fractional modified anomalous sub-diffusion equation, Appl. Math. Model. 40 (2016), pp. 3635–3654. doi: 10.1016/j.apm.2015.10.036
  • S. Esmaeili and R. Garrappa, A pseudo-spectral scheme for the approximate solution of a time-fractional diffusion equation, Int. J. Comput. Math. 92 (2015), pp. 980–994. doi: 10.1080/00207160.2014.915962
  • G. Gao, A.A. Alikhanov, and Z. Sun, 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 (2017), pp. 93–121. doi: 10.1007/s10915-017-0407-x
  • G. Gao, H. Sun, and Z. Sun, Some high-order difference schemes for the distributed-order differential equations, J. Comput. Phys. 298 (2015), pp. 337–359. doi: 10.1016/j.jcp.2015.05.047
  • X. Gu, T. Huang, C. Ji, B. Carpentieri, and A.A. Alikhanov, Fast iterative method with a second-order implicit difference scheme for time-space fractional convection–diffusion equation, J. Sci. Comput. 72 (2017), pp. 957–985. doi: 10.1007/s10915-017-0388-9
  • S. Guo, L. Mei, Z. Zhang, and Y. Jiang, Finite difference/spectral-Galerkin method for a two-dimensional distributed-order time-space fractional reaction–diffusion equation, Appl. Math. Lett. 85 (2018), pp. 157–163. doi: 10.1016/j.aml.2018.06.005
  • C. Ji, Z. Sun, and Z. Hao, Numerical algorithms with high spatial accuracy for the fourth-order fractional sub-diffusion equations with the first Dirichlet boundary conditions, J. Sci. Comput. 66 (2016), pp. 1148–1174. doi: 10.1007/s10915-015-0059-7
  • Z. Jiao, Y. Chen, and I. Podlubny, Distributed-Order Dynamic Systems: Stability, Simulation, Applications and Perspectives, Springer, London, 2012.
  • E. Kharazmi, M. Zayernouri, and G.E. Karniadakis, Petrov-Galerkin and spectral collocation methods for distributed order differential equations, SIAM J. Sci. Comput. 39 (2017), pp. A1003–A1037. doi: 10.1137/16M1073121
  • A. Kilbas, H.M. Srivastava, and J.J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam, 2006.
  • H.-L. Liao, D. Li, and J. Zhang, Sharp error estimate of the nonuniform L1 formula for linear reaction–subdiffusion equations, SIAM J. Numer. Anal. 56 (2018), pp. 1112–1133. doi: 10.1137/17M1131829
  • M. Li, X. Gu, C. Huang, M. Fei, and G. Zhang, A fast linearized conservative finite element method for the strongly coupled nonlinear fractional Schrödinger equations, J. Comput. Phys. 358 (2018), pp. 256–282. doi: 10.1016/j.jcp.2017.12.044
  • X. Li and H. Rui, Two temporal second-order H1-Galerkin mixed finite element schemes for distributed-order fractional sub-diffusion equations, Numer. Algor. 79 (2018), pp. 1107–1130. doi: 10.1007/s11075-018-0476-4
  • X. Li, H. Rui, and Z. Liu, Two alternating direction implicit spectral methods for two-dimensional distributed-order differential equation, Numer. Algor. doi:10.1007/s11075-018-0606-z.
  • X. Li and C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal. 47 (2009), pp. 2108–2131. doi: 10.1137/080718942
  • M. Li and Y. Zhao, A fast energy conserving finite element method for the nonlinear fractional Schrödinger equation with wave operator, Appl. Math. Comput. 338 (2018), pp. 758–773.
  • X. Liang and A.Q.M. Khaliq, An efficient Fourier spectral exponential time differencing method for the space-fractional nonlinear Schrödinger equations, Comput. Math. Appl. 75 (2018), pp. 4438–4457. doi: 10.1016/j.camwa.2018.03.042
  • Y. Liu, Y. Du, H. Li, S. He, and W. Gao, Finite difference/finite element method for a nonlinear timefractional fourth-order reaction–diffusion problem, Comput. Math. Appl. 70 (2015), pp. 573–591. doi: 10.1016/j.camwa.2015.05.015
  • Y. Liu, Z. Fang, H. Li, and S. He, A mixed finite element method for a time-fractional fourth-order partial differential equation, Appl. Math. Comput. 243 (2014), pp. 703–717.
  • F. Mainardi, G. Pagnini, and R. Gorenflo, Some aspects of fractional diffusion equations of single and distributed order, Appl. Math. Comput. 187 (2007), pp. 295–305.
  • K. Oldhan and J. Spainer, The Fractional Calculus, Academic Press, New York, 1974.
  • V.G. Pimenov, A.S. Hendy, and R.H. De Staelen, On a class of non-linear delay distributed order fractional diffusion equations, J. Comput. Appl. Math. 318 (2017), pp. 433–443. doi: 10.1016/j.cam.2016.02.039
  • I. Podlubny, Fractional Differential Equations, Academic Press, SanDiego, 1999.
  • M. Ran and C. Zhang, New compact difference scheme for solving the fourth-order time fractional sub-diffusion equation of the distributed order, Appl. Numer. Math. 129 (2018), pp. 58–70. doi: 10.1016/j.apnum.2018.03.005
  • J. Shen, Efficient spectral-Galerkin method I. direct solvers of second-and fourth-order equations using Legendre polynomials, SIAM J. Sci. Comput. 15 (1994), pp. 1489–1505. doi: 10.1137/0915089
  • J. Shen, T. Tang, and L. Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer Science & Business Media, Berlin, 2011.
  • S. Siddiqi and S. Arshed, Numerical solution of time-fractional fourth-order partial differential equations, Int. J. Comput. Math. 92 (2014), pp. 1496–1518. doi: 10.1080/00207160.2014.948430
  • W. Tian, H. Zhou, and W. Deng, A class of second order difference approximations for solving space fractional diffusion equations, Math. Comput. 84 (2015), pp. 1703–1727. doi: 10.1090/S0025-5718-2015-02917-2
  • P. Wang and C. Huang, An energy conservative difference scheme for the nonlinear fractional Schrödinger equations, J. Comput. Phys. 293 (2015), pp. 238–251. doi: 10.1016/j.jcp.2014.03.037
  • D. Wang, A. Xiao, and H Liu, Dissipativity and stability analysis for fractional functional differential equations, Fract. Calc. Appl. Anal. 18 (2015), pp. 1399–1422.
  • H. Wang and X. Zhang, A high-accuracy preserving spectral Galerkin method for the Dirichlet boundary-value problem of variable-coefficient conservative fractional diffusion equations, J. Comput. Phys. 281 (2015), pp. 67–81. doi: 10.1016/j.jcp.2014.10.018
  • L. Wei and Y. He, Analysis of a fully discrete local discontinuous Galerkin method for time-fractional fourth-order problems, Appl. Math. Model. 38 (2014), pp. 1511–1522. doi: 10.1016/j.apm.2013.07.040
  • X. Yang, H. Zhang, and D. Xu, WSGD-OSC Scheme for two-dimensional distributed order fractional reaction–diffusion equation, J. Sci. Comput. 76 (2018), pp. 1502–1520. doi: 10.1007/s10915-018-0672-3
  • F. Zeng, F. Liu, C. Li, K. Burrage, and I. Turner, A Crank–Nicolson ADI spectral method for a two-dimensional Riesz space fractional nonlinear reaction–diffusion equation, SIAM J. Numer. Anal. 52 (2014), pp. 2599–2622. doi: 10.1137/130934192
  • H. Zhang, X. Yang, and D. Xu, A high-order numerical method for solving the 2d fourth-order reaction–diffusion equation, Numer. Algor. doi:10.1007/s11075-018-0509-z.
  • P. Zhang and H. Pu, A second-order compact difference scheme for the fourth-order fractional sub-diffusion equation, Numer. Algor. 76 (2017), pp. 573–59. doi: 10.1007/s11075-017-0271-7
  • Y. Zhao, W. Bu, X. Zhao, and Y. Tang, Galerkin finite element method for two-dimensional space and time fractional Bloch-Torrey equation, J. Comput. Phys. 350 (2017), pp. 117–135. doi: 10.1016/j.jcp.2017.08.051

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.