508
Views
2
CrossRef citations to date
0
Altmetric
Research Article

An indirect collocation method for variable-order fractional wave equations on uniform or graded meshes and its optimal error estimates

, &
Pages 2296-2309 | Received 11 Apr 2020, Accepted 12 Feb 2021, Published online: 15 Mar 2021

References

  • R.A. Adams and J.J.F. Fournier, Sobolev Spaces, Elsevier, San Diego, 2003.
  • R. Bagley and P. Torvik, On the fractional calculus model of viscoelastic behavior, J. Rheol. 30 (1986), pp. 133–155.
  • H. Chen and H. Wang, Numerical simulation for conservative fractional diffusion equations by an expanded mixed formulation, J. Comput. Appl. Math. 296 (2016), pp. 480–498.
  • R. Du, A.A. Alikhanov, and Z.Z. Sun, Temporal second order difference schemes for the multi-dimensional variable-order time fractional sub-diffusion equations, Comput. Math. Appl. 79 (2020), pp. 2952–2972.
  • V. Ervin, N. Heuer, and J. Roop, Regularity of the solution to 1-D fractional order diffusion equations, Math. Comp. 87 (2018), pp. 2273–2294.
  • R. Garrappa, Trapezoidal methods for fractional differential equations: theoretical and computational aspects, Math. Comput. Simul. 110 (2015), pp. 96–112.
  • C. Ji, W. Dai, and Z. Sun, Numerical schemes for solving the time-fractional dual-phase-lagging heat conduction model in a double-layered nanoscale thin film, J. Sci. Comput. 81 (2019), pp. 1767–1800.
  • J. Jia and H. Wang, A fast finite volume method for conservative space-time fractional diffusion equations discretized on space-time locally refined meshes, Comput. Math. Appl. 78 (2019), pp. 1345–1356.
  • L. Jia, H. Chen, and H. Wang, Mixed-type Galerkin variational principle and numerical simulation for a generalized nonlocal elastic model, J. Sci. Comput. 71 (2017), pp. 660–681.
  • K. Kazmi and A.Q. Khaliq, An efficient split-step method for distributed-order space-fractional reaction-diffusion equations with time-dependent boundary conditions, Appl. Numer. Math. 147 (2020), pp. 142–160.
  • A.Q. Khaliq, T.A. Biala, S.S. Alzahrani, and K.M. Furati, Linearly implicit predictor-corrector methods for space-fractional reaction-diffusion equations with non-smooth initial data, Comput. Math. Appl 75 (2018), pp. 2629–2657.
  • N. Kopteva and M. Stynes, Analysis and numerical solution of a Riemann-Liouville fractional derivative two-point boundary value problem, Adv. Comput. Math 43 (2017), pp. 77–99.
  • C. Li and Z. Wang, The local discontinuous Galerkin finite element methods for caputo-type partial differential equations: numerical analysis, Appl. Numer. Math 140 (2019), pp. 1–22.
  • C. Lorenzo and T. Hartley, Variable-order and distributed-order fractional operators, Nonlinear Dyn29 (2002), pp. 57–98.
  • Z. Mao, S. Chen, and J. Shen, Efficient and accurate spectral method using generalized Jacobi functions for solving Riesz fractional differential equations, Appl. Numer. Math 106 (2016), pp. 165–181.
  • R. Metzler and J. Klafter, The random walk's guide to anomalous diffusion: A fractional dynamics approach, Phys. Rep 339 (2000), pp. 1–77.
  • I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
  • K. Sakamoto and M. Yamamoto, Initial value/boundary value problems for fractional diffusion-wave equations and applications to some inverse problems, J. Math. Anal. Appl 382 (2011), pp. 426–447.
  • S. Samko and B. Ross, Integration and differentiation to a variable fractional order, Integral Transform. Spec. Funct 1 (1993), pp. 277–300.
  • G. Slonimsky, Laws of mechanical relaxation processes in polymers, J. Polymer Sci. Part C 16 (1967), pp. 1667–1672.
  • M. Stynes, E. O'Riordan, and J.L. Gracia, Error analysis of a finite difference method on graded mesh for a time-fractional diffusion equation, SIAM J. Numer. Anal. 55 (2017), pp. 1057–1079.
  • H. Sun, A. Chang, Y. Zhang, and W. Chen, A review on variable-order fractional differential equations: mathematical foundations, physical models, numerical methods and applications, Fract. Calc. Appl. Anal 22 (2019), pp. 27–59.
  • F. Wang, H. Chen, and H. Wang, Finite element simulation and efficient algorithm for fractional Cahn-Hilliard equation, J. Comput. Appl. Math. 356 (2019), pp. 248–266.
  • F. Zeng, Z. Zhang, and G. Karniadakis, A generalized spectral collocation method with tunable accuracy for variable-order fractional differential equations, SIAM J. Sci. Comp. 37 (2015), pp. A2710–A2732.
  • X. Zheng and H. Wang, Wellposedness and regularity of a nonlinear variable-order fractional wave equation, Appl. Math. Lett. 95 (2019), pp. 29–35.
  • X. Zheng and H. Wang, An optimal-order numerical approximation to variable-order space-fractional diffusion equations on uniform or graded meshes, SIAM J. Numer. Anal. 58 (2020), pp. 330–352.
  • P. Zhuang, F. Liu, V. Anh, and I. Turner, Numerical methods for the variable-order fractional advection-diffusion equation with a nonlinear source term, SIAM J. Numer. Anal. 47 (2009), pp. 1760–1781.

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.