Taiwanese Journal of Mathematics

Numerical Methods for Solving the Time-fractional Telegraph Equation

Leilei Wei, Lijie Liu, and Huixia Sun

Full-text: Open access

Abstract

A flexible numerical method for the time-fractional telegraph equation is proposed and analyzed in this paper. The solution is discretized with a new finite difference scheme in time, and a local discontinuous Galerkin (LDG) method in space. We prove that the method is unconditionally stable and convergent with order $O(h^{k+1} + (\Delta t)^{3-\alpha})$, where $h$, $\Delta t$, $k$ are the space step size, time step size and degree of piecewise polynomial, respectively. Numerical experiments are carried out to illustrate the robustness, reliability, and accuracy of the method.

Article information

Source
Taiwanese J. Math., Volume 22, Number 6 (2018), 1509-1528.

Dates
Received: 17 September 2017
Revised: 25 February 2018
Accepted: 10 April 2018
First available in Project Euclid: 24 May 2018

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1527127364

Digital Object Identifier
doi:10.11650/tjm/180503

Mathematical Reviews number (MathSciNet)
MR3880238

Zentralblatt MATH identifier
07021702

Subjects
Primary: 65M12: Stability and convergence of numerical methods 65M06: Finite difference methods 35S10: Initial value problems for pseudodifferential operators

Keywords
fractional telegraph equation discontinuous Galerkin method stability convergence

Citation

Wei, Leilei; Liu, Lijie; Sun, Huixia. Numerical Methods for Solving the Time-fractional Telegraph Equation. Taiwanese J. Math. 22 (2018), no. 6, 1509--1528. doi:10.11650/tjm/180503. https://projecteuclid.org/euclid.twjm/1527127364


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References

  • T. S. Basu and H. Wang, A fast second-order finite difference method for space-fractional diffusion equations, Int. J. Numer. Anal. Model. 9 (2012), no. 3, 658–666.
  • L. Beghin and E. Orsingher, The telegraph process stopped at stable-distributed times and its connection with the fractional telegraph equation, Fract. Calc. Appl. Anal. 6 (2003), no. 2, 187–204.
  • J. Biazar, H. Ebrahimi and Z. Ayati, An approximation to the solution of telegraph equation by variational iteration method, Numer. Methods Partial Differential Equations 25 (2009), no. 4, 797–801.
  • A. R. Carella and C. A. Dorao, Least-squares spectral method for the solution of a fractional advection-dispersion equation, J. Comput. Phys. 232 (2013), 33–45.
  • J. Chen, F. Liu and V. Anh, Analytical solution for the time-fractional telegraph equation by the method of separating variables, J. Math. Anal. Appl. 338 (2008), no. 2, 1364–1377.
  • C.-m. Chen, F. Liu and K. Burrage, Finite difference methods and a Fourier analysis for the fractional reaction-subdiffusion equation, Appl. Math. Comput. 198 (2008), no. 2, 754–769.
  • H. Chen, S. Lü and W. Chen, A fully discrete spectral method for the nonlinear time fractional Klein-Gordon equation, Taiwanese J. Math. 21 (2017), no. 1, 231–251.
  • M. Cui, Compact finite difference method for the fractional diffusion equation, J. Comput. Phys. 228 (2009), no. 20, 7792–7804.
  • W. Deng, Finite element method for the space and time fractional Fokker-Planck equation, SIAM J. Numer. Anal. 47 (2008), no. 1, 204–226.
  • H. Ding and C. Li, Mixed spline function method for reaction-subdiffusion equations, J. Comput. Phys. 242 (2013), 103–123.
  • R. Du, W. R. Cao and Z. Z. Sun, A compact difference scheme for the fractional diffusion-wave equation, Appl. Math. Model. 34 (2010), no. 10, 2998–3007.
  • V. J. Ervin, N. Heuer and J. P. Roop, Numerical approximation of a time dependent, nonlinear, space-fractional diffusion equation, SIAM J. Numer. Anal. 45 (2007), no. 2, 572–591.
  • G. J. Fix and J. P. Roop, Least squares finite-element solution of a fractional order two-point boundary value problem, Comput. Math. Appl. 48 (2004), no. 7-8, 1017–1033.
  • K. M. Furati, O. S. Iyiola and M. Kirane, An inverse problem for a generalized fractional diffusion, Appl. Math. Comput. 249 (2014), 24–31.
  • G.-h. Gao and Z.-z. Sun, A compact finite difference scheme for the fractional sub-diffusion equations, J. Comput. Phys. 230 (2011), no. 3, 586–595.
  • J.-H. He and X.-H. Wu, Variational iteration method: New development and applications, Comput. Math. Appl. 54 (2007), no. 7-8, 881–894.
  • C. Huang, X. Yu, C. Wang, Z. Li and N. An, A numerical method based on fully discrete direct discontinuous Galerkin method for the time fractional diffusion equation, Appl. Math. Comput. 264 (2015), 483–492.
  • W. Jiang and Y. Lin, Representation of exact solution for the time-fractional telegraph equation in the reproducing kernel space, Commun. Nonlinear Sci. Numer. Simul. 16 (2011), no. 9, 3639–3645.
  • Y. Jiang and J. Ma, High-order finite element methods for time-fractional partial differential equations, J. Comput. Appl. Math. 235 (2011), no. 11, 3285–3290.
  • B. Jin, R. Lazarov, Y. Liu and Z. Zhou, The Galerkin finite element method for a multi-term time-fractional diffusion equation, J. Comput. Phys. 281 (2015), 825–843.
  • A. Q. M. Khaliq, X. Liang and K. M. Furati, A fourth-order implicit-explicit scheme for the space fractional nonlinear Schrödinger equations, Numer. Algorithms 75 (2017), no. 1, 147–172.
  • S. Kumar, A new analytical modelling for fractional telegraph equation via Laplace transform, Appl. Math. Model. 38 (2014), no. 13, 3154–3163.
  • S. Kumar, H. Kocak and A. Y\ild\ir\im, A fractional model of gas dynamics equations and its analytical approximate solution using Laplace transform, Z. Naturforsch. A 67 (2012), no. 6-7, 389–396.
  • T. A. M. Langlands and B. I. Henry, The accuracy and stability of an implicit solution method for the fractional diffusion equation, J. Comput. Phys. 205 (2005), no. 2, 719–736.
  • M. Li, X.-M. 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), 256–282.
  • X. Li and C. Xu, A space-time spectral method for the time fractional diffusion equation, SIAM J. Numer. Anal. 47 (2009), no. 3, 2108–2131.
  • C. Li and F. Zeng, The finite difference methods for fractional ordinary differential equations, Numer. Funct. Anal. Optim. 34 (2013), no. 2, 149–179.
  • S. Liao, Notes on the homotopy analysis method: some definitions and theorems, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), no. 4, 983–997.
  • Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, J. Comput. Phys. 225 (2007), no. 2, 1533–1552.
  • F. Liu, P. Zhuang and K. Burrage, Numerical methods and analysis for a class of fractional advection-dispersion models, Comput. Math. Appl. 64 (2012), no. 10, 2990–3007.
  • H. Lu and M. Zhang, The spectral method for long-time behavior of a fractional power dissipative system, Taiwanese J. Math. 22 (2018), no. 2, 453–483.
  • M. M. Meerschaert and C. Tadjeran, Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math. 172 (2004), no. 1, 65–77.
  • S. Momani, Analytic and approximate solutions of the space- and time-fractional telegraph equations, Appl. Math. Comput. 170 (2005), no. 2, 1126–1134.
  • S. Momani and Z. Odibat, Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations, Comput. Math. Appl. 54 (2007), no. 7-8, 910–919.
  • E. Orsingher and L. Beghin, Time-fractional telegraph equations and telegraph processes with Brownian time, Probab Theory Related Fields 128 (2004), no. 1, 141–60.
  • I. Podlubny, Fractional Differential Equations: An introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, Mathematics in Science and Engineering 198, Academic Press, San Diego, CA, 1999.
  • J. P. Roop, Computational aspects of FEM approximation of fractional advection dispersion equations on bounded domains in $\mathbb{R}^2$, J. Comput. Appl. Math. 193 (2006), no. 1, 243–268.
  • K. Wang and H. Wang, A fast characteristic finite difference method for fractional advection-diffusion equations, Adv. in Water Res. 34 (2011), no. 7, 810–816.
  • H. Wang, K. Wang and T. Sircar, A direct $O(N \log^2 N)$ finite difference method for fractional diffusion equations, J. Comput. Phys. 229 (2010), no. 21, 8095–8104.
  • L. Wei, Y. He and B. Tang, Analysis of the fractional Kawahara equation using an implicit fully discrete local discontinuous Galerkin method, Numer. Methods Partial Differential Equations 29 (2013), no. 5, 1441–1458.
  • L. Wei, Y. He, X. Zhang and S. Wang, Analysis of an implicit fully discrete local discontinuous Galerkin method for the time-fractional Schrödinger equation, Finite Elem. Anal. Des. 59 (2012), 28–34.
  • Y. Xia, Y. Xu and C.-W. Shu, Application of the local discontinuous Galerkin method for the Allen-Cahn/Cahn-Hilliard system, Commun. Comput. Phys. 5 (2009), no. 2-4, 821–835.
  • Y. Xu and C.-W. Shu, A local discontinuous Galerkin method for the Camassa-Holm equation, SIAM J. Numer. Anal. 46 (2008), no. 4, 1998–2021.
  • Q. Yang, I. Turner, F. Liu and M. Ilić, Novel numerical methods for solving the time-space fractional diffusion equation in two dimensions, SIAM J. Sci. Comput. 33 (2011), no. 3, 1159–1180.
  • A. Y\ild\ir\im, He's homotopy perturbation method for solving the space- and time-fractional telegraph equations, Int. J. Comput. Math. 87 (2010), no. 13, 2998–3006.
  • A. Y\ild\ir\im and H. Koçak, Homotopy perturbation method for solving the space-time fractional advection-dispersion equation, Adv. in Water Res. 32 (2009), no. 12, 1711–1716.
  • S. B. Yuste, Weighted average finite difference methods for fractional diffusion equations, J. Comput. Phys. 216 (2006), no. 1, 264–274.
  • M. Zayernouri and G. E. Karniadakis, Fractional spectral collocation methods for linear and nonlinear variable order FPDEs, J. Comput. Phys. 293 (2015), 312–338.
  • Q. Zhang and C.-W. Shu, Error estimates for the third order explicit Runge-Kutta discontinuous Galerkin method for a linear hyperbolic equation in one-dimension with discontinuous initial data, Numer. Math. 126 (2014), no. 4, 703–740.
  • X. Zhang, B. Tang and Y. He, Homotopy analysis method for higher-order fractional integro-differential equations, Comput. Math. Appl. 62 (2011), no. 8, 3194–3203.
  • X. Zhao and Z.-Z. Sun, Compact Crank-Nicolson schemes for a Class of fractional Cattaneo equation in inhomogeneous medium, J. Sci. Comput. 62 (2015), no. 3, 747–771.
  • Y. Zheng, C. Li and Z. Zhao, A note on the finite element method for the space-fractional advection diffusion equation, Comput. Math. Appl. 59 (2010), no. 5, 1718–1726.
  • P. Zhuang, F. Liu, V. Anh and I. Turner, New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation, SIAM J. Numer. Anal. 46 (2008), no. 2, 1079–1095.