High-order rotated grid point iterative method for solving 2D time fractional telegraph equation and its convergence analysis

Abstract

In this paper, the compact finite difference (CFD) and rotated four-point compact explicit decoupled group (CEDG) methods are proposed to solve the two-dimensional time-fractional telegraph equation. The CEDG method is derived from a rotated of CFD approximation formula combine with the arranging of the grid points in the form of a group. This method shows superior performance in the term of CPU timings and iteration compared to the CFD method on the standard grid but with the same order of accuracy. We have proved the stability and convergence of the proposed schemes using the Fourier analysis. The convergence order of the proposed methods is \(O\left( \tau +h_{x}^{4}+h_{y}^{4}\right) \). Some numerical experiments are performed to demonstrate the effectiveness of the proposed methods.

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References

  1. Abdullah AR (1991) The four point explicit decoupled group (edg) method: a fast Poisson solver. Int J Comput Math 38:61–70

    Article  Google Scholar 

  2. Ajmal A, Ali NHM (2019) On skewed grid point iterative method for solving 2d hyperbolic telegraph fractional differential equation. Adv Differ Equ 2019:1–29

    MathSciNet  Article  Google Scholar 

  3. Akram T, Abbas M, Iqbal A, Baleanu D, Asad JH (2020) Novel numerical approach based on modified extended cubic b-spline functions for solving non-linear time-fractional telegraph equation. Symmetry 12:1–19

    Google Scholar 

  4. Ali A, Ali NHM (2018) Explicit group iterative methods in the solution of two dimensional time-fractional diffusion-waves equation. Compusoft 7:2931–2938

    Google Scholar 

  5. Baleanu D, Lopes AM (2019) Handbook of fractional calculus with applications in engineering, life and social sciences, part A. Walter de Gruyter GmbH & Co KG, Berlin

    Google Scholar 

  6. Baleanu D, Lopes AM (2019) Handbook of fractional calculus with applications in engineering, life and social sciences, part B. Walter de Gruyter GmbH & Co KG, Berlin

    Google Scholar 

  7. Balasim AT, Hj Mohd Ali N (2017) New group iterative schemes in the numerical solution of the two-dimensional time fractional advection-diffusion equation. Cogent Math 4:1–19

    MathSciNet  Article  Google Scholar 

  8. Cascaval RC, Eckstein EC, Frota CL, Goldstein JA (2002) Fractional telegraph equations. J Math Anal Appl 276:145–159

    MathSciNet  Article  Google Scholar 

  9. Cui M (2012) Compact alternating direction implicit method for two-dimensional time fractional diffusion equation. J Comput Phys 231:2621–2633

    MathSciNet  Article  Google Scholar 

  10. Debnath L (2011) Nonlinear partial differential equations for scientists and engineers. Springer Science & Business Media, Birkhauser

    Google Scholar 

  11. Evangelista LR, Lenzi EK (2018) Fractional diffusion equations and anomalous diffusion. Cambridge University Press, Cambridge

    Google Scholar 

  12. Eltayeb H, Abdalla YT, Bachar I, Khabir MH (2019) Fractional telegraph equation and its solution by natural transform decomposition method. Symmetry 11:334–347

    Article  Google Scholar 

  13. Hosseini VR, Shivanian E, Chen W (2015) Local integration of 2-d fractional telegraph equation via local radial point interpolant approximation. Eur Phys J Plus 130:1–21

    Article  Google Scholar 

  14. Karatay I, Kale N, Bayramoglu SR (2013) A new difference scheme for time fractional heat equations based on the Crank–Nicholson method. Fract Calc Appl Anal 16:892–910

    MathSciNet  Article  Google Scholar 

  15. Kumar A, Bhardwaj A, Dubey S (2020) A local meshless method to approximate the time-fractional telegraph equation. Eng Comput 1–16

  16. Liang Y, Yao Z, Wang Z (2020) Fast high order difference schemes for the time fractional telegraph equation. Numer Methods Partial Differ Equ 36:154–172

    MathSciNet  Article  Google Scholar 

  17. Mehra M (2018) Wavelets theory and its applications. Springer, Singapore

    Google Scholar 

  18. Metaxas A, Meredith RJ (1983) Industrial microwave heating. IET

  19. Podlubny I (1998) Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications. Elsevier, Amsterdam

    Google Scholar 

  20. Saad Y (2003) Iterative methods for sparse linear systems. SIAM, Philadelphia

    Google Scholar 

  21. Saadatmandi A, Mohabbati M (2015) Numerical solution of fractional telegraph equation via the tau method. Math Rep 17:2–13

    MathSciNet  MATH  Google Scholar 

  22. Saha Ray S, Subhadarshan S (2018) Generalized fractional order differential equations arising in physical models. Chapman and Hall/CRC, Boca Raton

    Google Scholar 

  23. Shivanian E (2016) Spectral meshless radial point interpolation (smrpi) method to two-dimensional fractional telegraph equation. Math Methods Appl Sci 39:1820–1835

    MathSciNet  Article  Google Scholar 

  24. Sun Z, Wu X (2006) A fully discrete difference scheme for a diffusion-wave system. Appl Numer Math 56:193–209

    MathSciNet  Article  Google Scholar 

  25. Tarasov VE (2019) Handbook of fractional calculus with applications in physics, part A. Walter de Gruyter GmbH & Co KG, Berlin

    Google Scholar 

  26. Tarasov VE (2019) Handbook of fractional calculus with applications in physics, part B. Walter de Gruyter GmbH & Co KG, Berlin

    Google Scholar 

  27. Taghipour M, Aminikhah H (2020) A new compact alternating direction implicit method for solving two dimensional time fractional diffusion equation with Caputo–Fabrizio derivative. Filomat 34:3609–3626

    Google Scholar 

  28. Vyawahare VA, Nataraj P (2013) Analysis of fractional-order telegraph model for neutron transport in nuclear reactor with slab geometry. In: 2013 European control conference (ECC), pp 3476–3481

  29. Yang X, Wu L et al (2020) An efficient alternating segment parallel difference method for the time fractional telegraph equation. Adv Math Phys 2020:1–11

    MathSciNet  MATH  Google Scholar 

  30. Yaseen M, Abbas M (2020) An efficient cubic trigonometric b-spline collocation scheme for the time-fractional telegraph equation. Appl Math J Chin Univ 35:359–378

    MathSciNet  Article  Google Scholar 

  31. Youssri Y, Abd-Elhameed W (2018) Numerical spectral Legendre–Galerkin algorithm for solving time fractional telegraph equation. Rom J Phys 63:1–16

    Google Scholar 

  32. Zhao Z, Li C (2012) Fractional difference/finite element approximations for the time-space fractional telegraph equation. Appl Math Comput 219:2975–2988

    MathSciNet  MATH  Google Scholar 

  33. Zhuang P, Liu F, Anh V, Turner I (2008) New solution and analytical techniques of the implicit numerical method for the anomalous subdiffusion equation. SIAM J Numer Anal 46:1079–1095

    MathSciNet  Article  Google Scholar 

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Acknowledgements

We are very grateful to anonymous referees for their careful reading and valuable comments which led to the improvement of this paper. The authors are also very grateful to the Associate Editor, Professor Vasily E. Tarasov for managing the review process.

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Correspondence to H. Aminikhah.

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Abdi, N., Aminikhah, H. & Sheikhani, A.H.R. High-order rotated grid point iterative method for solving 2D time fractional telegraph equation and its convergence analysis. Comp. Appl. Math. 40, 54 (2021). https://doi.org/10.1007/s40314-021-01451-4

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Keywords

  • Crank–Nicolson method
  • Compact scheme
  • Rotated finite difference
  • Explicit decoupled group method
  • Caputo fractional derivative
  • Stability
  • Convergence

Mathematics Subject Classification

  • 34K37
  • 65M06
  • 65D07