10.1007/s40096-022-00503-y

A novel adaptive meshless method for solving the nonlinear time fractional telegraph equations on arbitrary domains

  • Lin Li1,
  • Zhong Chen*1,
  • Hong Du2,
  • Wei Jiang1,
  • Biao Zhang3,
  1. Department of Mathematics, Harbin Institute of Technology, Weihai, Shandong, CN
  2. College of Mathematics and Computer Science, GuangDong Ocean University, Zhanjiang, Guangdong, 524000, CN
  3. School of Mathematics, Harbin Institute of Technology, Heilongjiang, CN

Published in Issue 2023-01-02

How to Cite

Li, L., Chen, Z., Du, H., Jiang, W., & Zhang, B. (2023). A novel adaptive meshless method for solving the nonlinear time fractional telegraph equations on arbitrary domains. Mathematical Sciences, 18(2 (June 2024). https://doi.org/10.1007/s40096-022-00503-y
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Abstract

Abstract A novel adaptive meshless numerical method for tackling with the two-dimensional time fractional telegraph equations on arbitrary domains based on partitioning domains is presented in this paper. The proposed approach has many advantages. (i) When the exact solution varies dramatically in some parts of the domain and gently in others, our method can reduce the amount of calculation waste and improve computational speed. (ii) With the assistance of the extension theorem, the multiscale orthonormal bases of two-dimensional reproducing kernel space on rectangle regions are constructed to obtain the approximate solutions of the equations on arbitrary domains, which can achieve high accuracy. (iii) The convergence-order analysis of bicubic spline space can be employed to study the convergence order of the proposed strategy primely. Eventually, some numerical examples and comparisons graphically elucidate the implementation and capability of the adaptive meshless method, which is more accurate and efficient than some existing methods.

Keywords

  • Fractional telegraph equation,
  • Adaptive meshless technology,
  • Convergence order,
  • Arbitrary domain

References

  1. Nahin and Heaviside (2002) The John Hopkins University Press
  2. Ureña et al. (2020) Solving the telegraph equation in 2-D and 3-D using generalized finite difference method (GFDM) (pp. 13-24) https://doi.org/10.1016/j.enganabound.2019.11.010
  3. Mishra et al. (2021) Numerical approximation of fractional telegraph equation via legendre collocation technique 7(5) (pp. 1-27) https://doi.org/10.1007/s40819-021-01133-z
  4. Nikan et al. (2021) Numerical approximation of the nonlinear time-fractional telegraph equation arising in neutron transport https://doi.org/10.1016/j.cnsns.2021.105755
  5. Abdou (2005) Adomian decomposition method for solving the telegraph equation in charged particle transport 95(3) (pp. 407-414) https://doi.org/10.1016/j.jqsrt.2004.08.045
  6. Nikan, O., Avazzadeh, Z., Machado, J.A., Rasoulizadeh, M.N.: An accurate localized meshfree collocation technique for the telegraph equation in propagation of electrical signals. Eng. Comput-Germany. 1-18 (2022)
  7. Lock, C.G.L., Greeff, J.C., Joubert, S.V.: Modelling of telegraph equations in transmission lines. Buffelspoort TIME2008 Peer-reviewed Conference Proceedings. (2008)
  8. Orsingher (1985) Hyperbolic equations arising in random models (pp. 93-106) https://doi.org/10.1016/0304-4149(85)90379-5
  9. Chang and Werner (1952) A solution of the telegraph equation with application to two-dimensional supersonic shear flow 31(1–4) (pp. 91-101) https://doi.org/10.1002/sapm195231191
  10. Yang et al. (2021) An efficient alternating direction implicit finite difference scheme for the three-dimensional time-fractional telegraph equation (pp. 233-247) https://doi.org/10.1016/j.camwa.2021.10.021
  11. Shah et al. (2021) Fibonacci wavelet method for solving time-fractional telegraph equations with Dirichlet boundary conditions https://doi.org/10.1016/j.rinp.2021.104123
  12. Povstenko and Ostoja-Starzewski (2022) Fractional telegraph equation under moving time-harmonic impact https://doi.org/10.1016/j.ijheatmasstransfer.2021.121958
  13. Hashemi and Baleanu (2016) Numerical approximation of higher-order time-fractional telegraph equation by using a combination of a geometric approach and method of line (pp. 10-20) https://doi.org/10.1016/j.jcp.2016.04.009
  14. Singh et al. (2018) Application of Bernoulli matrix method for solving two-dimensional hyperbolic telegraph equations with Dirichlet boundary conditions 75(7) (pp. 2280-94) https://doi.org/10.1016/j.camwa.2017.12.003
  15. Prakash et al. (2019) A homotopy technique for a fractional order multi-dimensional telegraph equation via the Laplace transform 134(1) (pp. 1-18) https://doi.org/10.1140/epjp/i2019-12411-y
  16. Zhou et al. (2020) A hybrid meshless method for the solution of the second order hyperbolic telegraph equation in two space dimensions (pp. 21-27) https://doi.org/10.1016/j.enganabound.2020.02.015
  17. Shivanian (2016) Spectral meshless radial point interpolation (SMRPI) method to two-dimensional fractional telegraph equation 39(7) (pp. 1820-1835) https://doi.org/10.1002/mma.3604
  18. Kumar et al. (2021) A local meshless method to approximate the time-fractional telegraph equation 37(4) (pp. 3473-3488) https://doi.org/10.1007/s00366-020-01006-x
  19. Aslefallah and Rostamy (2019) Application of the singular boundary method to the two-dimensional telegraph equation on arbitrary domains 118(1) (pp. 1-14) https://doi.org/10.1007/s10665-019-10008-8
  20. Nikan and Avazzadeh (2021) Numerical simulation of fractional evolution model arising in viscoelastic mechanics (pp. 303-320) https://doi.org/10.1016/j.apnum.2021.07.008
  21. Nikan et al. (2021) Numerical approach for modeling fractional heat conduction in porous medium with the generalized Cattaneo model (pp. 107-124) https://doi.org/10.1016/j.apm.2021.07.025
  22. Cao et al. (2023) A localized meshless technique for solving 2D nonlinear integro-differential equation with multi-term kernels (pp. 140-156) https://doi.org/10.1016/j.apnum.2022.07.018
  23. Diethelm, K.: The analysis of fractional differential equations: an application-oriented exposition using differential operators of Caputo type. Lect, Notes Math (2010)
  24. Cui and Lin (2008) Nova Science Publishers
  25. Adams (2002) Academic Press
  26. Chen et al. (2006) Multilevel augmentation methods for differential equations 24(1) (pp. 213-238) https://doi.org/10.1007/s10444-004-4092-6
  27. Carlson and Hall (1973) Error bounds for bicubic spline interpolation (pp. 41-47) https://doi.org/10.1016/0021-9045(73)90050-6
  28. Li, L., Chen, Z.: A meshless method for solving nonlinear variable-order fractional Ginzburg-Landau equations on arbitrary domains. J. Appl. Math. Comput. 1–23 (2022)
  29. Jiang et al. (2020) Multi-scale orthogonal basis method for nonlinear fractional equations with fractional integral boundary value conditions
  30. Du et al. (2020) A stable least residue method in reproducing kernel space for solving a nonlinear fractional integro-differential equation with a weakly singular kernel (pp. 210-222) https://doi.org/10.1016/j.apnum.2020.06.004