A numerical approximation for the solution of a time-fractional telegraph equation based on the Crank–Nicolson method

Document Type : Research Article

Authors

• H. Hajinezhad ¹
• A.R. Soheili ²

¹ Department of Mathematics, Payame Noor University, Tehran, Iran.

² Department of Applied Mathematics, Faculty of Mathematical Sciences, Ferdowsi University of Mashhad, Mashhad, Iran.

10.22067/ijnao.2022.77142.1154

Abstract

In this paper, a two-dimensional time-fractional telegraph equation is considered with derivative in the sense of Caputo and $ 1<\beta<2$. The aim of this work is to extend the Crank--Nicolson method for this time-fractional telegraph equation. The stability and convergence of the numerical method are investigated. Also, the accuracy and efficiency of the proposed method are demonstrated by numerical experiments.

Keywords

• Time-Fractional Telegraph Equation
• Crank–Nicolson Method
• Stability
• Convergence

Main Subjects

• Numerical methods for differential equations and its applications

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References

1. Bagley, R.L. and Torvik, P. A theoretical basis for the application of fractional calculus to viscoelasticity. J. Rheol. 27(3) (1983) 201–210.
2. Banasiak, J. and Mika, J.R. Singularly perturbed telegraph equations with applications in the random walk theory. J. Appl. Math. Stochastic Anal. 11(1) (1998) 9–28.
3. Biazar, J., Ebrahimi, H. and Ayati, Z. An approximation to the solution of telegraph equation by variational iteration method. Numer. Methods Partial Differ. Equ. 25(4) (2009) 797–801.
4. Cen, Z. Huang, J. Xu, A. and Le, A. Numerical approximation of a time-fractional Black–Scholes equation. Comput. Math. Appl. 75(8) (2018) 2874–2887.

5. Chen, J., Liu, F. and Anh, V. Analytical solution for the time-fractional telegraph equation by the method of separating variables. J. Math. Anal. Appl. 338(2) (2008) 1364–1377.
6. Das, S. Vishal, K. Gupta, P.k. and Yildirim, A. An approximate analytical solution of time-fractional telegraph equation. Appl. Math. Comput. 217(18) (2011) 7405–7411.
7. Diethelm, k. Garrappa, R. and Stynes, M. Good (and not so good) practices in computational methods for fractional calculus. Mathematics, 8(3) (2020) 324.
8. Horn, R. A. and Johnson, C. R. Matrix analysis. Cambridge university press, 2012.
9. Hosseini, V.R. Chen, W. and Avazzadeh, Z. Numerical solution of fractional telegraph equation by using radial basis functions. Eng. Anal. Bound. Elem. 38 (2014) 31–39.
10. Hosseini, V.R. Shivanian, E. and Chen, W. Local integration of 2-D fractional telegraph equation via local radial point interpolant approximation. Eur. Phys. J. Plus, 130(2) (2015) p. 1–21.
11. Jiang, W. and Lin, Y. Representation of exact solution for the time-fractional telegraph equation in the reproducing kernel space. Commun. Nonlinear Sci. Numer. Simul. 16(9) (2011) 3639–3645.
12. Karatay, I. Kale, N. and Bayramoglu, S. A new difference scheme for time fractional heat equations based on the Crank-Nicholson method. Fract. Calc. Appl. Anal. 16(4) (2013) 892–910.
13. Kumar, A. Bhardwaj, A. and Dubey, S. A local meshless method to approximate the time-fractional telegraph equation. Eng. Comput. 37(4) (2021) 3473–3488.
14. Li, C. and Cao, J. A finite difference method for time-fractional tele-graph equation. in Proceedings of 2012 IEEE/ASME 8th IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications. 2012. IEEE 314–318.
15. Liang, Y. Yao, Z. and Wang, Z. Fast high order difference schemes for the time fractional telegraph equation. Numer. Methods Partial Differ. Equ. 36(1) (2020) 154–172.
16. Mohebbi, A. Abbaszadeh, M. and Dehghan, M. The meshless method of radial basis functions for the numerical solution of time fractional tele-graph equation. Internat. J. Numer. Methods Heat Fluid Flow 24 (2014), no. 8, 1636–1659.

17. Momani, S. Analytic and approximate solutions of the space- and time-fractional telegraph equations. Appl. Math. Comput. 170(2) (2005) 1126–1134.
18. Nemati, S. Lima, P.M. and Torres, D.F. A numerical approach for solving fractional optimal control problems using modified hat functions. Commun. Nonlinear Sci. Numer. Simul. 78 (2019) 104849.
19. Nikan, O. Avazzadeh, Z. and Machado, J.T. Numerical approximation of the nonlinear time-fractional telegraph equation arising in neutron transport. Commun. Nonlinear Sci. Numer. Simul. 99 (2021) 105755.
20. Quarteroni, A. and Valli, A. Numerical approximation of partial differential equations. Vol. 23. Springer Science and Business Media, 2008.
21. Reddy, B.D. Introductory functional analysis: with applications to boundary value problems and finite elements. Springer Science and Business Media, 2013.
22. Sepehrian, B. and Shamohammadi, Z. Numerical solution of nonlinear time-fractional telegraph equation by radial basis function collocation method. Iran. J. Sci. Technol. Trans. A: Sci. 42(4) (2018) 2091–2104.
23. Shivanian, E. Spectral meshless radial point interpolation (SMRPI) method to two-dimensional fractional telegraph equation. Math. Methods Appl. Sci. 39(7) (2016) 1820–1835.
24. Shivanian, E. Abbasbandy, S. Alhuthali, M.S. and Alsulami, H.H. Local integration of 2-D fractional telegraph equation via moving least squares approximation. Eng. Anal. Bound. Elem. 56 (2015) 98–105.
25. Sun, H. Zhang, Y. Baleanu, D. Chen, W. and Chen, Y. A new collection of real world applications of fractional calculus in science and engineering. Commun. Nonlinear Sci. Numer. Simul. 64 (2018) 213–231.
26. Uchaikin, V.V. Fractional derivatives for physicists and engineers. Vol. 2. 2013: Springer.
27. Vyawahare, V.A. and Nataraj, P. Fractional-order modeling of neutron transport in a nuclear reactor. Appl. Math. Model. 37(23) (2013) 9747–9767.
28. Vyawahare, V.A. and Nataraj, P. Analysis of fractional-order telegraph model for neutron transport in nuclear reactor with slab geometry. in 2013 European control conference (ECC). 2013. IEEE.
29. Wang, Y. and Mei, L. Generalized finite difference/spectral Galerkin approximations for the time-fractional telegraph equation. Adv. Difference Equ. 2017(1) (2017) 1–16.

30. Wei, L., Liu, L. and Sun, H. Numerical methods for solving the time-fractional telegraph equation. Taiwanese J. Math. 22(6) (2018) 1509–1528.
31. Yildirim, A. He’s homotopy perturbation method for solving the space-and time-fractional telegraph equations. Int. J. Comput. Math. 87(13) (2010) 2998–3006.
32. Zhao, Z. and Li, C. Fractional difference/finite element approximations for the time-space fractional telegraph equation. Appl. Math. Comput. 219(6) (2012) 2975–2988.