A Hybrid Numerical Scheme for the Time-Fractional Telegraph Equation

Document Type : Research article

Authors

Department of Mathematics, Faculty of Mathematics, Statistics and Computer Science, Semnan University, Semnan, Iran

Abstract

This study introduces a hybrid approach based on a finite difference scheme and radial basis functions (RBF) for solving the Caputo time-fractional telegraph equation numerically. To handle the temporal fractional derivatives, a finite difference formulation of the L1/L2 type is adopted, while the spatial derivatives are approximated using an RBF collocation technique. This combination results in a  discretized system characterized by a sparse matrix structure. Through the application of energy methods, the stability and convergence assessment of the temporal discretization is conducted, yielding a theoretical error estimate of $\mathcal{O}(\delta_t^{3-\alpha})$ in the time direction. The reliability and effectiveness of the proposed numerical scheme are verified through a representative test problem, confirming the capability of the proposed method to achieve high accuracy.

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References

[1] L. Kelvin, On the theory of the electric telegraph, Proceedings of the Royal Society of London, 7, 382–399, (1855).
[2] S. Goldstein, On diffusion by discontinuous movements, and on the telegraph equation, The Quarterly Journal of Mechanics and Applied Mathematics, 4(2), 129–156, (1951).
[3] F. Mainardi, Fractional relaxation-oscillation and fractional diffusion-wave phenomena, Chaos, Solitons and Fractals, 7(9), 1461–1477, (1996).
[4] R. Metzler and J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach, Physics Reports, 339(1), 1–77, (2000).
[5] Y. Povstenko, Fractional Thermoelasticity, in Fractional Thermoelasticity, Springer, Cham, 1–37, (2015).
[6] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.
[7] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier, Amsterdam, 2006.
[8] E. Orsingher and X. Zhao, The space-fractional telegraph equation and the related fractional telegraph process, Chinese Annals of Mathematics, 24(1), 45–56, (2003).
[9] R. Gorenflo, Y. Luchko, and F. Mainardi, Wright functions as scale-invariant solutions of the diffusion-wave equation, Journal of Computational and Applied Mathematics, 118(1–2), 175–191, (2000).
[10] Y. Lin and C. Xu, Finite difference/spectral approximations for the time-fractional diffusion equation, Journal of Computational Physics, 225(2), 1533–1552, (2007).
[11] C. Li and F. Zeng, Numerical Methods for Fractional Calculus, Chapman and Hall/CRC, Boca Raton, 2015.
[12] J. P. Boyd, Chebyshev and Fourier Spectral Methods, Second edition, Dover Publications, Mineola, 2001.
[13] M. H. Heydari, M. R. Hooshmandasl, and F. Mohammadi, Wavelet collocation method for solving multiorder fractional differential equations, Journal of Applied Mathematics, 2014, Article ID 487406, (2014).
[14] J. J. Mohan and G. V. S. R. Deekshitulu, Numerical solution of time fractional telegraph equation, Journal of Applied Mathematics and Informatics, 30(1–2), 203–215, (2012).
[15] G. B. Whitham, Linear and Nonlinear Waves, Wiley-Interscience, New York, 1974.
[16] G. H. Weiss, Some applications of persistent random walks and the telegraph equation, Physica A: Statistical Mechanics and its Applications, 311(1–2), 381–410, (2002).
[17] D. D. Joseph and L. Preziosi, Heat waves, Reviews of Modern Physics, 61(1), 41–73, (1989).
[18] K. B. Oldham and J. Spanier, The Fractional Calculus: Theory and Applications of Differentiation and Integration to Arbitrary Order, Academic Press, New York, 1974.
[19] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, 1993.
[20] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London, 2010.
[21] I. M. Sokolov, J. Klafter, and A. Blumen, Fractional kinetics, Physics Today, 55(11), 48–54, (2002).
[22] R. C. Cascaval, E. C. Eckstein, C. L. Frota, and J. A. Goldstein, Fractional telegraph equations, Journal of Mathematical Analysis and Applications, 276(1), 145–159, (2002).
[23] M. Naber, Time fractional Schrodinger equation, Journal of Mathematical Physics, 45(8), 3339–3352, (2004).
[24] R. Gorenflo and F. Mainardi, Fractional calculus: integral and differential equations of fractional order, in Fractals and Fractional Calculus in Continuum Mechanics, Springer, Vienna, 223–276, (1997).
[25] Z. Z. Sun and X. Wu, A fully discrete difference scheme for a diffusion-wave system, Applied Numerical Mathematics, 56(2), 193–209, (2006).
[26] G. H. Gao and Z. Z. Sun, A compact finite difference scheme for the fractional sub-diffusion equations, Journal of Computational Physics, 230(3), 586–595, (2011).
[27] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer, Berlin, 2006.
[28] F. Zhou and X. Xu, The third kind Chebyshev wavelets collocation method for solving the time-fractional convection diffusion equations with variable coefficients, Applied Mathematics and Computation, 280, 11–29, (2016).
[29] A. K. Singh and V. K. Singh, Chebyshev wavelet method for solving fractional differential equations, Computational and Applied Mathematics, 38(1), 1–21, (2019).
[30] S. Zhi-zhong and W. Xiaonan, A fully discrete difference scheme for a diffusion-wave system, Applied Numerical Mathematics, 56(2), 193–209, (2006). 
Analytical and Numerical Solutions for Nonlinear Equations
Volume 11, Issue 2
September 2026
Pages 311-323

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History

  • Receive Date: 18 February 2026
  • Revise Date: 22 March 2026
  • Accept Date: 27 May 2026
  • Publish Date: 01 September 2026

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