Abstract
In this work, we examine the numerical solution of the time-fractional telegraph equation by applying a higher-order numerical scheme via the tension spline method. The fractional order derivative of two different orders is analyzed using Caputo’s definition. Moreover, the numerical scheme formation was carried out using spline functions incorporating the tension parameter in the spatial direction and the discretization technique based on a finite difference approach in the temporal direction. The proposed scheme includes some parameters, and by a suitable choice of parameters, its order can be enhanced from two to four in the spatial direction. The present technique is proven unconditionally stable through meticulous analysis. The convergence analysis is also demonstrated using the Fourier series. Numerical results of two test examples are presented through tables and plots that show the proficiency of the proposed numerical scheme and validate our theoretical findings.
Similar content being viewed by others
Data availability
Data sharing does not apply to this article, as no data sets were generated or analyzed during the current study.
References
Abdou, M.: Adomian decomposition method for solving the telegraph equation in charged particle transport. J. Quant. Spectrosc. Radiat. Transf. 95, 407–414 (2005)
Aghamohamadi, M., Rashidinia, J., Ezzati, R.: Tension spline method for solution of non-linear Fisher equation. Appl. Math. Comput. 249, 399–407 (2014)
Agrawal, O.P.: Solution for a fractional diffusion-wave equation defined in a bounded domain. Nonlinear Dyn. 29, 145–155 (2002)
Bansu, H., Kumar, S.: Numerical solution of space and time fractional telegraph equation: a meshless approach. Int. J. Nonlinear Sci. Numer. Simul. 20, 325–337 (2019)
Barletta, A., Zanchini, E.: A thermal potential formulation of hyperbolic heat conduction. J. Heat Transfer 121, 166–169 (1999)
Biazar, J., Eslami, M.: Analytic solution for telegraph equation by differential transform method. Phys. Lett. A 374, 2904–2906 (2010)
Chawla, R., Deswal, K., Kumar, D., Baleanu, D.: Numerical simulation for generalized time-fractional Burgers’ equation with three distinct linearization schemes. J. Comput. Nonlinear Dyn. 18, 041001 (2023)
Chawla, R., Kumar, D.: Higher-order tension spline-based numerical technique for time fractional reaction-diffusion wave equation with dam**. Int J Dyn Control (2023). https://doi.org/10.1007/s40435-023-01222-5
Chen, J., Liu, F., Anh, V.: Analytical solution for the time-fractional telegraph equation by the method of separating variables. J. Math. Anal. Appl. 338, 1364–1377 (2008)
Cui, M.: Compact finite difference method for the fractional diffusion equation. J. Comput. Phys. 228, 7792–7804 (2009)
Das, S., Vishal, K., Gupta, P.K., Yildirim, A.: An approximate analytical solution of time-fractional telegraph equation. Appl. Math. Comput. 217, 7405–7411 (2011)
De Staelen, R.H., Hendy, A.S.: Numerically pricing double barrier options in a time-fractional Black–Scholes model. Comput. Math. Appl. 74, 1166–1175 (2017)
Dehghan, M., Shokri, A.: A numerical method for solving the hyperbolic telegraph equation. Numer. Methods Partial Differ. Equ. 24, 1080–1093 (2008)
Delić, A., Jovanović, Bo.ško S., Živanović, S.: Finite difference approximation of a generalized time-fractional telegraph equation. Comput. Appl. Math. 20, 595–607 (2020)
Esmaeili, S., Shamsi, M.: A pseudo-spectral scheme for the approximate solution of a family of fractional differential equations. Commun. Nonlinear Sci. Numer. Simul. 16, 3646–3654 (2011)
Ford, N.J., Rodrigues, M.M., **ao, J., Yan, Y.: Numerical analysis of a two-parameter fractional telegraph equation. J. Comput. Appl. Math. 249, 95–106 (2013)
Gregory, J.A., Sarfraz, M.: A rational cubic spline with tension. Comput. Aided Geom. Des. 7, 1–13 (1990)
Hashemi, M.S., Baleanu, D.: Numerical approximation of higher-order time-fractional telegraph equation by using a combination of a geometric approach and method of line. J. Comput. Phys. 316, 10–20 (2016)
Hesameddini, E., Asadolahifard, E.: A new spectral Galerkin method for solving the two dimensional hyperbolic telegraph equation. Comput. Math. Appl. 72, 1926–1942 (2016)
Heydari, M.H., Hooshmandasl, M.R., Mohammadi, F.: Two-dimensional Legendre wavelets for solving time-fractional telegraph equation. Adv. Appl. Math. Mech. 6, 247–260 (2014)
Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000)
Hosseini, V.R., Chen, W., Avazzadeh, Z.: Numerical solution of fractional telegraph equation by using radial basis functions. Eng. Anal. Bound. Elem. 38, 31–39 (2014)
Jordan, P.M., Puri, A.: Digital signal propagation in dispersive media. J. Appl. Phys. 85, 1273–1282 (1999)
Kanth, A.S.V., Sirswal, D.: Analysis and numerical simulation for a class of time fractional diffusion equation via tension spline. Numer. Algorithms 79, 479–497 (2018)
Khan, I., Aziz, T.: Tension spline method for second-order singularly perturbed boundary-value problems. Int. J. Comput. Math. 82, 1547–1553 (2005)
Khan, Y., Diblik, J., Faraz, N., Smarda, Z.: An efficient new perturbative Laplace method for space-time fractional telegraph equations. Adv. Differ. Equ. 2012, 204 (2012). https://doi.org/10.1186/1687-1847-2012-204
Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland, New York (2006)
Lakestani, M., Saray, B.N.: Numerical solution of telegraph equation using interpolating scaling functions. Comput. Math. Appl. 60, 1964–1972 (2010)
Lin, Y., Xu, C.: Finite difference/spectral approximations for the time-fractional diffusion equation. J. Comput. Phys. 225, 1533–1552 (2007)
Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equations. Wiley, New York (1993)
Momani, S., Odibat, Z., Alawneh, A.: Variational iteration method for solving the space- and time-fractional KdV equation. Numer. Methods Partial Differ. Equ. 24, 262–271 (2008)
Orsingher, E., Beghin, L.: Time-fractional telegraph equations and telegraph processes with Brownian time. Probab. Theory Relat. Fields 128, 141–160 (2004)
Perdikaris, P., Karniadakis, G.E.: Fractional-order viscoelasticity in one-dimensional blood flow models. Ann. Biomed. Eng. 42, 1012–1023 (2014)
Podlubny, I.: Fractional Differential Equations. Academic Press, San Diego (1999)
Saadatmandi, A., Dehghan, M.: Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method. Numer. Methods Partial Differ. Equ. 26, 239–252 (2010)
Saadatmandi, A., Dehghan, M.: A Tau approach for solution of the space fractional diffusion equation. Comput. Math. Appl. 62, 1135–1142 (2011)
Schumer, R., Benson, D.A., Meerschaert, M.M., Baeumer, B.: Fractal mobile/immobile solute transport. Water Resour. Res. 39(10), 1296 (2003). https://doi.org/10.1029/2003WR002141
Sharifi, S., Rashidinia, J.: Numerical solution of hyperbolic telegraph equation by cubic B-spline collocation method. Appl. Math. Comput. 281, 28–38 (2016)
Shekarabi, H.S., Rashidinia, J.: Three level implicit tension spline scheme for solution of convection-reaction-diffusion equation. Ain Shams Eng. J. 9, 1601–1610 (2018)
Wei, L., Liu, L., Sun, H.: Numerical methods for solving the time-fractional telegraph equation. Taiwan. J. Math. 22, 1509–1528 (2018)
Weston, V., He, S.: Wave splitting of the telegraph equation in \(R^3\) and its application to inverse scattering. Inverse Probl. 9, 789 (1993)
Yousefi, S.A., Lotfi, A., Dehghan, M.: The use of a Legendre multiwavelet collocation method for solving the fractional optimal control problems. J. Vib. Control 17, 2059–2065 (2011)
Zhang, Y., Qian, J., Papelis, C., Sun, P., Yu, Z.: Improved understanding of bimolecular reactions in deceptively simple homogeneous media: from laboratory experiments to Lagrangian quantification. Water Resour. Res. 50, 1704–1715 (2014)
Acknowledgements
The authors sincerely thank the reviewers for providing valuable comments to improve the manuscript.
Funding
No funding was received.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors state that they have no known competing financial interests or personal ties that could have influenced the research presented in this study.
Ethical approval
Not applicable.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Chawla, R., Kumar, D. A High Order Numerical Scheme for Time-Fractional Telegraph Equation Via Cubic Spline in Tension. Differ Equ Dyn Syst (2024). https://doi.org/10.1007/s12591-024-00678-x
Accepted:
Published:
DOI: https://doi.org/10.1007/s12591-024-00678-x