Analytical solution for the time-fractional telegraph equation. (English) Zbl 1190.35224

Summary: We discuss and derive the analytical solution for three basic problems of the so-called time-fractional telegraph equation. The Cauchy and Signaling problems are solved by means of superposition of of the Laplace and Fourier transforms in variable \(t\) and \(x,\) respectively. The appropriate structures and negative properties for their Green functions are provided. The boundary problem in a bounded space domain is also solved by the spatial Sine transform and temporal Laplace transform, whose solution is given in the form of a series.


35R11 Fractional partial differential equations
35A22 Transform methods (e.g., integral transforms) applied to PDEs
35C10 Series solutions to PDEs
35A08 Fundamental solutions to PDEs
Full Text: DOI EuDML


[1] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 1056.93542 · doi:10.1109/9.739144
[2] F. Mainardi, “Fractional calculus: some basic problems in continuum and statistical mechanics,” in Fractals and Fractional Calculus in Continuum Mechanics, A. Carpinteri and F. Mainardi, Eds., vol. 378 of CISM Courses and Lectures, pp. 291-348, Springer, Vienna, Austria, 1997. · Zbl 0917.73004
[3] R. C. Cascaval, E. C. Eckstein, C. L. Frota, and J. A. Goldstein, “Fractional telegraph equations,” Journal of Mathematical Analysis and Applications, vol. 276, no. 1, pp. 145-159, 2002. · Zbl 1038.35142 · doi:10.1016/S0022-247X(02)00394-3
[4] E. Orsingher and L. Beghin, “Time-fractional telegraph equations and telegraph processes with brownian time,” Probability Theory and Related Fields, vol. 128, no. 1, pp. 141-160, 2004. · Zbl 1049.60062 · doi:10.1007/s00440-003-0309-8
[5] J. Chen, F. Liu, and V. Anh, “Analytical solution for the time-fractional telegraph equation by the method of separating variables,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1364-1377, 2008. · Zbl 1138.35373 · doi:10.1016/j.jmaa.2007.06.023
[6] E. Orsingher and X. Zhao, “The space-fractional telegraph equation and the related fractional telegraph process,” Chinese Annals of Mathematics Series B, vol. 24, no. 1, pp. 45-56, 2003. · Zbl 1033.60077 · doi:10.1142/S0252959903000050
[7] S. Momani, “Analytic and approximate solutions of the space- and time-fractional telegraph equations,” Applied Mathematics and Computation, vol. 170, no. 2, pp. 1126-1134, 2005. · Zbl 1103.65335 · doi:10.1016/j.amc.2005.01.009
[8] R. Figueiredo Camargo, A. O. Chiacchio, and E. Capelas de Oliveira, “Differentiation to fractional orders and the fractional telegraph equation,” Journal of Mathematical Physics, vol. 49, no. 3, Article ID 033505, 12 pages, 2008. · Zbl 1153.81330 · doi:10.1063/1.2890375
[9] F. Mainardi, “Fractional relaxation-oscillation and fractional diffusion-wave phenomena,” Chaos, Solitons & Fractals, vol. 7, no. 9, pp. 1461-1477, 1996. · Zbl 1080.26505 · doi:10.1016/0960-0779(95)00125-5
[10] R. Gorenflo, Y. Luchko, and F. Mainardi, “Wright functions as scale-invariant solutions of the diffusion-wave equation,” Journal of Computational and Applied Mathematics, vol. 118, no. 1-2, pp. 175-191, 2000. · Zbl 0973.35012 · doi:10.1016/S0377-0427(00)00288-0
[11] F. Mainardi, Y. Luchko, and G. Pagnini, “The fundamental solution of the space-time fractional diffusion equation,” Fractional Calculus & Applied Analysis, vol. 4, no. 2, pp. 153-192, 2001. · Zbl 1054.35156
[12] W. R. Schneider and W. Wyss, “Fractional diffusion and wave equations,” Journal of Mathematical Physics, vol. 30, no. 1, pp. 134-144, 1989. · Zbl 0692.45004 · doi:10.1063/1.528578
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.