×

An approximate analytical solution of time-fractional telegraph equation. (English) Zbl 1216.65135

Summary: The powerful, easy-to-use and effective approximate analytical mathematical tool homotopy analysis method is used to solve the telegraph equation with fractional time derivative \(\alpha\) \((1 < \alpha \leqslant 2\)). By using initial values, explicit solutions of the telegraph equation for different particular cases are derived. The numerical solutions show that only a few iterations are needed to obtain accurate approximate solutions. The method performs extremely well in terms of efficiency and simplicity to solve this historical model.

MSC:

65M70 Spectral, collocation and related methods for initial value and initial-boundary value problems involving PDEs
35L20 Initial-boundary value problems for second-order hyperbolic equations
35R11 Fractional partial differential equations
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Mainardi, F., Fractional calculus: some basic problems in continuum and statistical mechanics, () · Zbl 0917.73004
[2] Podlubny, I., Fractional differential equations, (1999), Academic Press New York · Zbl 0918.34010
[3] R. Gorenflo, F. Mainardi, Fractional calculus: integral and differential equations of fractional order, in: A. Carpinteri, F. Mainardi (Eds.), Fractals and Fractional Calculus, New York, 1997. · Zbl 1030.26004
[4] A.Y. Luchko, R. Groreflo, The initial value problem for some fractional differential equations with the Caputo derivative, Preprint series A08-98, Fachbreich Mathematik und Informatik, Freic Universitat, Berlin, 1998.
[5] S.J. Liao, On the proposed homotopy analysis technique for nonlinear problems and its applications, Ph.D. Dissertation. Shanghai Jiao Tong University, Shanghai, 1992.
[6] Hashim, I.; Abdulaziz, O.; Momani, S., Homotopy analysis method for fractional ivps, Communications in nonlinear science and numerical simulation, 14, 674-684, (2009) · Zbl 1221.65277
[7] Mohebbi, A.; Dehaghan, M., High order compact solution of the one dimensional linear hyperbolic equation, Numerical method for partial differential equations, 24, 1122-1135, (2008)
[8] El-Azab, M.S.; El-Glamel, M., A numerical algorithm for the solution of telegraph equation, Applied mathematics and computation, 190, 757-764, (2007) · Zbl 1132.65087
[9] Yousefi, S.A., Legendre multi wavelet Galerkin method for solving the hyperbolic telegraph equation, Numerical method for partial differential equations, (2008), doi:10.1002/num
[10] Gao, F.; Chi, C., Unconditionally stable difference scheme for a one-space dimensional linear hyperbolic equation, Applied mathematics and computation, 187, 1272-1276, (2007) · Zbl 1114.65347
[11] Dehghan, M.; Ghesmati, A., Solution of the second-order one-dimensional hyperbolic telegraph equation by using the dual reciprocity boundary integral equation (DRBIE) method, Engineering analysis with boundary elements, 34, 51-59, (2010) · Zbl 1244.65137
[12] Das, S.; Gupta, P.K., Homotopy analysis method for solving fractional hyperbolic partial differential equations, International journal of computer mathematics, 88, 578-588, (2011) · Zbl 1211.65133
[13] Atanackovic, T.M.; Pilipovic, S.; Zorica, D., A diffusion wave equation with two fractional derivatives of different order, Journal of physics: mathematical and theoretical, 40, 5319-5333, (2007) · Zbl 1121.35069
[14] Orsingher, E.; Beghin, L., Time-fractional telegraph equations and telegraph processes with Brownian time, Probability theory and related fields, 128, 141-160, (2004) · Zbl 1049.60062
[15] Huang, F., Analytical solution for the time-fractional telegraph equation, Journal of applied mathematics, (2009), doi:10.1155/2009/890158 · Zbl 1190.35224
[16] Momani, S., Analytic and approximate solutions of the space- and time-fractional telegraph equations, Applied mathematics and computation, 170, 1126-1134, (2005) · Zbl 1103.65335
[17] Chen, J.; Liu, F.; Anh, V., Analytical solution for the time-fractional telegraph equation by the method of separating variables, Journal on mathematical analysis and applications, 338, 1364-1377, (2008) · Zbl 1138.35373
[18] Liao, S.J., Homotopy analysis method: a new analytical method for nonlinear problems, Applied mathematics and mechnics, 19, 957-962, (1998) · Zbl 1126.34311
[19] Liao, S.J., On the homotopy analysis method for nonlinear problems, Applied mathematics and computation, 147, 499-513, (2004) · Zbl 1086.35005
[20] Liao, S.J.; Tan, Y., A general approach to obtain series solutions of nonlinear differential equations, Studies in applications of mathematics, 119, 297-355, (2007)
[21] Liao, S.J., Notes on the homotopy analysis method: some definition and theorems, Communications in nonlinear science and numerical simulation, 14, 983-997, (2009) · Zbl 1221.65126
[22] Giona, M.; Roman, H.E., Fractional diffusion equation on fractals: one-dimensional case and asymptotic behavior, Journal of physics: mathematical and general, 25, 2093-2105, (1992) · Zbl 0755.60067
[23] Das, S., A note on fractional diffusion equations, Chaos, solitons and fractals, 42, 2074-2079, (2009) · Zbl 1198.65137
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.