# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 $\alpha$ $\left(1<\alpha ⩽2$). 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 (IVP of PDE) 35L20 Second order hyperbolic equations, boundary value problems 35R11 Fractional partial differential equations
##### References:
 [1] Mainardi, F.: Fractional calculus: some basic problems in continuum and statistical mechanics, Fractal and fractional calculus in continuum mechanics (1997) [2] Podlubny, I.: Fractional differential equations, (1999) [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. [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 · doi:doi:10.1016/j.cnsns.2007.09.014 [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 · doi:doi:10.1016/j.amc.2007.01.091 [9] Yousefi, S. A.: Legendre multi wavelet Galerkin method for solving the hyperbolic telegraph equation, Numerical method for partial differential equations (2008) [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 · doi:doi:10.1016/j.amc.2006.09.057 [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) [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 · doi:doi:10.1080/00207161003631901 [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 · doi:doi:10.1088/1751-8113/40/20/006 [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 · doi:doi:10.1007/s00440-003-0309-8 [15] Huang, F.: Analytical solution for the time-fractional telegraph equation, Journal of applied mathematics (2009) · Zbl 1190.35224 · doi:doi:10.1155/2009/890158 [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 · doi:doi:10.1016/j.amc.2005.01.009 [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 · doi:doi:10.1016/j.jmaa.2007.06.023 [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 · doi:doi:10.1007/BF02457955 [19] Liao, S. J.: On the homotopy analysis method for nonlinear problems, Applied mathematics and computation 147, 499-513 (2004) · Zbl 1086.35005 · doi:doi:10.1016/S0096-3003(02)00790-7 [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 · doi:doi:10.1016/j.cnsns.2008.04.013 [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 · doi:doi:10.1088/0305-4470/25/8/023 [23] Das, S.: A note on fractional diffusion equations, Chaos, solitons and fractals 42, 2074-2079 (2009) · Zbl 1198.65137 · doi:doi:10.1016/j.chaos.2009.03.163