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)
Generalized differential transform method to space-time fractional telegraph equation. (English) Zbl 1234.35299
Summary: We use the generalized differential transform method (GDTM) to derive the solution of the space-time fractional telegraph equation in closed form. The space and time fractional derivatives are considered in Caputo sense and the solution is obtained in terms of the Mittag-Leffler functions.

MSC:
35R11Fractional partial differential equations
35Q60PDEs in connection with optics and electromagnetic theory
33E12Mittag-Leffler functions and generalizations
WorldCat.org
Full Text: DOI
References:
[1] R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000. · Zbl 0998.26002 · doi:10.1142/9789812817747
[2] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier, Amsterdam, The Netherlands, 2006. · Zbl 1092.45003
[3] K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, A Wiley-Interscience Publication, John Wiley & Sons, New York, NY, USA, 1993. · Zbl 0789.26002
[4] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 0924.34008
[5] 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
[6] S. S. Ray and R. K. Bera, “Solution of an extraordinary differential equation by Adomian decomposition method,” Journal of Applied Mathematics, no. 4, pp. 331-338, 2004. · Zbl 1080.65069 · doi:10.1155/S1110757X04311010 · eudml:52065
[7] S. S. Ray and R. K. Bera, “An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method,” Applied Mathematics and Computation, vol. 167, no. 1, pp. 561-571, 2005. · Zbl 1082.65562 · doi:10.1016/j.amc.2004.07.020
[8] Q. Wang, “Homotopy perturbation method for fractional KdV-Burgers equation,” Chaos, Solitons and Fractals, vol. 35, no. 5, pp. 843-850, 2008. · Zbl 1132.65118 · doi:10.1016/j.chaos.2006.05.074
[9] Ahmet Yıldırım, “He’s homotopy perturbation method for solving the space- and time-fractional telegraph equations,” International Journal of Computer Mathematics, vol. 87, no. 13, pp. 2998-3006, 2010. · Zbl 1206.65239 · doi:10.1080/00207160902874653
[10] H. Jafari, C. Chun, S. Seifi, and M. Saeidy, “Analytical solution for nonlinear gas dynamic equation by homotopy analysis method,” Applications and Applied Mathematics, vol. 4, no. 1, pp. 149-154, 2009. · Zbl 1175.76124 · http://www.pvamu.edu/pages/5692.asp
[11] J.-H. He, “Approximate analytical solution for seepage flow with fractional derivatives in porous media,” Computer Methods in Applied Mechanics and Engineering, vol. 167, no. 1-2, pp. 57-68, 1998. · Zbl 0942.76077 · doi:10.1016/S0045-7825(98)00108-X
[12] A. Sevimlican, “An approximation to solution of space and time fractional telegraph equations by He’s variational iteration method,” Mathematical Problems in Engineering, vol. 2010, Article ID 290631, 10 pages, 2010. · Zbl 1191.65137 · doi:10.1155/2010/290631 · eudml:230924
[13] M. Garg and P. Manohar, “Numerical solution of fractional diffusion-wave equation with two space variables by matrix method,” Fractional Calculus & Applied Analysis, vol. 13, no. 2, pp. 191-207, 2010. · Zbl 1211.26004 · eudml:219635
[14] S. Momani, Z. Odibat, and V. S. Erturk, “Generalized differential transform method for solving a space- and time-fractional diffusion-wave equation,” Physics Letters. A, vol. 370, no. 5-6, pp. 379-387, 2007. · Zbl 1209.35066 · doi:10.1016/j.physleta.2007.05.083
[15] Z. Odibat and S. Momani, “A generalized differential transform method for linear partial differential equations of fractional order,” Applied Mathematics Letters, vol. 21, no. 2, pp. 194-199, 2008. · Zbl 1132.35302 · doi:10.1016/j.aml.2007.02.022
[16] Z. Odibat, S. Momani, and V. S. Erturk, “Generalized differential transform method: application to differential equations of fractional order,” Applied Mathematics and Computation, vol. 197, no. 2, pp. 467-477, 2008. · Zbl 1141.65092 · doi:10.1016/j.amc.2007.07.068
[17] J. K. Zhou, Differential Transformation and Its Applications for Electrical Circuits, Huazhong University Press, Wuhan, China, 1986.
[18] J. Biazar and M. Eslami, “Differential transform method for systems of Volterra integral equations of the first kind,” Nonlinear Science Letters A, vol. 1, pp. 173-181, 2010.
[19] A. El-Said, M. El-Wakil, M. Essam Abulwafa, and A. Mohammed, “Extended weierstrass transformation method for nonlinear evolution equations,” Nonlinear Science Letters A, vol. 1, 2010.
[20] Y. Keskin and G. Oturanc, “The reduced differential transform method: a new approach to fractional partial differential equations,” Nonlinear Science Letters A, vol. 1, pp. 207-217, 2010. · Zbl 1209.65100
[21] L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhäauser, Boston, Mass, USA, 1997. · Zbl 0892.35001
[22] A. C. Metaxas and R. J. Meredith, Industrial Microwave Heating, Peter Peregrinus, London, UK, 1993.
[23] E. C. Eckstein, J. A. Goldstein, and M. Leggas, “The mathematics of suspensions: Kac walks and asymptotic analyticity,” in Proceedings of the 4th Mississippi State Conference on Difference Equations and Computational Simulations, vol. 3, pp. 39-50. · Zbl 0963.76090
[24] J. Biazar, H. Ebrahimi, and Z. Ayati, “An approximation to the solution of telegraph equation by variational iteration method,” Numerical Methods for Partial Differential Equations, vol. 25, no. 4, pp. 797-801, 2009. · Zbl 1169.65335 · doi:10.1002/num.20373
[25] Radu C. Cascaval, E. C. Eckstein, 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
[26] D. Kaya, “A new approach to the telegraph equation: an application of the decomposition method,” Bulletin of the Institute of Mathematics. Academia Sinica, vol. 28, no. 1, pp. 51-57, 2000. · Zbl 0958.35006
[27] Z. Odibat and S. Momani, “A generalized differential transform method for linear partial differential equations of fractional order,” Applied Mathematics Letters, vol. 21, no. 2, pp. 194-199, 2008. · Zbl 1132.35302 · doi:10.1016/j.aml.2007.02.022
[28] 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
[29] 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
[30] M. Caputo, Elasticita e Dissipazione, Zanichelli, Bologna, Italy, 1969.
[31] G. M. Mittag-Leffler, “Sur la nouvelle fonction E\alpha (x),” Comptes rendus de l’ Académie des Sciences Paris, no. 137, pp. 554-558, 1903. · Zbl 34.0435.01
[32] A. Wiman, “Über den fundamentalsatz in der teorie der funktionen E\alpha (x),” Acta Mathematica, vol. 29, no. 1, pp. 191-201, 1905. · Zbl 36.0471.01 · doi:10.1007/BF02403202
[33] M. Garg and A. Sharma, “Solution of space-time fractional telegraph equation by Adomian decomposition method,” Journal of Inequalities and Special Functions, vol. 2, no. 1, pp. 1-7, 2011. · Zbl 1312.35178