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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:

35R11 Fractional partial differential equations
35Q60 PDEs in connection with optics and electromagnetic theory
33E12 Mittag-Leffler functions and generalizations
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[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
[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
[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
[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
[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. · JFM 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. · JFM 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
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