International Journal of Differential Equations

Generalized Differential Transform Method to Space-Time Fractional Telegraph Equation

Mridula Garg, Pratibha Manohar, and Shyam L. Kalla

Full-text: Open access

Abstract

We use generalized differential transform method (GDTM) to derive the solution of 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 Mittag-Leffler functions.

Article information

Source
Int. J. Differ. Equ., Volume 2011 (2011), Article ID 548982, 9 pages.

Dates
Received: 28 May 2011
Revised: 20 July 2011
Accepted: 23 July 2011
First available in Project Euclid: 25 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1485313240

Digital Object Identifier
doi:10.1155/2011/548982

Mathematical Reviews number (MathSciNet)
MR2843508

Zentralblatt MATH identifier
1234.35299

Citation

Garg, Mridula; Manohar, Pratibha; Kalla, Shyam L. Generalized Differential Transform Method to Space-Time Fractional Telegraph Equation. Int. J. Differ. Equ. 2011 (2011), Article ID 548982, 9 pages. doi:10.1155/2011/548982. https://projecteuclid.org/euclid.ijde/1485313240


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References

  • R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier, Amsterdam, The Netherlands, 2006.
  • 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.
  • I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  • 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.
  • 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.
  • 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.
  • Q. Wang, “Homotopy perturbation method for fractional KdV-Burgers equation,” Chaos, Solitons and Fractals, vol. 35, no. 5, pp. 843–850, 2008.
  • Ahmet Y\ild\ir\im, “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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • J. K. Zhou, Differential Transformation and Its Applications for Electrical Circuits, Huazhong University Press, Wuhan, China, 1986.
  • 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.
  • 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.
  • 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.
  • L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhäauser, Boston, Mass, USA, 1997.
  • A. C. Metaxas and R. J. Meredith, Industrial Microwave Heating, Peter Peregrinus, London, UK, 1993.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • 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.
  • M. Caputo, Elasticita e Dissipazione, Zanichelli, Bologna, Italy, 1969.
  • 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.
  • A. Wiman, “Über den fundamentalsatz in der teorie der funktionen ${E}_{\alpha }(x)$,” Acta Mathematica, vol. 29, no. 1, pp. 191–201, 1905.
  • 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.