International Journal of Differential Equations

A New Technique of Laplace Variational Iteration Method for Solving Space-Time Fractional Telegraph Equations

Fatima A. Alawad, Eltayeb A. Yousif, and Arbab I. Arbab

Full-text: Open access

Abstract

In this paper, the exact solutions of space-time fractional telegraph equations are given in terms of Mittage-Leffler functions via a combination of Laplace transform and variational iteration method. New techniques are used to overcome the difficulties arising in identifying the general Lagrange multiplier. As a special case, the obtained solutions reduce to the solutions of standard telegraph equations of the integer orders.

Article information

Source
Int. J. Differ. Equ., Volume 2013 (2013), Article ID 256593, 10 pages.

Dates
Received: 7 July 2013
Accepted: 8 November 2013
First available in Project Euclid: 20 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.ijde/1484881354

Digital Object Identifier
doi:10.1155/2013/256593

Mathematical Reviews number (MathSciNet)
MR3149764

Zentralblatt MATH identifier
1294.35184

Citation

Alawad, Fatima A.; Yousif, Eltayeb A.; Arbab, Arbab I. A New Technique of Laplace Variational Iteration Method for Solving Space-Time Fractional Telegraph Equations. Int. J. Differ. Equ. 2013 (2013), Article ID 256593, 10 pages. doi:10.1155/2013/256593. https://projecteuclid.org/euclid.ijde/1484881354


Export citation

References

  • J. Sabatier, O. P. Agrawal, and J. A. T. Machado, Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, Springer, New York, NY, USA, 2007.MR2432163
  • L. Debnath, “Recent applications of fractional calculus to science and engineering,” International Journal of Mathematics and Mathematical Sciences, no. 54, pp. 3413–3442, 2003.MR2025566
  • M. Bhatti, “Fractional Schrödinger wave equation and fractional uncertainty principle,” International Journal of Contemporary Mathematical Sciences, vol. 2, no. 19, pp. 943–950, 2007.MR2371326
  • Z. Z. Ganji, D. D. Ganji, and Y. Rostamiyan, “Solitary wave solutions for a time-fraction generalized Hirota-Satsuma coupled KdV equation by an analytical technique,” Applied Mathematical Modelling, vol. 33, no. 7, pp. 3107–3113, 2009.MR2502734
  • R. E. Gutirrez, J. M. Rosario, and J. T. Machado, “Fractional order calculus: basic concepts and engineering applications,” Mathematical Problems in Engineering, vol. 2010, Article ID 375858, 19 pages, 2010.
  • 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.MR2610512
  • D. Campos and V. Méndez, “Different microscopic interpretations of the reaction-telegrapher equation,” Journal of Physics A, vol. 42, no. 7, Article ID 075003, 13 pages, 2009.MR2525453
  • 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.MR2827220
  • M. Garg, P. Manohar, and S. L. Kalla, “Generalized differential transform method to space-time fractional telegraph equation,” International Journal of Differential Equations, vol. 2011, Article ID 548982, 9 pages, 2011.MR2843508
  • A. I. Arbab, “Derivation of Dirac, Klein-Gordon, Schrodinger, diffusion and quantum heat transport equations from the universal wave equation,” Europhysics Letters, vol. 92, no. 4, Article ID 40001, 2010.
  • J. Chen, F. Liu, and V. Anh, “Analytical solution for the time-fractional telegraph equation by the method of separating variables,” Journal of Mathematical Analysis and Applications, vol. 338, no. 2, pp. 1364–1377, 2008.MR2386503
  • A. Ansari, “Fractional exponential operators and time-fractional telegraph equation,” Boundary Value Problems, vol. 2012, article 125, 2012.MR3016677
  • F. Huang, “Analytical solution for the time-fractional telegraph equation,” Journal of Applied Mathematics, vol. 2009, Article ID 890158, 9 pages, 2009.MR2565356
  • S. Das, K. Vishal, P. K. Gupta, and A. Yildirim, “An approximate analytical solution of time-fractional telegraph equation,” Applied Mathematics and Computation, vol. 217, no. 18, pp. 7405–7411, 2011.MR2784584
  • W. Jiang and Y. Lin, “Representation of exact solution for the time-fractional telegraph equation in the reproducing kernel space,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 9, pp. 3639–3645, 2011.MR2787810
  • Y. Khan, J. Diblik, N. Faraz, and Z. Smarda, “An efficient new perturbation Laplace method for space-time fractional telegraph equations,” Advance in Difference Equations, vol. 2012, article 204, 2012.
  • J. H. He, “An aproximation to solution of space and time fractional telegraph equations by the variational iteration method,” Mathematical Problems in Engineering, vol. 2012, Article ID 394212, 2 pages, 2012.MR2983802
  • T. A. Abassy, M. A. El-Tawil, and H. El-Zoheiry, “Exact solutions of some nonlinear partial differential equations using the variational iteration method linked with Laplace transforms and the Padé technique,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 940–954, 2007.MR2395632
  • Z. Hammouch and T. Mekkaoui, “A Laplace-variational iteration method for solving the homogeneous Smoluchowski coagulation equation,” Applied Mathematical Sciences, vol. 6, no. 18, pp. 879–886, 2012.MR2891394
  • A. S. Arife and A. Yildirim, “New modified variational iteration transform method (MVITM) for solving eighth-order boundary value problems in one step,” World Applied Sciences Journal, vol. 13, no. 10, pp. 2186–2190, 2011.
  • I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999.MR1658022
  • L. Debnath and D. Bhatta, Integral Transforms and Their Applications, Taylor & Francis Group,LLC, 2nd edition, 2007.MR2253985
  • H. J. Haubold, A. M. Mathai, and R. K. Saxena, “Mittag-Leffler functions and their applications,” Journal of Applied Mathematics, vol. 2011, Article ID 298628, 51 pages, 2011.MR2800586
  • M. N. Berberan-Santos, “Properties of the Mittag-Leffler relaxation function,” Journal of Mathematical Chemistry, vol. 38, no. 4, pp. 629–635, 2005.MR2210207
  • J. H. He, “Variational iteration method–-a kind of nonlinear analytical technique: some examples,” International Journal of Nonlinear Mechanics, vol. 34, no. 4, pp. 699–708, 1999.
  • A. M. Wazwaz, Partial Differential Equations and Solitary Waves Theory, Nonlinear Physical Science, Higher Education Press, Beijing, China; Springer, Heidelberg, Germany, 2009.MR2548288
  • J. H. He and X. H. Wu, “Variational iteration method: new development and applications,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 881–894, 2007.MR2395625
  • S. A. Khuri and A. Sayfy, “A Laplace variational iteration strategy for the solution of differential equations,” Applied Mathematics Letters, vol. 25, no. 12, pp. 2298–2305, 2012.MR2967833 \endinput