About this Journal Submit a Manuscript Table of Contents
Mathematical Problems in Engineering
Volume 2010 (2010), Article ID 290631, 10 pages
http://dx.doi.org/10.1155/2010/290631
Research Article

An Approximation to Solution of Space and Time Fractional Telegraph Equations by He's Variational Iteration Method

Department of Mathematics, Faculty of Arts and Sciences, Dokuz Eylül University, Tınaztepe, Buca 35160, Izmir, Turkey

Received 10 November 2009; Accepted 27 January 2010

Academic Editor: Massimo Scalia

Copyright © 2010 Ali Sevimlican. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

He's variational iteration method (VIM) is used for solving space and time fractional telegraph equations. Numerical examples are presented in this paper. The obtained results show that VIM is effective and convenient.

1. Introduction

In recent years, there has been a great deal of interest in fractional differential equations since there have been a wide variety of applications in physics and engineering. The space and time fractional telegraph equations have been studied by Orsingher and Zhao [1] and Orsingher and Beghin [2]. The telegraph equation is used in signal analysis for transmission and propagation of electrical signals and also used modeling reaction diffusion [3, 4]. In the papers by Momani [5] and Yildirim [6], Adomian decomposition method (ADM) and homotopy perturbation method (HPM) were used for solving the space and time fractional telegraph equations, respectively. Variational iteration method was used for solving linear telegraph equation in [7]. In this paper we will use variational iteration method (VIM) for solving the space and time fractional telegraph equations. The variational iteration method (VIM) which was developed in 1999 by He [8] has been applied to a wide variety of differential equations by many authors. He [9] used the variational iteration method (VIM) for solving seepage flow with fractional derivatives in porous media. Momani and Odibat [10] constructed numerical solutions of the space-time fractional advection dispersion equation by decomposition method and variational iteration method. Momani and Odibat [11] also compared homotopy perturbation method (HPM) and VIM for linear fractional partial differential equations. Drăgănescu [12] used VIM for viscoelastic models with fractional derivatives. Yulita at al. [13] applied the variational iteration method for fractional heat and wave-like equations. Dehghan at al. [14] studied telegraph and space telegraph equations using variational iteration method. In [14], space fractional telegraph equation was considered for . However, in this paper space fractional telegraph equation has been considered for and also variational iteration method has been applied for time fractional telegraph equation.

We note that the space and time fractional derivatives are considered in Caputo sense in this paper. The main objective of the present paper is to extend the application of the variational iteration method (VIM) to obtain approximate solution of the space and time fractional telegraph equations.

2. He's Variational Iteration Method

We will give a brief description of He's variational iteration method. The basic concepts of the variational iteration method can be expressed as follows. Consider the differential equation of the form where is a linear operator, is a nonlinear operator, and is the inhomogeneous term. According to the variational iteration method, a correction functional for (2.1) can be constructed as follows:

where is a general Lagrange multiplier, which can be identified optimally via the variational theory [15, 16], the subscript denotes the th approximations, and is considered as restricted variation [17, 18], that is, The successive approximations , of the solution can be obtained after finding the Lagrange multiplier and by using the selective function which is usually selected from initial conditions.

3. Space and Time Fractional Derivatives in Caputo Sense

For to be the smallest integer that exceeds , the Caputo time-fractional derivative operator of order is defined as

and for to be the smallest integer that exceeds , the Caputo space-fractional derivative operator of order is defined as

where is the Gamma function.

Further information about fractional derivatives and its properties can be found in [19, 20].

4. Application to Space-Time Fractional Telegraph Equations

In this section we will obtain iteration formulas for space-time fractional telegraph equations. We first consider the following space fractional telegraph equation for :

where and are given constants, given function. The correctional functional for (4.1) can be approximately expressed as follows

Making the correctional functional in (4.2) stationary, and noticing that , we obtain the following stationary conditions for :

Lagrange multiplier can be identified from (4.3) as

Substituting the above obtained Lagrange multiplier into (4.2), we get the following iteration formula:

Now consider the following time fractional telegraph equation for :

where and are given constants, and given function.. The correctional functional for the equation (4.6) can be approximately expressed as follows:

Making the correctional functional in (4.7) stationary, and noticing that , we obtain the following stationary conditions for :

Lagrange multiplier can be identified from (4.8) as

Substituting the above obtained Lagrange multiplier into (4.7), we get the following iteration formula:

5. Numerical Examples

We will give the following three examples to illustrate variational iteration method for solving the space and time fractional telegraph equations.

Example 5.1. We first consider the following one-dimensional initial and boundary value problem of space-fractional homogeneous telegraph equation for , (see [5, 21]) It follows from (4.5) for and ; the iteration formula for (5.1) can be written in the following form: We start with initial approximation: and by the iteration formula (5.4) we obtain the first two approximations as and so on; in the same manner further approximations of the iteration formula (5.4) can be obtained by Mapple. We observe that, setting in the approximations yields the exact solution as . In Figures 1(a), 1(b), and 1(c) exact and second-order approximate solutions of (5.1)–(5.3) are given. Figures 2(a) and 2(b) show the evolution results for the second-order approximate solutions of (5.1)–(5.3) obtained for different values of using the variational iteration method.

fig1
Figure 1: The surfaces related with the solution of (5.1)–(5.3) for .
fig2
Figure 2: The surfaces show the second-order approximate solutions of (5.1)–(5.3).

Example 5.2. We now consider the following one-dimensional initial and boundary value problem of space-fractional inhomogeneous telegraph equation for , (see, [5, 21]): The iteration formula for (5.7) can be written in the following form: We start with initial approximation: and by iteration formula (5.10) we obtain the first two approximations as and so on; in the same manner further approximations of the iteration formula (5.10) can be obtained by Mapple. We observe that, setting in the approximations and canceling noise terms yields the exact solution as . In Figures 3(a), 3(b), and 3(c) exact and second-order approximate solutions of (5.7)–(5.9) are given. Figures 4(a) and 4(b) show the evolution results for the second-order approximate solutions of (5.7)–(5.9) obtained for different values of using the variational iteration method.

fig3
Figure 3: The surfaces related with the solution of (5.7)–(5.9) for .
fig4
Figure 4: The surfaces show the second-order approximate solutions of (5.7)–(5.9).

Example 5.3. We last consider the following initial and boundary value problem of time fractional telegraph equation of order , (see, [5, 21]): It follows from (4.10) for and ; the iteration formula for (5.13) can be written in the following form: We start with the following initial approximation: and by the iteration formula (5.17), we get and so on; in the same manner further approximations of the iteration formula (5.17) can be obtained by Mapple.

6. Conclusion

The variational iteration method has been successfully applied for finding the solution of space and time fractional telegraph equations. The space and time fractional derivatives are considered in the Caputo sense. We have achieved a very good agreement between the approximate solution obtained by He's VIM and the exact solution. The results of the examples show that He's variational iteration method is reliable and efficient method for solving space and time fractional telegraph equations and also other equations.

References

  1. E. Orsingher and X. Zhao, “The space-fractional telegraph equation and the related fractional telegraph process,” Chinese Annals of Mathematics B, vol. 24, no. 1, pp. 45–56, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  2. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. L. Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, Birkhäuser, Boston, Mass, USA, 1997. View at MathSciNet
  4. A. C. Metaxas and R. J. Meredith, Industrial Microwave Heating, Peter Peregrinus, London, UK, 1993.
  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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. A. Yıldırım, “He's homtopy perturbation method for solving the space and time fractional telegraph equations,” International Journal of Computer Mathematics. In press.
  7. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. J.-H. He, “Variational iteration method—a kind of non-linear analytical technique: some examples,” International Journal of Non-Linear Mechanics, vol. 34, no. 4, pp. 699–708, 1999. View at Scopus
  9. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. S. Momani and Z. Odibat, “Numerical solutions of the space-time fractional advection-dispersion equation,” Numerical Methods for Partial Differential Equations, vol. 24, no. 6, pp. 1416–1429, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. S. Momani and Z. Odibat, “Comparison between the homotopy perturbation method and the variational iteration method for linear fractional partial differential equations,” Computers & Mathematics with Applications, vol. 54, no. 7-8, pp. 910–919, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. G. E. Drăgănescu, “Application of a variational iteration method to linear and nonlinear viscoelastic models with fractional derivatives,” Journal of Mathematical Physics, vol. 47, no. 8, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. R. Yulita Molliq, M. S. M. Noorani, and I. Hashim, “Variational iteration method for fractional heat- and wave-like equations,” Nonlinear Analysis: Real World Applications, vol. 10, no. 3, pp. 1854–1869, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. M. Dehghan, S. A. Yousefi, and A. Lotfi, “The use of He's variational iteration method for solving the telegraph and fractional telegraph equations,” Communications in Numerical Methods in Engineering. In press. View at Publisher · View at Google Scholar
  15. J.-H. He, “Variational iteration method—some recent results and new interpretations,” Journal of Computational and Applied Mathematics, vol. 207, no. 1, pp. 3–17, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. M. Inokuti, H. Sekine, and T. Mura, “General use of the Lagrange multiplier in nonlinear mathematical physics,” in Variational Method in the Mechanics of Solids, S. Nemat-Nasser, Ed., pp. 159–162, Pergamon Press, New York, NY, USA, 1978.
  17. J.-H. He, “A new approach to nonlinear partial differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 2, no. 4, pp. 115–123, 2000.
  18. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, Yverdon, Switzerland, 1993. View at MathSciNet
  20. 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. View at MathSciNet
  21. A. Yıldırım and H. Koçak, “Homotopy perturbation method for solving the space-time fractional advection-dispersion equation,” Advances in Water Resources, vol. 32, no. 12, pp. 1711–1716, 2009. View at Publisher · View at Google Scholar