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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 819268, 9 pages
http://dx.doi.org/10.1155/2013/819268
Research Article

Variational Approximate Solutions of Fractional Nonlinear Nonhomogeneous Equations with Laplace Transform

1School of Mathematical Sciences, Dezhou University, Dezhou 253023, China
2The Center of Data Processing and Analyzing, Dezhou University, Dezhou 253023, China

Received 11 May 2013; Accepted 14 August 2013

Academic Editor: Rodrigo Lopez Pouso

Copyright © 2013 Yanqin Liu et al. 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

A novel modification of the variational iteration method is proposed by means of Laplace transform and homotopy perturbation method. The fractional lagrange multiplier is accurately determined by the Laplace transform and the nonlinear one can be easily handled by the use of He’s polynomials. Several fractional nonlinear nonhomogeneous equations are analytically solved as examples and the methodology is demonstrated.

1. Introduction

Recently, systems of fractional nonlinear partial differential equations [13] have attracted much attention in a variety of applied sciences. With the development of nonlinear sciences, some numerical [46], semianalytical [712], and analytical methods [1315] have been developed for fractional differential equations. So, the semianalytical methods have largely been used to solve fractional equations. Most of these methods have their inbuilt deficiencies like the calculation of Adomian’s polynomials, the Lagrange multiplier, divergent results, and huge computational work. Recently, some improved homotopy perturbation methods [16, 17] and improved variational iteration methods, [18, 19] have been used by many researches.

The variational iteration method (VIM) [8, 9, 20] was extended to initial value problems of differential equations and has been one of the methods used most often. The key problem of the VIM is the correct determination of the Lagrange multiplier when the method is applied to fractional equations; combined with the Laplace transform, the crucial point of this method is solved efficiently by Wu and Baleanu [21, 22]. Laplace transform overcomes principle drawbacks in application of the VIM to fractional equations.

Motivated and inspired by the ongoing research in this field, we give a new modification of variational iteration method, combined with the Laplace transform and the homotopy perturbation method. The fractional Lagrange multiplier is accurately determined by the Laplace transform and the nonlinear one can be easily handed by the use of He’s polynomials. In this work, we will use this new method to obtain approximate solutions of the fractional nonlinear equations, and the fractional derivatives are described in the Caputo sense.

2. Description of the Method

In order to illustrate the basic idea of the technique, consider the following general nonlinear system: where , is the term of the highest-order derivative, is the source term, represents the general nonlinear differential operator, and is the linear differential operator.

Now, we consider the application of the modified VIM [21, 22]. Taking the above Laplace transform to both sides of (1) and (2), then the linear part is transformed into an algebraic equation as follows: where . The iteration formula of (3) can be used to suggest the main iterative scheme involving the Lagrange multiplier as

Considering as restricted terms, one can derive a Lagrange multiplier as

With (5) and the inverse-Laplace transform , the iteration formula (4) can be explicitly given as is an initial approximation of (1), and

In order to deal with the nonlinear term in the iteration formula (6), combining with the homotopy perturbation method, we give a new modification of the above method [21, 22]. In the homotopy method, the basic assumption is that the solutions can be written as a power series in : and the nonlinear term can be decomposed as where is an embedding parameter. is He’s polynomials [16, 23] can be generated by

This new modified method is obtained by the elegant coupling of correction function (6) of variational iteration method with He’s polynomials and is given by represents the term arising from the source term and the prescribed initial conditions. Equating the terms with identical powers in , we obtain the following approximations:

The best approximations for the solution are

This new modified method here transfers the problem into the partial differential equation in the Laplace -domain, removes the differentiation with respect to time, and uses He’s polynomials to deal with the nonlinear term. This new method basically illustrates how three powerful algorithms, variational iteration method, Laplace transform method, and homotopy perturbation method, can be combined and used to approximate the solutions of nonlinear equation. In this work, we will use this method to solve fractional nonlinear equations.

3. Illustrative Examples

We will apply the new modified VIM to both PDEs and FDEs. All the results are calculated by using the symbolic calculation software Mathematica.

3.1. Partial Differential Equations

Example 1. Consider the following nonhomogeneous nonlinear Gas Dynamic equation [24] with the initial condition

After taking the Laplace transform to both sides of (14) and (15), we get the following iteration formula:

Considering as restricted terms, Lagrange multiplier can be defined as ; with the inverse-Laplace transform, the approximate solution of (16) can be given as where is an initial approximation of (14), and

Combining with the homotopy perturbation method, one has where is He’s polynomials that represent nonlinear term ; we have a few terms of the He’s polynomials for which are given by

Comparing the coefficient with identical powers in , and so on; in this manner the rest of component of the solution can be obtained. The solution of (14) and (15) in series form is given by which is the exact solution. For this equation, the first-order approximate solution is justly the exact solution, and this proposed new method provides the solution in a rapid convergent. Furthermore, the new modified method can be easily extended to FDEs and this is the main purpose of our work.

3.2. Fractional Differential Equations

Let us consider the time fractional equation as follows: where is the source term, represents the general nonlinear differential operator, and is the linear differential operator. And the Caputo timefractional derivative operator of order is defined as where denotes the Gamma function.

Now, we consider the application of the modified VIM [21, 22]. The following Laplace transform of the term holds: where . The detailed properties of fractional calculus and Laplace transform can be found in [1, 2]; we have chosen to the Caputo fractional derivative because it allows traditional initial and boundary conditions to be included in the formulation of the problem. Taking the above Laplace transform to both sides of (23) and (24), the iteration formula of (23) can be constructed as

Considering as restricted terms, one can derive a Lagrange multiplier as

With (28) and the inverse-Laplace transform , the iteration formula (27) can be explicitly given as is an initial approximation of (23), and

In the homotopy method, the basic assumption is that the solutions can be written as a power series in : and the nonlinear term can be decomposed as where is an embedding parameter. is He’s polynomials [16, 23] that can be generated by

The variational homotopy perturbation method is obtained by the elegant coupling of correction function (29) of variational iteration method with He’s polynomials and is given by represents the term arising from the source term and the prescribed initial conditions. Equating the terms with identical powers in , we obtain the following approximations:

The best approximations for the solution are . Let us apply the above method to solve fractional nonlinear equations of Caputo type.

Example 2. Consider the following nonlinear space time fractional equation [25]: where , and the time-space fractional derivatives defined here are in Caputo sense. The Caputo space-fractional derivative operator of order is defined as

After taking the Laplace transform on both sides of (36) and (37), we get the following iteration formula:

As a result, after the identification of a Lagrange multiplier , and with the inverse-Laplace transform, one can derive is an initial approximation of (36), and

Applying the variational homotopy perturbation method, one has where is He’s polynomials that represent nonlinear term ; we have a few terms of the He’s polynomials for which are given by

Comparing the coefficient with identical powers in , one has

The solution of (36) and (37) is given as . If we take , one has

The noise terms between the components and can be canceled and the remaining term of still satisfies the equation. For this special case, the exact solution is therefore which was given in [25].

Example 3. Consider the following timefractional nonlinear system arising in thermoelasticity [26]: where , and the time fractional derivatives defined here are in Caputo sense. ,   and   are defined by

and the right-hand side of (46) is replaced by where , and are defined above and with the initial conditions thus the exact solution of system is . After taking the Laplace transform to both sides of (46) and (50), we get the following iteration formula: where . As a result, after the identification of a Lagrange multiplier and with the inverse-Laplace transform, one can derive the following iteration formula:

is an initial approximation of (46), and

Applying the variational homotopy perturbation method, one has where , is He’s polynomials that represent nonlinear terms , respectively; we have a few terms of the He’s polynomials for these nonlinear terms which are given by

Comparing the coefficient with identical powers in , one has and so on; in this manner the rest of components of the solution can be obtained using the Mathematica symbolic computation software for purpose of simlification, the approximate solutions are not listed here.

4. Conclusion

In this paper, a new modification of variational iteration method is considered, which is based on Laplace transform and homotopy perturbation method. The fractional lagrange multiplier is accurately determined by the Laplace transform and the nonlinear one can be easily handled by the use of He’s polynomials. Several fractional nonlinear nonhomogeneous equations are analytically solved as examples and the methodology is demonstrated. Examples 1, 2, and 3 have been successfully solved. And the results show that this method is a powerful and reliable method for finding the solution of the fractional nonlinear equations.

Acknowledgment

The authors express their thanks to the referees for their fruitful advices and comments.

References

  1. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. View at Zentralblatt MATH · View at MathSciNet
  2. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific Publishing, River Edge, NJ, USA, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. R. Metzler and J. Klafter, “The random walk's guide to anomalous diffusion: a fractional dynamics approach,” Physics Reports, vol. 339, no. 1, p. 77, 2000. View at Publisher · View at Google Scholar · View at MathSciNet
  4. K. Diethelm and N. J. Ford, “Multi-order fractional differential equations and their numerical solution,” Applied Mathematics and Computation, vol. 154, no. 3, pp. 621–640, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. F. W. Liu, V. Anh, and I. Turner, “Numerical solution of the space fractional Fokker-Planck equation,” Journal of Computational and Applied Mathematics, vol. 166, no. 1, pp. 209–219, 2004.
  6. D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional Calculus: Models and Numerical Methods, vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientific, Boston, Mass, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  7. J. S. Duan, R. Rach, D. Buleanu, and A. M. Wazwaz, “A review of the Adomian decomposition method and its applicaitons to fractional differential equations,” Communications in Fractional Calculus, vol. 3, pp. 73–99, 2012.
  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. A.-M. Wazwaz, “The variational iteration method for analytic treatment of linear and nonlinear ODEs,” Applied Mathematics and Computation, vol. 212, no. 1, pp. 120–134, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  10. V. S. Erturk, S. Momani, and Z. Odibat, “Application of generalized differential transform method to multi-order fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 13, no. 8, pp. 1642–1654, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. J.-H. He, “Homotopy perturbation method: a new nonlinear analytical technique,” Applied Mathematics and Computation, vol. 135, no. 1, pp. 73–79, 2003. View at Publisher · View at Google Scholar · View at MathSciNet
  12. Z. Odibat and S. Momani, “Modified homotopy perturbation method: application to quadratic Riccati differential equation of fractional order,” Chaos, Solitons & Fractals, vol. 36, no. 1, pp. 167–174, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  13. X. Y. Jiang and H. T. Qi, “Thermal wave model of bioheat transfer with modified Riemann-Liouville fractional derivative,” Journal of Physics, vol. 45, no. 48, Article ID 485101, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  14. J. H. Ma and Y. Q. Liu, “Exact solutions for a generalized nonlinear fractional Fokker-Planck equation,” Nonlinear Analysis. Real World Applications, vol. 11, no. 1, pp. 515–521, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. Y.-Q. Liu and J.-H. Ma, “Exact solutions of a generalized multi-fractional nonlinear diffusion equation in radical symmetry,” Communications in Theoretical Physics, vol. 52, no. 5, pp. 857–861, 2009. View at Publisher · View at Google Scholar · View at Scopus
  16. Y. Q. Liu, “Approximate solutions of fractional nonlinear equations using homotopy perturbation transformation method,” Abstract and Applied Analysis, vol. 2012, Article ID 752869, 14 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. Y. Q. Liu, “Study on space-time fractional nonlinear biological equation in radial symmetry,” Mathematical Problems in Engineering, vol. 2013, Article ID 654759, 6 pages, 2013. View at MathSciNet
  18. Y. Q. Liu, “Variational homotopy perturbation method for solving fractional initial boundary value problems,” Abstract and Applied Analysis, vol. 2012, Article ID 727031, 10 pages, 2012. View at Zentralblatt MATH · View at MathSciNet
  19. S. M. Guo, L. Q. Mei, and Y. Li, “Fractional variational homotopy perturbation iteration method and its application to a fractional diffusion equation,” Applied Mathematics and Computation, vol. 219, no. 11, pp. 5909–5917, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  20. 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
  21. G.-C. Wu and D. Baleanu, “Variational iteration method for fractional calculus—a universal approach by Laplace transform,” Advances in Difference Equations, vol. 2013, article 18, 9 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  22. G. C. Wu, “Laplace tranform overcoming principle drawbacks in applicaiton of the variation iteration method to fractional heat equations,” Thermal Science, vol. 16, no. 4, pp. 1257–1261, 2012.
  23. A. Ghorbani, “Beyond Adomian polynomials: He polynomials,” Chaos, Solitons and Fractals, vol. 39, no. 3, pp. 1486–1492, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  24. H. Jafari, M. Alipour, and H. Tajadodi, “Two-dimensional differential transform method for solving nonlinear partial differential equaitons,” International Journal of Research and Reviews in Applied Sciences, vol. 2, no. 1, pp. 47–52, 2010.
  25. S. Momani and Z. Odibat, “A novel method for nonlinear fractional partial differential equations: combination of DTM and generalized Taylor's formula,” Journal of Computational and Applied Mathematics, vol. 220, no. 1-2, pp. 85–95, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  26. N. H. Sweilam and M. M. Khader, “Variational iteration method for one dimensional nonlinear thermoelasticity,” Chaos, Solitons & Fractals, vol. 32, no. 1, pp. 145–149, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet