Abstract and Applied Analysis

Volume 2014 (2014), Article ID 760957, 9 pages

http://dx.doi.org/10.1155/2014/760957

## Variational Iteration Transform Method for Fractional Differential Equations with Local Fractional Derivative

School of Mathematics and Statistics, Nanyang Normal University, Nanyang 473061, China

Received 25 March 2014; Revised 15 May 2014; Accepted 16 May 2014; Published 18 June 2014

Academic Editor: Ali H. Bhrawy

Copyright © 2014 Yong-Ju Yang and Liu-Qing Hua. 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

We propose the variational iteration transform method in the sense of local fractional derivative, which is derived from the coupling method of local fractional variational iteration method and differential transform method. The method reduces the integral calculation of the usual variational iteration computations to more easily handled differential operation. And the technique is more orderly and easier to analyze computing result as compared with the local fractional variational iteration method. Some examples are illustrated to show the feature of the presented technique.

#### 1. Introduction

Fractional differential equation has been considered with great importance due to its demonstrated applications in various areas such as electrical networks, fluid flow, biology, and dynamical systems [1–8]. And many classical differential equations possess a fractional analogy. As a result, a substantial number of methods for solving the fractional differential equations are developed to solve fractional differential equations [9–25].

Recently, local fractional derivative and calculus theory has been introduced by the researcher in [18, 19], which is set up on fractal geometry and which is the best method for describing the nondifferential function defined on Cantor sets. The physical explanation of the local fractional derivative can be seen in [26, 27]. A great deal of research work has been directed for the nondifferentiable phenomena in fractal domain concerning local fractional derivative, for example, [11, 12, 15, 18–25].

Motivated by the ongoing research method of local fractional differential equation, we present variational iteration transform method, which is the coupling method of local fractional variational iteration method and differential transform method. It is worth mentioning that since the method makes the complex iteration calculation more orderly, it may be considered as an efficient modification of variational iteration method. The rest of the paper is organized as follows. In Section 2, the basic mathematical fundamentals are presented briefly. In Section 3, the local fractional function differential transform method for solving the differential equations with local fractional derivative is investigated. In Section 4, several examples are illustrated. Finally, in Section 5, the conclusion is given.

#### 2. Mathematical Fundamentals

In this section, we introduce the basic notions of local fractional derivative, local fractional integral, and local fractional differential transform method.

##### 2.1. Local Fractional Derivatives and Integrals

The local fractional derivative of of order at is defined by [18, 19] where .

In the interval , local fractional integral of of order is given by [18, 19] where , and , , and is a partition of the interval .

##### 2.2. Local Fractional Differential Transform Method

Similarly in [28–34], the local fractional different transform (or , resp.) of the function (or , resp.) is defined by the following formula: Obviously If we let and let then we have

In Table 1, we list a few operations concerning local fractional differential transform, where is a nonnegative integer.

#### 3. Local Fractional Variational Iteration Transform Method

To introduce the basic ideas of the variational iteration transform method, we consider the following nonlinear fractional differential equation in the sense of local fractional derivative: where is the linear local fractional operator of order less than , is the nonlinear local fractional operator of order less than , and is a source term of the nondifferential function.

According to the local fractional variational iteration method [20, 35], we can construct local fractional variational iteration algorithm: where is a fractal Lagrange multiplier.

Suppose is a restricted local fractional variation; that is, [20, 35]; we get [21] Substituting (10) into (9), a fractional iteration procedure is constructed as follows:

Applying local fractional differential transform with respect to on both sides of (11) and using local fractional derivative algorithm [18], we obtain and also where is an initial value.

By virtue of (5), (12), (13), and the operation displayed in Table 1, we can yield where and where

So, we have the exact solution of (8) in the form:

Now we discuss the solution of the following quadratic linear fractional differential equations which is the simple example of (8): where , , , and are all constant numbers.

According to (9), the correction iteration algorithm is given as follows: Taking local fractional differential transform with respect to on (19), we deduce where and where By using (20), (21), (22), and the results included in Table 1, we obtain and also If we let then By (23), (24), and (26), we have According to (14) and (27), we get Taking the limit, we arrive at

Hence, under the condition equation (25), the exact solution of (18) reads as follows:

#### 4. Illustrative Examples

In porous media, there are fractals [35]. In fractal media, there are lots of mathematical physical models by the local differential fractional equation, for example, [36, 37]. In this section, we will present the solution of several equations or system of equations to assess the validity of variational iteration transform method.

*Case 1. *Consider the following nonlinear local fractional wave-like differential equation with variable coefficients:
where
subject to the initial conditions

From (33) we take the initial value, which reads By using (31), we structure a local fractional iteration procedure as follows: where

Let

According to (14), we can derive Proceeding in this manner, we get

Thus, the exact solution of (31) is given in the form:

*Case 2. *Consider the following local fractional equation:
subject to the initial conditions

According to (41), we structure the following iteration formula: By using (25) and (42), we take an initial value as According to (23) and (43), we can get and moreover Thus we have Continuing in this manner, we obtain Thus, the expression of the exact solution of (41) is given by

According to the ideas of local fractional variational iteration transform method, we can also deal with fractional differential equation system with a similar method.

*Case 3. *Consider the system of local fractional coupled Helmholtz equations

with the initial conditions

Now we can structure the similar local fractional iteration procedure.

Using (51), we take an initial value as

According to variational iteration method [38], we can similarly construct local fractional variational iteration formula

Applying local fractional differential transform and integral transformation on (53), we can easily get Analyzing (52) and (54), we can deduce According to (54) and (55), we have From (52), we can deduce Then we get By virtue of (58), we have Then we derive the following result: Obviously Thus, the expression of the final solution of (50) reads as follows:

*Case 4. *Consider the system of local fractional coupled Burger’s equations
with the initial conditions

Now we can structure the similar local fractional iteration procedure.

Using (64), we take an initial value as

According to variational iteration method [38], we can similarly construct local fractional variational iteration formula

Applying local fractional differential transform and integral transformation on (66), we can easily get Analyzing (64) and (67), we can deduce According to (67) and (68), we have From (64), we can deduce Then we get By virtue of (71), we have Then we derive the following result: Obviously Thus, the expression of the final solution of (63) reads as follows:

#### 5. Conclusions

In this work, we proposed the local fractional iteration transform method. The applications of the methods for solving fractional differential equations with local fractional derivative are discussed in several cases. The method provides the variational iteration formula in the form of polynomial or Maclaurin’s series, which is more easily to deal with convergence and approximate problem. Of course, the new problems are beyond the scope of the present work.

#### Conflict of Interests

The authors declare that they have no conflict of interests regarding this paper.

#### Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 609040410), the Foundation and Advanced Technology Research Program of Henan province (112300410300), and the Nanyang Normal University (NYNU2005k37). The authors would like to thank the editor and the reviewers for their constructive comments and suggestions to improve the quality of the paper.

#### References

- R. D. Driver,
*Ordinary and Delay Differential Equations*, Springer, Berlin, Germany, 1977. View at MathSciNet - I. Podlubny,
*Fractional Differential Equations*, Academic Press, New York, NY, USA, 1999. View at MathSciNet - 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 - A. M. Wazwaz,
*Differential Equations: Methods and Applications*, Balkema, Rotterdam, The Netherlands, 2002. - I. Podlubny, “Geometric and physical interpretation of fractional integration and fractional differentiation,”
*Fractional Calculus & Applied Analysis*, vol. 5, no. 4, pp. 367–386, 2002. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - N. Heymans and I. Podlubny, “Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives,”
*Rheologica Acta*, vol. 45, no. 5, pp. 765–771, 2006. View at Publisher · View at Google Scholar · View at Scopus - D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo,
*Fractional Calculus: Models and Numerical Methods*, Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing, Boston, Mass, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - H. Karami, A. Babakhani, and D. Baleanu, “Existence results for a class of fractional differential equations with periodic boundary value conditions and with delay,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 176180, 8 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet - A. H. Bhrawy and A. S. Alofi, “The operational matrix of fractional integration for shifted Chebyshev polynomials,”
*Applied Mathematics Letters*, vol. 26, no. 1, pp. 25–31, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - D. Baleanu, A. H. Bhrawy, and T. M. Taha, “Two efficient generalized Laguerre spectral algorithms for fractional initial value problems,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 546502, 10 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - Y.-J. Yang, D. Baleanu, and X.-J. Yang, “A local fractional variational iteration method for laplace equation within local fractional operators,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 202650, 6 pages, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Y.-J. Yang, D. Baleanu, and X.-J. Yang, “Analysis of fractal wave equations by local fractional Fourier series method,”
*Advances in Mathematical Physics*, vol. 2013, Article ID 632309, 6 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - A. H. Bhrawy and M. A. Alghamdi, “A shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractional Langevin equation involving two fractional orders in different intervals,”
*Boundary Value Problems*, vol. 2012, article 62, 2012. View at Publisher · View at Google Scholar · View at Scopus - A. H. Bhrawy, M. M. Alghamdi, and T. M. Taha, “A new modified generalized Laguerre operational matrix of fractional integration for solving fractional differential equations on the half line,”
*Advances in Difference Equations*, vol. 2012, article 179, 2012. View at Publisher · View at Google Scholar · View at Scopus - A.-M. Yang, X.-J. Yang, and Z.-B. Li, “Local fractional series expansion method for solving wave and diffusion equations on Cantor sets,”
*Abstract and Applied Analysis*, vol. 2013, Article ID 351057, 5 pages, 2013. View at Publisher · View at Google Scholar · View at Scopus - A. H. Bhrawy and M. M. Al-Shomrani, “A shifted Legendre spectral method for fractional-order multi-point boundary value problems,”
*Advances in Difference Equations*, vol. 2012, article 8, 2012. View at Publisher · View at Google Scholar · View at Scopus - M. Yaseen and M. Samraiz, “The modified new iterative method for solving linear and nonlinear Klein-Gordon equations,”
*Applied Mathematical Sciences*, vol. 6, no. 57–60, pp. 2979–2987, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - X. J. Yang,
*Fractional Functional Analysis and Its Applications*, Asian Academic, Hong Kong, 2011. - X. J. Yang,
*Local Fractional Calculus and Its Applications*, World Science Publisher, New York, NY, USA, 2012. - X.-J. Yang and D. Baleanu, “Fractal heat conduction problem solved by local fractional variation iteration method,”
*Thermal Science*, vol. 17, no. 2, pp. 625–628, 2013. View at Publisher · View at Google Scholar · View at Scopus - C.-F. Liu, S.-S. Kong, and S.-J. Yuan, “Reconstructive schemes for variational iteration method within Yang-Laplace transform with application to fractal heat conduction problem,”
*Thermal Science*, vol. 17, no. 3, pp. 715–721, 2013. View at Publisher · View at Google Scholar - Y.-J. Hao, H. M. Srivastava, H. Jafari, and X.-J. Yang, “Helmholtz and diffusion equations associated with local fractional derivative operators involving the Cantorian and Cantor-type cylindrical coordinates,”
*Advances in Mathematical Physics*, vol. 2013, Article ID 754248, 5 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S.-Q. Wang, Y.-J. Yang, and H. K. Jassim, “Local fractional function decomposition method for solving inhomogeneous wave equations with local fractional derivative,”
*Abstract and Applied Analysis*, vol. 2014, Article ID 176395, 7 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet - M.-S. Hu, R. P. Agarwal, and X.-J. Yang, “Local fractional Fourier series with application to wave equation in fractal vibrating string,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 567401, 15 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - Y. Zhang, A. Yang, and X.-J. Yang, “1-D heat conduction in a fractal medium: a solution by the local fractional Fourier series method,”
*Thermal Science*, vol. 17, no. 3, pp. 953–956, 2013. View at Publisher · View at Google Scholar - J.-H. He, S. K. Elagan, and Z. B. Li, “Geometrical explanation of the fractional complex transform and derivative chain rule for fractional calculus,”
*Physics Letters A: General, Atomic and Solid State Physics*, vol. 376, no. 4, pp. 257–259, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - J.-H. He, “A new fractal derivation,”
*Thermal Science*, vol. 15, supplement 1, pp. S145–S147, 2011. View at Publisher · View at Google Scholar · View at Scopus - A. S. V. R. Kanth and K. Aruna, “Solution of singular two-point boundary value problems using differential transformation method,”
*Physics Letters A: General, Atomic and Solid State Physics*, vol. 372, no. 26, pp. 4671–4673, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - F. Ayaz, “Solutions of the system of differential equations by differential transform method,”
*Applied Mathematics and Computation*, vol. 147, no. 2, pp. 547–567, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - S.-H. Chang and I.-L. Chang, “A new algorithm for calculating one-dimensional differential transform of nonlinear functions,”
*Applied Mathematics and Computation*, vol. 195, no. 2, pp. 799–808, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - M. El-Shahed, “Application of differential transform method to non-linear oscillatory systems,”
*Communications in Nonlinear Science and Numerical Simulation*, vol. 13, no. 8, pp. 1714–1720, 2008. View at Publisher · View at Google Scholar · View at Scopus - S. Momani and V. S. Ertürk, “Solutions of non-linear oscillators by the modified differential transform method,”
*Computers & Mathematics with Applications*, vol. 55, no. 4, pp. 833–842, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - S. Momani, “Analytical approximate solutions of nonlinear oscillators by the modified decomposition method,”
*International Journal of Modern Physics C*, vol. 15, no. 7, pp. 967–979, 2004. View at Publisher · View at Google Scholar · View at Scopus - Z. M. Odibat, “Differential transform method for solving Volterra integral equation with separable kernels,”
*Mathematical and Computer Modelling*, vol. 48, no. 7-8, pp. 1144–1149, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus - B. Yu and J. Li, “Some fractal characters of porous media,”
*Fractals*, vol. 9, no. 3, pp. 365–372, 2001. View at Publisher · View at Google Scholar · View at Scopus - V. Z. Povstenko, “Fractional heat conduction equation and associated thermal stress,”
*Journal of Thermal Stresses*, vol. 28, no. 1, pp. 83–102, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus - J. Hristov, “Approximate solutions to fractional subdiffusion equations,”
*European Physical Journal: Special Topics*, vol. 193, no. 1, pp. 229–243, 2011. View at Publisher · View at Google Scholar · View at Scopus - J.-H. He, “Variational iteration method for autonomous ordinary differential systems,”
*Applied Mathematics and Computation*, vol. 114, no. 2-3, pp. 115–123, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus