Journal of Applied Mathematics

Volume 2013 (2013), Article ID 428079, 9 pages

http://dx.doi.org/10.1155/2013/428079

## A General Iteration Formula of VIM for Fractional Heat- and Wave-Like Equations

Department of Software, College of Computer, National University of Defense Technology, Changsha, Hunan 410073, China

Received 24 December 2012; Accepted 2 March 2013

Academic Editor: Zhongxiao Jia

Copyright © 2013 Fukang Yin 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 general iteration formula of variational iteration method (VIM) for fractional heat- and wave-like equations with variable coefficients is derived. Compared with previous work, the Lagrange multiplier of the method is identified in a more accurate way by employing Laplace’s transform of fractional order. The fractional derivative is considered in Jumarie’s sense. The results are more accurate than those obtained by classical VIM and the same as ADM. It is shown that the proposed iteration formula is efficient and simple.

#### 1. Introduction

Fractional differential equations (FDEs) have been proved to be a valuable tool in the modelling of many phenomena in the fields of applied sciences. This is because of the fact that fractional derivatives provide an excellent instrument to describe the memory and hereditary properties of various materials and processes. With the increasing applications of FDEs, considerable attentions have been paid to provide efficient methods for finding the exact and numerical solutions of FDEs. Recently, some approximate methods such as Adomian’s decomposition method [1–5], homotopy perturbation method [6–10], variational iteration method [11–27], and homotopy analysis method [28, 29] are given to find an analytical approximation to FDEs.

The variational iteration method (VIM) was first proposed by He et al. [11–15] and has been shown to be efficient for handling nonlinear problems. Thus, there has been a great deal of interest in FDEs by using the VIM [17–27]. For VIM, the key points are the construction of correct function and the identification of the Lagrange multiplier. However, the previous work either avoids the term of fractional derivative and handles them as a restricted variation or identifies the Lagrange multipliers by an approximate method, resulting in a poor convergence and inaccuracy. So, it is urgent to find a new method which can identify the Lagrange multiplier in a more accurate way.

In this paper, we will use a correct function described in [27] and give a new method to evaluate the Lagrange multipliers by employing Laplace’s transform of fractional order. Then, we develop a new framework of VIM for the three-dimensional fractional heat- and wave-like equations of the form [3]: subject to the boundary conditions and the initial conditions where is a parameter describing the fractional derivative. In the case of , (1) reduces to the fractional heat-like equation with variable coefficients and to the fractional wave-like equation that models anomalous diffusive and subdiffusive systems in the case of . The approximate solutions of (1) have been studied by using the ADM [3], VIM [17], FVIM [23], and modified VIM [24].

The remainder of the paper is organized as follows. In Section 2, we describe some necessary preliminaries of the fractional calculus and the Laplace transform for our subsequent development. Section 3 is devoted to the derivation of general iteration formula for the fractional heat- and wave-like equations. In Section 4, four examples are given to demonstrate our conclusions. Finally, a brief summary is presented.

#### 2. Preliminaries

In this section, we cover some preliminaries. First, we list some basic definitions about fractional calculus. Second, the Laplace transforms of fractional integral and derivative are described. For more details, see [30–39].

##### 2.1. Fractional Calculus

*Definition 1. *A real function , , is said to be in the space , , if there exist a real number , such that , where , and it is said to be in the space if and only if , .

*Definition 2. *The Riemann-Liouville fractional integral operator () of order , of a function , , is defined as
when , .

If we denote the Riemann-Liouville fractional derivative by , then the next equation define the Riemann-Liouville fractional derivative of order where , .

Jumarie (see [31–37]) proposed a simple alternative definition to the Riemann-Liouville derivative. Since his modified Riemann-Liouville derivative is defined for arbitrary continuous (nondifferentiable) functions and the fractional derivative of a constant is equal to zero, it has the advantages of both the standard Riemann-Liouville and the Caputo fractional derivatives. Now, we present some notations, definitions, and preliminary facts of the modified Riemann-Liouville derivative which will be used later in this work.

*Definition 3. *Jumarie’s fractional derivative is a modified Riemann-Liouville derivative defined by the expression [37, 38]
For positive , one will set

In addition, we want to give some properties of Jumarie’s fractional derivative.

Theorem 4 (the fractional Leibniz product rule [36]). *If and are two continuous functions on , then
*

Theorem 5 (the fractional Barrow’s formula [37]). *For a continuous function , one has
**
where .*

From Theorems 4 and 5, the formula of integration by parts is given as

*Definition 6. *Fractional derivative of compounded functions [37, 38] is defined as

*Definition 7. *The integral with respect to [37, 38] is defined as the solution of the fractional differential equation

Lemma 8. *Let denote a continuous function [37, 38], then the solution , of (8) is defined by the equality
*

Lemma 9. *Let , , , , , then
*

##### 2.2. The Laplace Transform

*Definition 10. *The Laplace transform , , of a function is defined by the integral

Lemma 11. *The Laplace transform of the fractional derivative (the Riemann-Liouville derivative) is
*

*Definition 12. *The inverse Laplace transform is defined by the complex integral
where the integration is done along the vertically in the complex plane such that is greater than the real part of all singularities of . This ensures that the contour path is in the region of convergence. If all singularities are in the left half-plane, or is a smooth function on (i.e., no singularities), then can be set to zero and the above inverse integral formula becomes identical to the inverse Fourier transform.

Now, we will introduce the definition of the fractional Laplace transform derived by Jumarie [38] for the first time, and some results of the fractional Laplace transform are also presented.

*Definition 13. *Let denote a function which vanishes for negative values of . Its Laplace's transform of order (or its th fractional Laplace's transform) is defined by the following expression, when it is finite:
where and is the Mittag-Leffler function .

Lemma 14. *If one defines the convolution of order of the two functions by the expression
**
Then, one has the equality
*

Corollary 15. *Given the Laplace transform that one has the inversion formula
**
where is the period of the complex-valued Mittag-Leffler function defined by the equality .*

Lemma 16. *The fractional Laplace transform of the fractional derivative (the modified Riemann-Liouville derivative) is
** Some properties of the fractional Laplace transform are given as follows [38]:
*

#### 3. Fractional Variational Iteration Method

In order to illustrate the solution procedure of the variational iteration method, we consider the following fractional differential equation: where is the linear operator, is the nonlinear operator, and is the modified Rieman-Liouville derivative of order .

Subject to the initial condition, Let , .

Before our solution, we will describe the fractional variational iteration method described in [23, 25, 26], which construct a correct function for (24) as

Then, by using Lemma 8 proposed by Jumarie [37, 38] for the first time, one gets a correction function But, unfortunately, (27) holds true only when . In the case of , , some modifications must be made. For example, replacing the fractional order () by the order ), whether there is a general iteration formula for (24). Certainly, there is. In the remainder section, we will use two methods to derive a general iteration formula of VIM for (24). Unlike the previous work, which calculates the Lagrange multiplier by some approximate methods, we will use a more accurate way by employing the properties of Laplace’s transform.

##### 3.1. The Laplace Transform Method

According to VIM [27], we can construct a correction functional as follows: where is the Riemann-Liouville fractional integral operator of order , that is, , with respect to the variable , and is a general Lagrange multiplier, and is a restricted variation, that is, .

By taking the Laplace transform on the both sides of (28), we have where is the operator of the Laplace transform.

From the Lemma 11, we get

Taking the variation derivative on the both sides of (31), we can derive

Without loss of generality, assuming that , then we have

Set the coefficient of to zeros, we obtain

By employing the inverse Laplace transform, we have

Substituting (35) into (28), we get the iteration formula as follows:

##### 3.2. The Fractional Laplace Transform Method

In order to illustrate our method, we replace the fractional order by the order . According to VIM [27], we can construct a correction function as follows:

Taking the fractional Laplace transform on the both sides of (37), we have where is the fractional Laplace transform of order .

By assuming that has the form as and using (10), then we get

From (20), we have

Substituting (40) into (38) and then taking the variation derivative on the both sides of (35), we can derive

Without loss of generality, assuming that , then we have

Set the coefficient of to zeros, we have

By employing the inverse fractional Laplace transform, we have

Substituting (40) into (26), we get the iteration formula as follows:

After replacing the fractional order by the order , we could find that (36) and (45) are the same. So, we could get a general iteration formula for (1) as follows:

By using the Lemma 11, we finally have

#### 4. Applications and Results

*Example 17. *Consider the following fractional model for heat conduction in polar bear hairs proposed by Qing-Li et al. [40]:
where is the modified Rieman-Liouville derivative and is a constant, with the initial condition
where is the body temperature, is the environment temperature, and is a constant. When , the exact solution of (46) is

According to the general iteration formula of VIM, one can get the iteration formulation as follows:

For the convenience, let and , then

Starting with the initial value as shown in (52), we can derive
and then
So, the solution is
which is the exact solution.

*Example 18. *Consider the one-dimensional fractional heat-like equation:
subject to the initial condition
when and the exact solution is .

According to the general iteration formula of VIM, one can get the following formula of iteration:

Starting with an initial approximation , one can obtain

The solution in a series form is given by where denotes the two-parameter Mittag-Leffler function. The result obtained in (56) is exactly the same result, obtained by Momani [3] and Faraz et al. [23].

*Example 19. *Consider the one-dimensional fractional heat-like equation:
subject to the boundary conditions
and the initial condition
The exact solution was found to be [41]

According to the general iteration formula of VIM, one can get the following formula of iteration:

Starting with an initial approximation , one can obtain the following successive approximate:

The solution in a series form is given by
where denotes the two-parameter Mittag-Leffler function.

*Example 20. *Consider the one-dimensional fractional heat-like equation:
subject to the boundary conditions
and the initial condition
The exact solution was found to be [41]

According to the general iteration formula of VIM, one can get the following formula of iteration:

Starting with an initial approximation,
which was given by [41]. One can obtain the following successive approximate:
when and the exact solution for (68) is
From above procedure of solution, one can conclude that the result obtained by the fractional variational iteration method (FVIM) is the same as the decomposition method [3].

#### 5. Conclusion

VIM has been known as a powerful tool for solving many fractional differential equations. In this paper, we derive a general iteration formula of VIM for fractional heat- and wave-like equations with variable coefficients. There are two points to make here. First, the Lagrange multiplier of the method is identified in a more accurate way by employing the Laplace transform. Second, our iteration formula still holds true in the case of . All the examples show that the results of the proposed method are more accurate than those obtained by the classical VIM and the same as the ADM but not require the calculation of Adomian’s polynomials.

#### Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant no. 41105063). The authors are very grateful to reviewers for carefully reading the paper and for their comments and suggestions which have improved the paper.

#### References

- Z. Odibat and S. Momani, “Numerical methods for nonlinear partial differential equations of fractional order,”
*Applied Mathematical Modelling*, vol. 32, no. 1, pp. 28–39, 2008. View at Publisher · View at Google Scholar · View at Scopus - S. Momani and Z. Odibat, “Analytical approach to linear fractional partial differential equations arising in fluid mechanics,”
*Physics Letters A*, vol. 355, no. 4-5, pp. 271–279, 2006. View at Publisher · View at Google Scholar · View at Scopus - S. Momani, “Analytical approximate solution for fractional heat-like and wave-like equations with variable coefficients using the decomposition method,”
*Applied Mathematics and Computation*, vol. 165, no. 2, pp. 459–472, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Momani and Z. Odibat, “Analytical solution of a time-fractional Navier-Stokes equation by Adomian decomposition method,”
*Applied Mathematics and Computation*, vol. 177, no. 2, pp. 488–494, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G.-C. Wu and J.-H. He, “Fractional adomian decomposition method,” http://arxiv.org/abs/1006.5264.
- S. Abbasbandy, “Numerical solutions of the integral equations: homotopy perturbation method and Adomian's decomposition method,”
*Applied Mathematics and Computation*, vol. 173, no. 1, pp. 493–500, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I.-C. Liu and A. M. Megahed, “Homotopy perturbation method for thin film flow and heat transfer over an unsteady stretching sheet with internal heating and variable heat flux,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 418527, 12 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. H. Hosseinnia, A. Ranjbar, and S. Momani, “Using an enhanced homotopy perturbation method in fractional differential equations via deforming the linear part,”
*Computers & Mathematics with Applications*, vol. 56, no. 12, pp. 3138–3149, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Abbasbandy, “Homotopy perturbation method for quadratic Riccati differential equation and comparison with Adomian's decomposition method,”
*Applied Mathematics and Computation*, vol. 172, no. 1, pp. 485–490, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Abbasbandy, “Modified homotopy perturbation method for nonlinear equations and comparison with Adomian decomposition method,”
*Applied Mathematics and Computation*, vol. 172, no. 1, pp. 431–438, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - 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 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 - 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 - J. H. He, “Asymptotic methods for solitary solutions and compactons,”
*Abstract and Applied Analysis*, vol. 2012, Article ID 916793, 130 pages, 2012. View at Publisher · View at Google Scholar - J. H. He, “Some applications of nonlinear fractional differential equations and their approximations,”
*Bulletin of Science and Technology*, vol. 15, no. 2, pp. 86–90, 1999. View at Google Scholar - T. A. Nofal, “Approximate solutions for nonlinear initial value problems using the modified variational iteration method,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 370843, 19 pages, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Y. M. R, 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 - Z. M. Odibat and S. Momani, “Application of variational iteration method to nonlinear differential equations of fractional order,”
*International Journal of Nonlinear Sciences and Numerical Simulation*, vol. 7, no. 1, pp. 27–34, 2006. View at Google Scholar · View at Scopus - 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, Article ID 082902, 9 pages, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. D. Ganji and S. H. Hashemi Kachapi, “Analysis of nonlinear equations in fluids,”
*Progress in Nonlinear Science*, vol. 2, pp. 1–293, 2011. View at Google Scholar - D. D. Ganji and S. H. Hashemi Kachapi, “Analytical and numerical methods in engineering and applied sciences,”
*Progress in Nonlinear Science*, vol. 3, pp. 1–579, 2011. View at Google Scholar - G.-C. Wu, “Variational iteration method for $q$-difference equations of second order,”
*Journal of Applied Mathematics*, vol. 2012, Article ID 102850, 5 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet - N. Faraz, Y. Khan, H. Jafari, A. Yildirim, and M. Madani, “Fractional variational iteration method via modified Riemann-Liouville derivative,”
*Journal of King Saud University-Science*, vol. 23, no. 4, pp. 413–417, 2011. View at Google Scholar - E. J. Ali, “Modified treatment of initial boundary value problems for one dimensional heat-like and wave-like equations using variational iteration method,”
*Applied Mathematical Sciences*, vol. 6, no. 33-36, pp. 1613–1626, 2012. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G.-C. Wu and E. W. M. Lee, “Fractional variational iteration method and its application,”
*Physics Letters A*, vol. 374, no. 25, pp. 2506–2509, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G.-C. Wu, “A fractional variational iteration method for solving fractional nonlinear differential equations,”
*Computers & Mathematics with Applications*, vol. 61, no. 8, pp. 2186–2190, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Z. Odibat and S. Momani, “The variational iteration method: an efficient scheme for handling fractional partial differential equations in fluid mechanics,”
*Computers & Mathematics with Applications*, vol. 58, no. 11-12, pp. 2199–2208, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Liao, “On the homotopy analysis method for nonlinear problems,”
*Applied Mathematics and Computation*, vol. 147, no. 2, pp. 499–513, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Abbasbandy and A. Shirzadi, “Homotopy analysis method for multiple solutions of the fractional Sturm-Liouville problems,”
*Numerical Algorithms*, vol. 54, no. 4, pp. 521–532, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - I. Podlubny,
*Fractional Differential Equations: An Introduction To Fractional Derivatives, Fractional Differential Equations, To Methods of Their Solution and Some of Their Applications*, vol. 198 of*Mathematics in Science and Engineering*, Academic Press, San Diego, Calif, USA, 1998. - G. Jumarie, “Stochastic differential equations with fractional Brownian motion input,”
*International Journal of Systems Science*, vol. 24, no. 6, pp. 1113–1131, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Jumarie, “On the representation of fractional Brownian motion as an integral with respect to ${(\text{d}t)}^{a}$,”
*Applied Mathematics Letters*, vol. 18, no. 7, pp. 739–748, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Jumarie, “On the solution of the stochastic differential equation of exponential growth driven by fractional Brownian motion,”
*Applied Mathematics Letters*, vol. 18, no. 7, pp. 817–826, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Jumarie, “Modified Riemann-Liouville derivative and fractional Taylor series of nondifferentiable functions further results,”
*Computers & Mathematics with Applications*, vol. 51, no. 9-10, pp. 1367–1376, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Jumarie, “New stochastic fractional models for Malthusian growth, the Poissonian birth process and optimal management of populations,”
*Mathematical and Computer Modelling*, vol. 44, no. 3-4, pp. 231–254, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Jumarie, “Stock exchange fractional dynamics defined as fractional exponential growth driven by (usual) Gaussian white noise. Application to fractional Black-Scholes equations,”
*Insurance Mathematics & Economics*, vol. 42, no. 1, pp. 271–287, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Jumarie, “Table of some basic fractional calculus formulae derived from a modified Riemann-Liouville derivative for non-differentiable functions,”
*Applied Mathematics Letters*, vol. 22, no. 3, pp. 378–385, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - G. Jumarie, “Laplace's transform of fractional order via the Mittag-Leffler function and modified Riemann-Liouville derivative,”
*Applied Mathematics Letters*, vol. 22, no. 11, pp. 1659–1664, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Kexue and P. Jigen, “Laplace transform and fractional differential equations,”
*Applied Mathematics Letters*, vol. 24, no. 12, pp. 2019–2023, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - W. Qing-Li, J. H. He, and L. Zheng-Biao, “Fractional model for heat conduction in polar bear hairs,”
*Thermal Science*, vol. 16, no. 2, pp. 339–342, 2012. View at Google Scholar - D.-H. Shou and J.-H. He, “Beyond Adomian method: the variational iteration method for solving heat-like and wave-like equations with variable coefficients,”
*Physics Letters A*, vol. 372, no. 3, pp. 233–237, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet