- Research
- Open Access

# Variational iteration method for fractional calculus - a universal approach by Laplace transform

- Guo-Cheng Wu
^{1, 2}Email author and - Dumitru Baleanu
^{3, 4, 5}Email author

**2013**:18

https://doi.org/10.1186/1687-1847-2013-18

© Wu and Baleanu; licensee Springer 2013

**Received:**19 October 2012**Accepted:**6 January 2013**Published:**21 January 2013

## Abstract

A novel modification of the variational iteration method (VIM) is proposed by means of the Laplace transform. Then the method is successfully extended to fractional differential equations. Several linear fractional differential equations are analytically solved as examples and the methodology is demonstrated.

**MSC:** 39A08, 65K10, 34A12.

## Keywords

- variational iteration method
- fractional calculus
- Laplace transform
- symbolic computation

## 1 Introduction

The Lagrange multiplier technique [1] was widely used to solve a number of nonlinear problems which arise in mathematical physics and other related areas, and it was developed into a powerful analytical method, *i.e.*, the variational iteration method [2, 3] for solving differential equations. The method has been applied to initial boundary value problems [4–9], fractal initial value problems [10, 11], *q*-difference equations [12] and fuzzy equations [13–15], *etc.*

Generally, in applications of VIM to initial value problems of differential equations, one usually follows the following three steps: (a) establishing the correction functional; (b) identifying the Lagrange multipliers; (c) determining the initial iteration. The step (b) is very crucial. Applications of the method to fractional differential equations (FDEs) mainly and directly used the Lagrange multipliers in ordinary differential equations (ODEs) which resulted in poor convergences. This point of view needs some explanations will elucidate the target of the suggested improvement, among them:

(1) When the Riemann-Liouville (RL) integral emerges in the constructed correctional functional, the integration by parts is difficult to apply;

(2) To avoid this problem, the RL integral is replaced by an integer one which allows the integration by parts. This is a very strong simplification but it affects the next steps of the application of the method;

(3) Therefore, the Lagrange multiplier is determined by a simplification not reasonably explained in the literature, so far.

To overcome this drawback, the present article conceives a method how the Lagrange multiplier has to be defined from Laplace transform. The technique can be readily and universally extended to solve both differential equations and FDEs with initial value conditions.

## 2 Basics of the variation iteration method

where $u=u(t)$, *R* is a linear operator, *N* is a nonlinear operator and $g(t)$ is a given continuous function and ${d}^{m}u/d{t}^{m}$ is the term of the highest-order derivative.

where $\lambda (t,\tau )$ is called the general Lagrange multiplier [1–3] and ${u}_{n}$ is the *n* th order approximate solution.

where $v=v(t)$, ${}_{0}{}^{C}D_{t}^{\alpha}$ and ${}_{0}I_{t}^{\alpha}$ are the notations of the Caputo derivative and the RL integration, respectively. That’s why the VIM was not so successful as other analytical methods such as the Adomian decomposition method (ADM) [17–19] and the homotopy perturbation method (HPM) [20–22] in fractional calculus. For this reason, we consider the following reconstruction of the method using the Laplace transform.

## 3 New identification of the Lagrange multipliers

*δ*is the classical variational operator. From (4) and (5), for a given initial value ${x}_{0}$, we can find the approximate solution ${x}_{n+1}$ by the iterative scheme for (5)

This algorithm is well known as the Newton-Raphson method and has quadratic convergence.

where $U(s)=L[u(t)]={\int}_{0}^{\mathrm{\infty}}{e}^{-st}u(t)\phantom{\rule{0.2em}{0ex}}dt$.

Eq. (11) also explained why the initial iteration in the classical VIM is determined by the Taylor series.

This modified VIM here transfers the problem into the partial differential equation in the Laplace *s*-domain and removes the differentiation with respect to time. This idea has been used in other analytical methods such as the Laplace ADM [23, 24] and the Laplace HPM [25], respectively.

## 4 Illustrative examples

We now consider the applications of the modified VIM to both ODEs and FDEs.

### 4.1 Ordinary differential equations

**Example 1**

which has the exact solution $u(t)={u}_{0}{e}^{-t}$.

For $n\to \mathrm{\infty}$, ${u}_{n}(t)$ tends to the exact solution ${u}_{0}{e}^{-t}$.

**Example 2**The logistic differential equation [26]

The same solutions using the classical VIM can be found in [26].

*u*when determining the Lagrange multiplier, we can derive a Lagrange multiplier explicitly

There can be various choices of ${u}_{0}(t)$ and $\lambda (s)$ which affect the speed of the convergence. We note that the integration by parts is not used and the calculation of the Lagrange multiplier here is much simpler. Furthermore, the VIM can be easily extended to FDEs and this is the main purpose of our work.

### 4.2 Fractional differential equations

*i.e.*,

where ${}_{0}{}^{C}D_{t}^{\alpha}$ is the Caputo derivative [27] and $g(\tau ,{u}_{n})$ is a nonlinear term.

Now, we consider the application of the modified VIM.

Let us apply the above VIM to solve FDEs of Caputo type.

**Example 3**

with the exact solution ${E}_{\alpha}({(-\omega t)}^{\alpha})$ [30], where ${E}_{\alpha}({(-\omega t)}^{\alpha})$ denotes the Mittag-Leffler function.

${u}_{n}(t)$ rapidly tends to the exact solution of Eq. (24) for $n\to \mathrm{\infty}$.

**Example 4**

The VIM solution of the fractional semi-derivative equation was developed by Das [31]. Other methods applied to this equation are available in [32] and the monographs [33, 34] in the fractional calculus.

and $\lambda (s)=-\frac{1}{{s}^{\alpha}}$ is used as earlier.

where ${k}^{i}={x}^{2}+(1+{x}^{2})({3}^{i}-1)$.

The method’s efficiency for a nonlinear differential equation with variable coefficients is illustrated in [35]. For other applications of a new modified VIM to ODEs and FDEs, readers are also referred to [36–38].

**Remarks**

(a) The conceived modification of the VIM is a universal approach to both ODEs and FDEs. As a result, it becomes possible to design a ‘universal’ software package in future work.

(b) Now one can consider implementing other linearized techniques, *i.e.*, the Adomian series and the homotopy series to handle the nonlinear terms and improve the accuracy of the approximate solutions.

(c) This modified VIM can also be used to solve the FDEs of RL type.

## 5 Conclusions

A new approach is proposed to identify the Lagrange multipliers of the VIM and a concept of the Laplace-Lagrange multipliers is proposed from the Laplace transform. Especially for the FDEs, to the best of our knowledge, there is no effective method to identify the Lagrange multipliers. With the approach given in this paper, we can easily derive Lagrange multipliers without tedious calculation and new variational iteration formulae can be derived. Some FDEs with the Caputo derivatives are illustrated. The results show the modified method’s efficiency compared with other versions of the VIM in fractional calculus.

## Declarations

### Acknowledgements

The authors would like to express their deep gratitude to the referees for their valuable suggestions and comments. The work is financially supported by the NSFC (11061028) and the key program of the NSFC (51134018).

## Authors’ Affiliations

## References

- Inokuti M, Sekine H, Mura T:
**General use of the Lagrange multiplier in nonlinear mathematical physics.**In*Variational Methods in the Mechanics of Solids*. Edited by: Nemat-Nasser S. Pregman Press, New York; 1978:156-162.Google Scholar - He JH:
**Approximate analytical solution for seepage flow with fractional derivatives in porous media.***Comput. Methods Appl. Mech. Eng.*1998, 167(1-2):57-68. 10.1016/S0045-7825(98)00108-XView ArticleGoogle Scholar - He JH:
**Variational iteration method - a kind of non-linear analytical technique: some examples.***Int. J. Non-Linear Mech.*1999, 34(4):699-708. 10.1016/S0020-7462(98)00048-1View ArticleGoogle Scholar - He JH:
**An elementary introduction to recently developed asymptotic methods and nanomechanics in textile engineering.***Int. J. Mod. Phys. B*2008, 22(21):3487-3578. 10.1142/S0217979208048668View ArticleGoogle Scholar - Abbasbandy S:
**A new application of He’s variational iteration method for quadratic Riccati differential equation by using Adomian’s polynomials.***J. Comput. Appl. Math.*2007, 207(1):59-63. 10.1016/j.cam.2006.07.012MathSciNetView ArticleGoogle Scholar - Noor MA, Mohyud-Din ST:
**Variational iteration technique for solving higher order boundary value problems.***Appl. Math. Comput.*2007, 189(2):1929-1942. 10.1016/j.amc.2006.12.071MathSciNetView ArticleGoogle Scholar - Noor MA, Mohyud-Din ST:
**Variational iteration method for solving higher-order nonlinear boundary value problems using He’s polynomials.***Int. J. Nonlinear Sci. Numer. Simul.*2008, 9(2):141-156.View ArticleGoogle Scholar - Yusufoglu E:
**The variational iteration method for studying the Klein-Gordon equation.***Appl. Math. Lett.*2008, 21(7):669-674. 10.1016/j.aml.2007.07.023MathSciNetView ArticleGoogle Scholar - Yıldırım A, Öziş T:
**Solutions of singular IVPs of Lane-Emden type by the variational iteration method.***Nonlinear Anal., Theory Methods Appl.*2009, 70(6):2480-2484. 10.1016/j.na.2008.03.012View ArticleGoogle Scholar - Wu GC:
**New trends in variational iteration method.***Commun. Fract. Calc.*2011, 2(2):59-75.Google Scholar - Wu GC, Wu KT:
**Variational approach for fractional diffusion-wave equations on Cantor sets.***Chin. Phys. Lett.*2012., 29(6): Article ID 060505Google Scholar - Wu GC: Variational iteration method for
*q***-difference equations of second order.***J. Appl. Math.*2012., 2012: Article ID 102850Google Scholar - Allahviranloo T, Abbasbandy S, Rouhparvar H:
**The exact solutions of fuzzy wave-like equations with variable coefficients by a variational iteration method.***Appl. Soft Comput.*2011, 11(2):2186-2192. 10.1016/j.asoc.2010.07.018View ArticleGoogle Scholar - Jafari H, Khalique CM:
**Homotopy perturbation and variational iteration methods for solving fuzzy differential equations.***Commun. Fract. Calc.*2012, 3(1):38-48.Google Scholar - Jafari H, Saeidy M, Baleanu D: The variational iteration method for solving
*n***-th order fuzzy differential equations.***Cent. Eur. J. Phys.*2012, 10(1):76-85. 10.2478/s11534-011-0083-7Google Scholar - He JH, Wu XH:
**Variational iteration method: new development and applications.***Comput. Math. Appl.*2007, 54(7-8):881-894. 10.1016/j.camwa.2006.12.083MathSciNetView ArticleGoogle Scholar - Shawagfeh NT:
**Analytical approximate solutions for nonlinear fractional differential equations.***Appl. Math. Comput.*2002, 131(2-3):517-529. 10.1016/S0096-3003(01)00167-9MathSciNetView ArticleGoogle Scholar - Ray SS, Bera RK:
**An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method.***Appl. Math. Comput.*2005, 167(1):561-571. 10.1016/j.amc.2004.07.020MathSciNetView ArticleGoogle Scholar - Duan JS, Rach R, Baleanu D, Wazwaz AM:
**A review of the Adomian decomposition method and its applications to fractional differential equations.***Commun. Fract. Calc.*2012, 3(2):73-99.Google Scholar - Wang Q:
**Homotopy perturbation method for fractional KdV-Burgers equation.***Chaos Solitons Fractals*2008, 35(5):843-850. 10.1016/j.chaos.2006.05.074MathSciNetView ArticleGoogle Scholar - Momani S, Odibat Z:
**Homotopy perturbation method for nonlinear partial differential equations of fractional order.***Phys. Lett. A*2007, 365(5-6):345-350. 10.1016/j.physleta.2007.01.046MathSciNetView ArticleGoogle Scholar - Kadem A, Baleanu D:
**Homotopy perturbation method for the coupled fractional Lotka-Volterra equations.***Rom. J. Phys.*2011, 56(3-4):332-338.MathSciNetGoogle Scholar - Tsai PY, Chen CK:
**An approximate analytic solution of the nonlinear Riccati differential equation.***J. Franklin Inst.*2010, 347(10):1850-1862. 10.1016/j.jfranklin.2010.10.005MathSciNetView ArticleGoogle Scholar - Zeng DQ, Qin YM:
**The Laplace-Adomian-Pade technique for the seepage flows with the Riemann-Liouville derivatives.***Commun. Fract. Calc.*2012, 3(1):26-29.Google Scholar - Javidi M, Raji MA:
**Combination of Laplace transform and homotopy perturbation method to solve the parabolic partial differential equations.***Commun. Fract. Calc.*2012, 3(1):10-19.Google Scholar - Wazwaz AM:
**The variational iteration method for analytic treatment for linear and nonlinear ODEs.***Appl. Math. Comput.*2009, 212(1):120-134. 10.1016/j.amc.2009.02.003MathSciNetView ArticleGoogle Scholar - Oldham KB, Spanier J:
*The Fractional Calculus*. Academic Press, New York; 1974.Google Scholar - Podlubny I:
*Fractional Differential Equations*. Academic Press, San Diego; 1999.Google Scholar - Kilbas AA, Srivastav HM, Trujillo JJ:
*Theory and Applications of Fractional Differential Equations*. Elsevier, New York; 2006.Google Scholar - Mainardi F:
**Fractional relaxation-oscillation and fractional diffusion-wave phenomena.***Chaos Solitons Fractals*1996, 7(9):1461-1477. 10.1016/0960-0779(95)00125-5MathSciNetView ArticleGoogle Scholar - Das S:
**Analytical solution of a fractional diffusion equation by variational iteration method.***Comput. Math. Appl.*2009, 57(3):483-487. 10.1016/j.camwa.2008.09.045MathSciNetView ArticleGoogle Scholar - Hristov J:
**Heat-balance integral to fractional (half-time) heat diffusion sub-model.***Therm. Sci.*2010, 14(2):291-316. 10.2298/TSCI1002291HView ArticleGoogle Scholar - Diethelm K:
*The Analysis of Fractional Differential Equations*. Springer, Berlin; 2010.View ArticleGoogle Scholar - Baleanu D, Diethelm K, Scalas E, Trujillo JJ:
*Fractional Calculus Models and Numerical Methods*. World Scientific, Singapore; 2012.Google Scholar - Wu GC:
**Challenge in the variational iteration method - a new approach to the identification of the Lagrange multipliers.***J. King Saud Univ., Sci.*2012. doi:10.1016/j.jksus.2012.12.002 (in press)Google Scholar - Hristov J:
**An exercise with the He’s variation iteration method to a fractional Bernoulli equation arising in transient conduction with non-linear heat flux at the boundary.***Int. Rev. Chem. Eng.*2012, 4(5):489-497.Google Scholar - Wu GC, Baleanu D:
**Variational iteration method for the Burgers’ flow with fractional derivatives-new Lagrange multipliers.***Appl. Math. Model.*2012. doi:10.1016/j.apm.2012.12.018 (in press)Google Scholar - Wu GC:
**Variational iteration method for solving the time-fractional diffusion equations in porous medium.***Chin. Phys. B*2012., 21(12): Article ID 120504Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.