- Open Access
Variational iteration method for fractional calculus - a universal approach by Laplace transform
© Wu and Baleanu; licensee Springer 2013
- Received: 19 October 2012
- Accepted: 6 January 2013
- Published: 21 January 2013
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.
- variational iteration method
- fractional calculus
- Laplace transform
- symbolic computation
The Lagrange multiplier technique  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  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.
where , R is a linear operator, N is a nonlinear operator and is a given continuous function and is the term of the highest-order derivative.
where , and 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.
This algorithm is well known as the Newton-Raphson method and has quadratic convergence.
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 , respectively.
We now consider the applications of the modified VIM to both ODEs and FDEs.
4.1 Ordinary differential equations
which has the exact solution .
For , tends to the exact solution .
The same solutions using the classical VIM can be found in .
There can be various choices of and 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
where is the Caputo derivative  and 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.
with the exact solution , where denotes the Mittag-Leffler function.
rapidly tends to the exact solution of Eq. (24) for .
The VIM solution of the fractional semi-derivative equation was developed by Das . Other methods applied to this equation are available in  and the monographs [33, 34] in the fractional calculus.
and is used as earlier.
The method’s efficiency for a nonlinear differential equation with variable coefficients is illustrated in . For other applications of a new modified VIM to ODEs and FDEs, readers are also referred to [36–38].
(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.
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.
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).
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