New applications of the variational iteration method - from differential equations to q-fractional difference equations
© Wu and Baleanu; licensee Springer 2013
Received: 11 September 2012
Accepted: 5 January 2013
Published: 24 January 2013
The non-classical calculi such as q-calculus, fractional calculus and q-fractional calculus have been hot topics in both applied and pure sciences. Then some new linear and nonlinear models have appeared. This study mainly concentrates on the analytical aspects, and the variational iteration method is extended in a new way to solve an initial value problem.
MSC: 39A13, 74H10.
Keywordsvariational iteration method fractional calculus time scales q-calculus Laplace transform symbolic computation
Recently, q-fractional calculus has been paid much attention to [1–7], i.e., q-factional modeling, linear q-fractional systems, q-special functions etc. As is well known, both fractional calculus (FC) and q-calculus (QC) are not new as they appeared in 1695 and about 1920s, respectively. Fractional q-calculus (FQC) serves as a bridge between FC and QC. The early developments of q-fractional calculus can be found in [8–10]. Now, various q-fractional initial value problems are proposed in [3, 11–16].
The variational iteration method (VIM) [17–20] has been one of the often used nonlinear methods in initial boundary value problems of differential equations. In this study, the extension of the method into FQC is undertaken and the Caputo q-fractional initial value problems are investigated. Our study is organized as follows. In Section 2, the basic idea of the VIM is illustrated. In Section 3, the VIM is extended to q-difference equations, and the Lagrange multipliers of the method are presented for the equations of high-order q-derivatives. In Section 4, recent development of the method in fractional calculus is introduced. Following Section 4, the application of the VIM in q-fractional calculus is considered. Then the method is applied to the Caputo q-fractional initial value problem.
2 The VIM in ordinary calculus
where , R is a linear operator, N is a nonlinear operator, is a given continuous function and is the term of the highest-order derivative.
where is called the Lagrange multiplier which can be identified optimally by variational calculus and is the n th term approximate solution.
Following the above steps, we can design a Maple-program which contains three parameters: ICs, Eqs and n. ICs reads the value of initial points. Eqs contains information of the linear terms, the nonlinear terms and the interval functions. n means the approximate solution’s truncated order.
Comparisons between , and the exact solution
3 The VIM in q-calculus
Definition 3.1 (q-calculus)
Lemma 3.4 ()
one of the Lagrange multipliers is .
Lemma 3.5 ()
where denotes the q-factorial and for the integer k.
More generally, one can derive the following Theorem 3.6.
where for the integer m.
Here the initial iteration value can be determined via the q-Taylor series .
Recall that the limit is an exact solution of (22). Here is one of the q-exponential functions.
4 The VIM in fractional calculus
Let be a real-valued function defined on a closed interval .
One can check the formula (26) results in a poor convergence even for a linear FDE. Such difficulty can be overcome by the Laplace transform [31–33]. The following iteration formula is initially proposed in [34, 35]. Let us revisit the proof.
where the function is a Lagrange multiplier for any order α.
Setting the Lagrange multiplier , Eq. (30) can be considered as a convolution of the function and the term .
This completes the proof. □
For , tends to which is an exact solution of (41).
Remarks Our simplest iteration formula (34) can reduce to the Volterra integral equation. See the analysis of the convergence and existence in  and the references therein. However, regarding Eq. (36), the VIM transforms it into a more general Volterra integral equation from which one can obtain approximate solutions of higher accuracies.
FDEs have been proven to be a useful tool to describe the nonlocal behaviors or long range interactions of dynamical systems. The previous applications of the VIM just ‘guessed’ the Lagrange multipliers or directly used the one in ordinary differential equations. In this study, various Lagrange multipliers are identified more explicitly and the variational approach for FDEs is systematically developed now.
5 The Caputo q-fractional initial value problem
where Z is the set of integers.
where and .
Now, we introduce the q-Laplace transform and some properties.
where , .
Lemma 5.4 ()
Lemma 5.6 ()
The existence and uniqueness of the solutions of the Caputo q-initial value problems have been discussed in .
One of the Lagrange multipliers can be identified as .
From Lemma 5.6, we set , where . Then the Lagrange multiplier is ‘good’ enough so that the product of and is similar as the function in (51b) and becomes a convolution (51b).
where is the Laplace transform of some function.
Readers are referred to the recent development in the application of the VIM for solving fuzzy equations [41–43] and the calculus of variations on time scales [44–48]. Since this study only concentrates on the applications of the VIM, other numerical methods in FC can be found in [49–51].
We aim at some new applications of the VIM from differential equations to q-fractional difference equations, and the following main contributions of this study are obtained:
(a) Designing a maple program of the VIM for differential equations. Now, there is no need for one to obtain approximate solutions of high order by hand. The efficiency and accuracy are improved;
(b) Correcting the popularly used variational iteration formulae in FC and explicitly identifying some new Lagrange multipliers from the Laplace transform. The FDEs are transformed into generalized Volterra integral equations;
(c) Applying the VIM in q-difference equations and identifying a Lagrange multiplier of q-difference equations of m th order;
(d) Extending the VIM to FQC and investigating the initial value problems analytically. The obtained variational iteration formula in FQC can reduce to those in FC and QC.
Due to the rapid development of advanced applied sciences, non-classical tools of calculus, i.e., fractional calculus, q-calculus, etc., have been becoming more active and have been found useful in describing important physical phenomena. This study discusses some new applications of the VIM and provides a potential tool to analytically investigate such models. There is still some other work needed to consider, i.e., maple-packages or the symbolic computation of the VIM in FC even in FQC, other numerical methods based on the VIM, etc. The authors believe, in not far future, the VIM can play the same crucial role as that in ordinary 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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