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New applications of the variational iteration method - from differential equations to q-fractional difference equations
Advances in Difference Equations volume 2013, Article number: 21 (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.
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
We illustrate its basic idea through the following nonlinear system:
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.
Then we construct the following correction function for Eq. (1):
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.
Example 2.1 Consider the following Riccati equations :
We input , , n can be set as and so on. We set and , respectively. Since here is very tedious, we only give the analytical approximate solutions , , and as follows:
The comparisons between , and the exact solution are listed in Table 1.
Example 2.2 The second example is a system representing a nonlinear reaction :
The information of the system (5) reads
The fourth-order approximation can be presented as
We can calculate , , even higher-order approximation. Noting that tends to for , we only compare and with the numerical results from the Runge-Kutta method (RKM) in Figure 1 and Figure 2, respectively. Obviously, and have higher accuracies than and . With symbolic computation, if the computer is excellent enough, higher accuracies can be obtained.
3 The VIM in q-calculus
Definition 3.1 (q-calculus)
Let be a real continuous function. The q-derivative is defined by
The partial q-derivative is defined as
Property 3.2 q-Leibniz product law is
Property 3.3 q-integration by parts holds
Lemma 3.4 ()
For the first-order q-difference equation,
one of the Lagrange multipliers is .
Lemma 3.5 ()
For the q-difference equation of second order,
one of the Lagrange multipliers can be identified as
We only use the leading term , while other terms are restricted variations
Through the integration by parts (9), we can have
where δ is the variation operator and ‘′’ denotes the q-derivative with respect to t. As a result, the system of the Lagrange multiplier can be obtained:
from which we can get
Furthermore, Kong  gave the Lagrange multiplier for the q-difference equations of third order,
where denotes the q-factorial and for the integer k.
More generally, one can derive the following Theorem 3.6.
Theorem 3.6 For the q-difference equation of mth order,
establish the correctional functional for Eq. (17) as
If is considered as a restricted variation, one can derive a q-analogue Lagrange multiplier
where for the integer m.
Substituting (20) into (18), one can obtain a q-variational iteration formula
Here the initial iteration value can be determined via the q-Taylor series .
Example 3.7 Consider the simple linear q-difference equation of second order 
subject to the initial conditions
The iteration formula and the initial iteration can be determined as
The successive solution can be given as
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 .
Definition 4.1 The R-L integration of α order is defined as
The left Caputo derivative is defined by
Definition 4.3 The α th Riemann-Liouville (R-L) derivative of a function is defined by
In Sections 2 and 3, we note that the integration by parts plays an important role and is often used in the derivation of the Lagrange multipliers in ordinary calculus. But in FC, the similar integration by parts cannot hold. That’s the main reason why the applications of the VIM were not very successful for fractional differential equations (FDEs). The popular iteration formulae of the VIM directly employed the so-called Lagrange multiplier . For the generalized FDE
the variational iteration formula was suggested
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.
Theorem 4.4 For the generalized FDE, one can have the variational iteration formula
where the function is a Lagrange multiplier for any order α.
We can construct a correction functional through the R-L integration
Take the Laplace transform L to both sides of (28)
Assuming the terms and are restricted variations, respectively, we only need to consider the term
Setting the Lagrange multiplier , Eq. (30) can be considered as a convolution of the function and the term .
Making the correction functional of Eq. (29) stationary, we can get
From Eq. (16), with the inverse Laplace transform , we can have
For , the Lagrange multiplier can be explicitly identified as
As a result, the iteration formula is given as
This completes the proof. □
We only derive the simplest Lagrange multiplier here. In fact, more explicit Lagrange multipliers can be identified if more terms in (if they exist) are used. For example, we can derive a variational iteration formula
for a multi-order FDE
We conclude the following useful Lagrange multipliers for FDEs
Example 4.5 Let us consider the linear fractional Schrodinger equation 
The variational iteration formula for (40) reads
Starting from the initial iteration , the successive approximate solutions can be given as
For , tends to which is an exact solution of (41).
Example 4.6 As the second example, consider the nonlinear fractional Schrodinger equation 
The corresponding iteration formula reads
As a result, the approximate solutions can be obtained
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
We employ some notations of the q-fractional derivative and integral in . For , let be the time scale
where Z is the set of integers.
Definition 5.1 More generally, if α is a nonnegative real number, then we define the time scale as follows:
Definition 5.2 The q-fractional integral of α order is defined by
and the left Caputo q-fractional derivative is defined as
When v is not a positive integer, the q-factorial function is defined by
The fractional q-derivative of the q-factorial function with respect to t is
where and .
Now, we introduce the q-Laplace transform and some properties.
Definition 5.3 The q-Laplace transform was defined by Hahn  in 1949 as follows:
where , .
Lemma 5.4 ()
Let be an analytic function and assume on , where . Then the following convolution theorem can hold:
where the convolution is defined as
Lemma 5.5 For the Caputo q-derivative of , the following Laplace transform holds:
Lemma 5.6 ()
The existence and uniqueness of the solutions of the Caputo q-initial value problems have been discussed in .
Lemma 5.7 Considering the initial value problems of the Caputo q-fractional equations,
we construct a q-fractional correction functional
One of the Lagrange multipliers can be identified as .
Proof Take the Laplace transform of both sides of (54)
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).
Then we get the following equation:
where is the Laplace transform of some function.
Considering as a restricted variation so that after taking the classical variational derivative to both sides of (56), we can obtain
from which we can derive
The inverse Laplace transform of is
As a result, we can get
Substituting into (54), the variational iteration formula is determined as
Example 5.8 Now consider the application in the initial value problems of the Caputo q-fractional difference equations ,
We have the following variational iteration formula:
Starting from the initial iteration
the successive solutions can be given as
For , tends to the exact solution
where is the discrete Mittag-Leffler function defined by 
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.
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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).
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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Wu, GC., Baleanu, D. New applications of the variational iteration method - from differential equations to q-fractional difference equations. Adv Differ Equ 2013, 21 (2013). https://doi.org/10.1186/1687-1847-2013-21
- variational iteration method
- fractional calculus
- time scales
- Laplace transform
- symbolic computation