We employ some notations of the q-fractional derivative and integral in [38]. For , let be the time scale
(43)
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:
(44)
The q-fractional derivative and integral have been defined in earlier work [8–10].
Definition 5.2 The q-fractional integral of α order is defined by
(45)
and the left Caputo q-fractional derivative is defined as
(46)
When v is not a positive integer, the q-factorial function is defined by
(47)
The fractional q-derivative of the q-factorial function with respect to t is
(48)
and
(49)
where and .
Now, we introduce the q-Laplace transform and some properties.
Definition 5.3 The q-Laplace transform was defined by Hahn [39] in 1949 as follows:
(50)
where , .
Lemma 5.4 ([38])
Let be an analytic function and assume on , where . Then the following convolution theorem can hold:
(51a)
where the convolution is defined as
(51b)
Lemma 5.5 For the Caputo q-derivative of , the following Laplace transform holds:
(52a)
and
(52b)
Lemma 5.6 ([38])
, .
The existence and uniqueness of the solutions of the Caputo q-initial value problems have been discussed in [40].
Lemma 5.7 Considering the initial value problems of the Caputo q-fractional equations,
(53)
we construct a q-fractional correction functional
(54)
One of the Lagrange multipliers can be identified as .
Proof Take the Laplace transform of both sides of (54)
(55)
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:
(56)
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
(57)
from which we can derive
(58)
The inverse Laplace transform of is
(59)
As a result, we can get
Substituting into (54), the variational iteration formula is determined as
(61)
□
Example 5.8 Now consider the application in the initial value problems of the Caputo q-fractional difference equations [12],
(62)
We have the following variational iteration formula:
(63)
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 [12]
(64)
and
(65)
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].