Example 3.1 Consider the following equation:
(14)
where is the unknown function and .
We propose to show that (14) has a unique analytic solution by using the Banach fixed point theorem. By assuming
as a formal solution, where , calculations imply
and
Therefore, satisfies
which is equivalent to
(15)
where
Now is a contraction mapping whenever ; therefore, in view of the Banach fixed point theorem, Eq. (15) has a unique analytic solution in the unit disk and consequently the problem (14).
To calculate the fractal index for the equation
(16)
we assume the transform and the solution can be expressed in a series in the form
(17)
where are constants. Substituting (17) into Eq. (16) yields
(18)
Since
then the computation imposes the relation
with , and consequently we obtain
Thus, we have the following solution:
which is equivalent to
where is a Mittag-Leffler function. The last assertion is the exact solution for the problem (16) and consequently for (14).
Example 3.2 Consider the following equation:
(19)
where is the unknown function and . In the same manner of Example 3.1, we let
as a formal solution, where and
Estimations imply
and
Therefore, satisfies
which is equivalent to
(20)
where
Now, to show that is a contraction mapping,
Thus, in view of the Banach fixed point theorem, Eq. (20) has a unique analytic solution in the unit disk and consequently the problem (19).
To evaluate the fractal index for the equation
(21)
we assume the transform and the solution can be articulated as in (17). Substituting (17) into Eq. (21), we have
(22)
where
Hence, the computation imposes the relation
with , and consequently we obtain
where in terms of a gamma function. If we let , then the solution approximates to
which is equivalent to
The last assertion is the exact solution for the problem (21) and consequently for (19).
Next, we consider the Cauchy problem by employing the generalized fractional differential operator (4). We shall show that the solution of such a problem can be determined in terms of the Fox-Wright function [37]:
where for all , for all , and for suitable values and , are complex parameters.
Example 3.3 Consider the Cauchy problem in terms of the differential operator (4)
(23)
where is analytic in u and is analytic in the unit disk. Thus, F can be expressed by
Let . Then the solution can be formulated as follows:
(24)
where are constants. Substituting (24) into Eq. (23) implies
(25)
Since
then the calculation yields the relation
consequently, we obtain
Thus, we have the following solution:
which is equivalent to
Since ϕ is an arbitrary constant, we assume that
Thus, for a suitable α, we present
or
where .