In this section, we establish some theorems on the fractional exponential operator which can be useful for solving PFDEs. First, we derive an integral representation for the operator , which can be considered as a generalized representation of the relations (1.3) and (1.4).
Theorem 2.1 The following identity holds true for :
(2.1)
where the function
is presented by
(2.2)
Proof By the definition of the inverse of the Mellin transform for a function , we have
(2.3)
The above relation implies that the Mellin transform of the last integral is equal to the function , that is,
By setting , we get the relation (2.1). □
Theorem 2.2 (The Schouten-Van der Pol theorem for the Laplace transform [12])
Let c be a suitable real constant such that and are analytic functions in the half-plane and is the Laplace transform of . Then the inverse of the Laplace transform is given by
(2.4)
Proof Using the definition of the Laplace transform for
replacing in the inverse of the Laplace transform
and changing the order of integration, we get the relation (2.4). □
Corollary 2.3 It is obvious that by setting , , in the relations (2.4) and using the relation (1.6) for the inverse of the Laplace transform , the inverse of the Laplace transform can be presented by
(2.5)
Corollary 2.4 By setting and combining the relations (2.1) and (2.5), we get a new integral representation for the fractional exponential equation
(2.6)
where the function
is given by
(2.7)
In view of the theorems of a fractional exponential operator expressed in this section, we may apply this operator to PFDEs in the next section.