Fractional exponential operators and time-fractional telegraph equation
© Ansari; licensee Springer. 2012
Received: 27 July 2012
Accepted: 12 October 2012
Published: 29 October 2012
In this paper, the Bromwich integral for the inverse Mellin transform is used for finding an integral representation for a fractional exponential operator. This operator can be considered as an approach for solving partial fractional differential equations. Also, application of this operator for obtaining a formal solution of the time-fractional telegraph equation is discussed.
MSC:26A33, 35A22, 44A10.
KeywordsLaplace transform Mellin transform partial fractional differential equation Wright function
1 Introduction and problem
By the above exponential operator, Dattoli et al. found solutions of some boundary value problems arising in mathematical physics in terms of integral transforms type; see [2, 3] and references therein. Also, they used this operational technique to describe properties of some special polynomials and functions [4–6]; also see .
In this paper, in a general case we obtain an integral representation for , , with order one for s, and then we show how this operator can be applied to find the formal solutions of partial fractional differential equations (PFDEs).
This problem for integral representation is referred to as the inverse of the Mellin transform of , , and in Section 2, we state main theorems and corollaries related to it. In Section 3, as an application of this technique, we find formal solutions of the space-fractional Moshinskii’s equation and the time-fractional telegraph equation. Finally, in Section 4 the main conclusions are drawn.
2 Main theorems and corollaries
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).
By setting , we get the relation (2.1). □
Theorem 2.2 (The Schouten-Van der Pol theorem for the Laplace transform )
and changing the order of integration, we get the relation (2.4). □
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.
3 Application to partial fractional differential equations
with the Cauchy-type initial condition as .
where we used the relations (1.1) and (1.2) by choosing the functions and .
with initial and asymptotic conditions , .
where we used the relation (2.6) for the linearization of a fractional exponential operator , and then we applied the relations (1.1) and (1.2) by substituting and .
This paper provides some new results in the theory of fractional derivative. These results show the flexible operational technique can be used in a fairly wide context beside the integral transforms for obtaining the formal solutions of PFDEs.
Also, this technique can be considered as a promising approach for many applications in applied sciences.
The author was partially supported by the Center of Excellence for Mathematics, University of Shahrekord.
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