Abstract
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.
Highlights
Introduction and problemWe consider the exponential operator eλ[q(x) d dx +v(x)]f (x) = f x(λ) g(λ), ( . )where x(λ), g(λ) are specified by the system of first-order differential equations [ ] ⎧ ⎨d dλ x(λ) q(x(λ)), x( ) = x ⎩
For the fractional exponential operator e–λαsα, < α
In a general case we obtain an integral representation for eλαsα, α >, with order one for s, and we show how this operator can be applied to find the formal solutions of partial fractional differential equations (PFDEs)
Summary
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 [ , ] When we encounter an exponential operator of higher order eλαsα , where α is integer or non-integer and s v(x), it is of interest to have an integral representation to reduce the order and apply the relation For the fractional exponential operator e–λαsα , < α < , it may occur that this operator can be written as the Laplace transform of the Wright function [ – ]
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