Pseudo-fractional differential equations and generalized g-Laplace transform

Abstract

In this article, we introduce a generalized g-Laplace transform and discuss some essential results of integral transform theory, in particular, involving a ψ-Hilfer pseudo-fractional derivative and function convolution. In this sense, we investigated the existence and uniqueness of known solutions for a pseudo-fractional differential equation.

Time-fractional telegraph equation with ψ-Hilfer derivatives

“…In this work, some integral transforms are used, namely, the ψ-Laplace, the Fourier, and the Mellin transforms. The ψ-Laplace transform of a real valued function f with respect to ψ is defined by (see [45,Def. 13])…”

“…The ψ-Laplace transform may be written as the following operator composition involving the classical Laplace transform (cf. [45,Thm. 4])…”

“…This is important in our work so that the ψ-Hilfer derivative covers the largest number of fractional derivatives. When the ψ-Laplace transform is applied to the ψ-Hilfer derivative, we get (see [45,Thm. 6])…”

“…f (a + ) are evaluated at the limit t → a + . The ψ-Laplace convolution of two functions is defined by (see [45,Def. 15])…”

“…During the last years fractional differential equations with Hilfer and ψ-Hilfer derivatives were studied by several authors, see e.g. [12,30,41,45,48]. Moreover, the arbitrariness in the function ψ allows to construct different kernels for the fractional derivative with different properties of relaxation and memory processes, which are important to better describe the dynamics of a given problem and to model more complex physical phenomena.…”