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Some applications of the generalized Laplace transform and the representation of a solution to Sobolev-type evolution equations with the generalized Caputo derivative

Aydin Mustafa, Mahmudov Nazim I.

Bulletin of the Polish Academy of Sciences. Technical Sciences

2024

Vol. 72, nr 2

art. no. e149170

We introduce the Sobolev-type multi-term μ-fractional evolution with generalized fractional orders with respect to another function. We make some applications of the generalized Laplace transform. In the sequel, we propose a novel type of Mittag-Leffler function generated by noncommutative linear bounded operators with respect to the given function and give a few of its properties. We look for the mild solution formula of the Sobolev-type evolution equation by building on the aforementioned Mittag-Leffler-type function with the aid of two different approaches. We share new special cases of the obtained findings.

Some closed form series solutions for the time-fractional diffusion-wave equation in polar coordinates with a generalized Caputo fractional derivative

Elkott Ibrahim, Abdel-Latif Mohamed S., El-Kalla Ibrahim L., Abdel Kader Abass H.

Journal of Applied Mathematics and Computational Mechanics

2023

Vol. 22, nr 2

5--14

In this paper, we obtain some closed form series solutions for the time fractional diffusion-wave equation (TFDWE) with the generalized time-fractional Caputo derivative (GTFCD) associated with a source term in polar coordinates. These solutions are found using generalized Laplace and Hankel transforms. We obtained the closed form series solutions in the form of the Polygamma function. The effect of the fractional order derivative on the diffusion-wave variable is illustrated graphically.