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Fourier, Laguerre, Laplace Transforms with applications

Aghili Arman

Journal of Mathematics and Applications

2021

Vol. 44

5--17

In this article, the author considered certain time fractional equations using joint integral transforms. Transform method is a powerful tool for solving singular integral equations, integral equation with retarded argument, evaluation of certain integrals and solution of partial fractional differential equations. The obtained results reveal that the transform method is very convenient and effective. Illustrative examples are also provided.

Null controllability from the exterior of a one-dimensional nonlocal heat equation

Warma Mahamadi, Zamorano Sebastián

Control and Cybernetics

2019

Vol. 48, No. 3

417--438

We consider the null controllability problem from the exterior for the one dimensional heat equation on the interval (−1, 1), associated with the fractional Laplace operator (−∂2 x)s, where 0 < s < 1. We show that there is a control function, which is localized in a nonempty open set O ⊂ (R \ (−1, 1)), that is, at the exterior of the interval (−1, 1), such that the system is null controllable at any time T > 0 if and only if 1/2 < s < 1.

Fractional heat conduction in a sphere under mathematical and physical Robin conditions

Kukla S., Siedlecka U.

Journal of Theoretical and Applied Mechanics

2018

Vol. 56 nr 2

339--349

In this paper, the effect of a fractional order of time-derivatives occurring in fractional heat conduction models on the temperature distribution in a composite sphere is investigated. The research concerns heat conduction in a sphere consisting of a solid sphere and a spherical layer which are in perfect thermal contact. The solution of the problem with a classical Robin boundary condition and continuity conditions at the interface in an analytical form has been derived. The fractional heat conduction is governed by the heat conduction equation with the Caputo time-derivative, a Robin boundary condition and a heat flux continuity condition with the Riemann-Liouville derivative. The solution of the problem of non-local heat conduction by using the Laplace transform technique has been determined, and the temperature distribution in the sphere by using a method of numerical inversion of the Laplace transforms has been obtained.