In this paper we consider a Timoshenko beam with variable physical parameters, we prove that the model can be stabilize by one control force for both internal and boundary cases.

In this paper we consider the question of nonexistence of nontrivial solutions for nonlinear elliptic systems involving fractional diffusion operators. Using a weak formulation approach and relying on a suitable choice of test functions, we derive sufficient conditions in terms of space dimension and systems parameters. Also, we present three main results associated to three different classes of systems.

Direct and inverse initial-boundary value problems of a time-fractional heat equation with involution perturbation are considered using both local and nonlocal boundary conditions. Results on existence of formal solutions to these problems are presented. Solutions are expressed in a form of series expansions using appropriate orthogonal basis obtained by separation of variables. Convergence of series solutions are obtained by imposing certain conditions on the given data. Uniqueness of the obtained solutions are also discussed. The obtained general solutions are illustrated by an example using an appropriate choice of the given data.

In the present work, we discuss a unique solvability of an inversesource problem with integral transmitting condition for time-fractional mixed type equation in a rectangular domain, where the unknown source term depends on space variable only.The method of solution based on a series expansion using biorthogonal basis of space corresponding to a nonself-adjoint boundary value problem. Under certain regularity conditions on the given data, we prove the uniqueness and existence of the solution for the given problem. Influence of transmitting condition on the solvability of the problem is shown as well. Precisely, two different cases were considered; a case of full integral form (0 < γ < 1) and a special case (γ = 1) of transmitting condition.In order to simplify the bulky expressions appearing in the proof of the main result, we have established a new property of the recently introduced Mittag-Leffler type function of two variables (see Lemma 2.1).

Inverse initial and inverse source problems of a time-fractional differential equation with Bessel operator are considered. Results on existence and uniqueness of solutions to these problems are presented. The solution method is based on series expansions using a set of Bessel functions of order zero. Convergence of the obtained series solutions is also discussed.