Fractional thermoelastic Timoshenko systems of type I and III

Thermoelasticity has attracted many researchers for a long period of time due to its numerous applications in different fields of science and engineering. In this proposal we are interested in a Timoshenko system coupled with a heat equation. We shall study it in both contexts of thermoelasticity of type I (the classical one) and thermoelasticity of type III. These cases have been investigated in fact by many researchers and a huge number of publication exist in the market. Our intention here is to consider the fractional cases of these systems. The integer-order derivatives are replaced by non-integer order derivatives of Caputo type. These systems model phenomena in complex media. Random diffusion of microscopic particles has been observed in a large number of processes. It varies from the localized diffusion (normal diffusion) all the way to the ballistic diffusion passing by the sub-diffusion and the super-diffusion. The classification depends on an exponent in the expression of the mean square displacement. Exponent one corresponds to the ordinary diffusion whereas exponent two is for the ballistic diffusion. The intermediary exponents correspond to fractional derivatives. Therefore, in complex media, it is appropriate to be in the fractional calculus context. We propose to investigate the stability of these two problems. In particular, we will seek sufficient conditions on the different parameters involved in the problems which are able to ensure the stabilization of the problem to its rest state. This includes finding a reasonable if not minimum damping mechanism. The thermal coupling is one of them. One of the main challenges we expect to face is the non-availability of some known properties in the integer order such as the power rule or the chain rule for fractional derivatives. This, in addition to the difficulties encountered in estimating certain arsing terms and the nonlocal character of the fractional derivatives. As a matter of fact, even the well-posedness is not clear for such problems in the absence of semi-groups and difficulty to deals with certain types of convergences. Therefore, there is a need to modify significantly the common arguments or come up with new techniques capable of treating these nonlocal problems.