Fractional calculus and viscoelasticity theory have played and continue to play a vital role in the study of many problems arising in applications. Of concern in this proposal are two problems modeling a fractional viscoelastic wave equation and a fractional Timoshenko beam. The equation and the system of equations are found by replacing the second order derivatives by derivatives of non-integer orders. Namely, they will be fractional derivatives, of Caputo type, between 1 and 2. This is in line with the description of anomalous diffusion as in porous media. The material is assumed to be viscoelastic and therefore takes into account the whole prehistory of motion. This results in a memory term in convolution form. The existing techniques for Fickian laws are not easily applicable for the case of non-Darcys laws. Therefore there is a need to modify significantly the common arguments or come up with new techniques capable of treating these nonlocal problems. While the viscoelastic problem with integer order derivative (second order derivative, that is the well-known wave equation) has been extensively studied, one cannot find many studies on the fractional case even without the viscoelastic term. 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 (including the relaxation function) which are able to ensure the stabilization of the problem to its rest state. One of the main challenges we expect to face is the non-availability of a power rule and neither a chain rule for fractional derivatives. This, in addition to the difficulties encountered in estimating certain terms involving the product of fractional derivatives with the memory terms.
|Effective start/end date
|15/04/18 → 15/04/20
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