Mittag-Leffler stability for a fractional thermoelastic-porous system arising in swelling soils problem

The study of swelling soils is of considerable importance in geotechnical engineering and soil mechanics due to their capacity to undergo significant volumetric changes in response to moisture variation. These volumetric changes can affect the mechanical stability of structures such as foundations, retaining walls, and pavements. Traditional thermoelastic models often fall short in capturing the complex interaction between thermal, mechanical, and poroelastic effects in such materials, especially when memory and hereditary behavior are prominent. This necessitates the development of more sophisticated models that integrate porous effects with nonlocal time operators such as fractional derivatives. In this paper, we consider a fractional thermoelastic swelling system consisting of two-wave equations in one-dimensional argument with a heat conduction equation based on Fourier's law. The coupling occurs with the first component of the system, describing the displacement of the fluid. We prove that the system is Mittag-Leffler is stable under the usual physical condition on the coefficients of the system and the only dissipation resulting from the heat. Our result is obtained by using the multiplier method and some properties in fractional calculus.