Mittag–Leffler stability for a fractional swelling soil problem

Swelling (also called expansive) soils are characterized by a swell in the soil's volume when subjected to moisture. The clay minerals in soil naturally attract and absorb water. When water is introduced to swelling soils, the water molecules are pulled into the gaps between the soil plates. As more water is absorbed, the plates are forced further apart, leading to an increase in soil pore pressure and consequently swelling soils significantly leading to geotechnical and structural challenges. In this paper, we consider a fractional swelling soil system damped by only a viscoelastic term. It turns out that investigating this problem the same way as done for the integer case is not possible. This is mainly due to the nonlinear character of the fractional derivative (in addition to the viscoelastic term), the difficulty caused by singular kernels, and the weak damping implied by the viscoelasticity. We prove that the system is Mittag–Leffler stable when the relaxation function itself is decaying in a Mittag–Leffler fashion. Our result is obtained using the multiplier method and some properties in fractional calculus. In addition, we present a numerical example to prove the validity of the theoretical stability results for this system.