The boundary value problem for the generalized moisture transfer equation

In mathematical modeling of continuous media with memory, we deal with equations that describe a new type of wave motion, something between ordinary wave diffusion and classical wave propagation. There are fractional differential equations, which are the basis for the most mathematical models describing a wide class of physical and chemical processes in the fractal geometry of the Nature. The paper presents a new moisture transfer equation with a fractional Riemann–Liouville derivative that generalize the Aller–Lykov equation. The first boundary value problem for the generalized moisture transfer equation is considered. To prove the uniqueness of a solution we employ the energy inequalities method; an a priori estimate is obtained in terms of the fractional Riemann–Liouville derivative. The existence of the solution for the problem is proved by the Fourier method.