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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.

Three-dimensional analogue of the problem of Tricomi with non-characteristic parallel planes of change of types of the equation is investigated for a parabolic-hyperbolic equation. The uniqueness of the solution of the problem is proved by the method of a priori estimates, and the existence of the solution of the problem is enlightened to the existence of a solution of the system of the second type Voltaire integral equation.