On a boundary value problem for a mixed equation with three planes of type change in an infinite prismatic domain

В данной работе в бесконечной призматической области формулируется и изучается одна задача для параболо-гиперболического уравнения c тремя плоскостями изменения типа. Основным методам исследования поставленной задачи является преобразование Фурье. Доказана единственность и существование решения поставленной задачи In this paper, in an infinite prismatic domain, one problem is formulated and studied for a parabolic-hyperbolic equation with three planes of type change. The main methods for studying the problem posed is the Fourier transform. The uniqueness and existence of a solution to the problem is proved.

On Boundary Value Problems for a Mixed Type Fractional Differential Equation with Caputo Operator

This article is devoted to study the boundary value problems of the first and second kind with respect to the spatial variable for a mixed inhomogeneous differential equation of parabolic-hyperbolic type with a fractional Caputo operator in a rectangular domain. In the study of such boundary value problems, we abandoned the boundary value condition with respect to the first argument and instead it is used additional gluing condition. In this case, in the justification of the unique solvability of the problems, the conditions on the boundary domain are removed. This allowed us to weaken the criterion for the unique solvability of boundary value problems under consideration. The solution is constructed in the form of Fourier series with eigenfunctions corresponding to homogeneous spectral problems. Estimates for the convergence of Fourier series are obtained as a regular solution of this mixed equation.

A MIXED EQUATION OF TRICOMI–KELDYSH TYPE

We study a mixed type equation, which is analogous, in parts, to Tricomi equation and Keldysh equation. For this equation, the characteristics in the hyperbolic region are transversal to the transition locus in a subset of the locus, but is tangential to it in another subset. We propose a formulation of a closed boundary value problem for such a mixed type equation, and prove that it admits a unique H1 solution with some higher regularity inside the domain. Our result shows the variety and the complexity of the boundary value problems for mixed-type equations.