On a nonlocal problem for a fourth-order mixed-type equation with the Hilfer operator

The present work is devoted to the study of the solvability questions for a nonlocal problem with an integrodifferential conjugation condition for a fourth-order mixed-type equation with a generalized RiemannLiouville operator. Under certain conditions on the given parameters and functions, we prove the theorems of uniqueness and existence of the solution to the problem. In the paper, given example indicates that if these conditions are violated, the formulated problem will have a nontrivial solution. To prove uniqueness and existence theorems of a solution to the problem, the method of separation of variables is used. The solution to the problem is constructed as a sum of an absolutely and uniformly converging series in eigenfunctions of the corresponding one-dimensional spectral problem. The Cauchy problem for a fractional equation with a generalized integro-differentiation operator is studied. A simple method is illustrated for finding a solution to this problem by reducing it to an integral equation equivalent in the sense of solvability. The authors of the article also establish the stability of the solution to the considered problem with respect to the nonlocal condition.

THE BOUNDARY VALUE PROBLEM OF FINDING A RIGHT-HAND SIDE FOR MIXED TYPE EQUATION WITH CHAPLYGIN’S TYPE OPERATOR

In this paper, we study the inverse problem for a mixed-type equation with power degeneracy on a transition line by definition of its right-hand side, depending on the spatial coordinate. The theory of identity has been proved. In the case of degree degeneracy, the uniqueness criterion for the solution of the problem is proved, and the solution itself is con- structed in the form of a sum of orthogonal series. The consistency of series in the class of solutions of the given equation is justified and the validity of the solution with respect to the boundary conditions is proved.

A boundary value problem for mixed type equation with singular coefficient

В работе изучается краевая задача для уравнения смешанного типа с сингулярным коэффициентом в области, эллиптической частью которой является первая четверть плоскости, а гиперболической частью — характеристический треугольник. Методами интегральных уравнений и принципа экстремума доказывается однозначная разрешимость рассматриваемой задачи. In this paper we study a boundary value problem for a mixed type equation in a domain whose elliptic part is the first quadrant of the plane and the hyperbolic part is the characteristic triangle. With the help of the method of integral equations and the principle of extremum we prove the unique solvability of the considered problem

The Keldysh problem for a mixed-type three-dimensional equation with three singular coefficients

В данной статье изучена задача Келдыша для трехмерного уравнения смешанного типа с тремя сингулярными коэффициентами в прямоугольном параллелепипеде. На основании свойства полноты систем собственных функций двух одномерных спектральных задач, доказана теорема единственности. Решение поставленной задачи построено в виде суммы двойного ряда Фурье-Бесселя. In this article, we study the Keldysh problem for a three-dimensional mixed-type equation with three singular coefficients in a rectangular parallelepiped. Based on the completeness property of systems of eigenfunctions of two one-dimensional spectral problems, a uniqueness theorem is proved. The solution to the problem posed is constructed as the sum of a double Fourier-Bessel series.