In this paper we consider the Gellerstedt problem for the Lavrentiev-Bitsadze equation with boundary conditions on parallel characteristics in the hyperbolic part of the equation.

The problem for the Bessel equation of an integer order with complex physical and spectral parameters in the boundary condition is considered. The spectral parameter enters the boundary condition quadratically. The question of the basis property of the system of eigenfunctions in the case of the appearance of a multiple eigenvalue is studied.

Sobolev-type equations can describe non-stationary processes in a semiconductor, plasma, hydrodynamic phenomena, and others. Widespread interest in them has been observed since the second half of the 20th century. They contain terms with mixed derivatives with respect to time and high order spatial variables. In this paper, we study one Sobolev-type equation, which describes a non-stationary process in a semiconductor medium. The Painleve test is an effective tool for constructing general solutions to ODEs and PDEs. It allows one to construct a general solution of an equation in the form of a Laurent series or to prove that this is impossible. In this paper, we apply the Painleve test to find a solution to the problem. As a result, it was found that when a certain relation is satisfied for the parameters of the problem, the Painleve test is passed and the solution exists in the required one.

Families of exact solutions of a nonlinear equation from the theory of spin waves are constructed that describe a nonstationary process in a magnetic medium with spatial dispersion. The group properties of this equation and the corresponding stationary equation are investigated. A theorem on the nonuniqueness of the classical solution of the Cauchy problem for the nonlinear equation is proved.

In this paper, we study the problem of boundary control of string oscillations, which is carried out over a period of time less than the critical time. The control is performed by a displacement of one end of the string, whereas at the other end a uniform boundary condition with oblique derivative is given, and the direction of this derivative does not coincide with characteristics. The classical statement of the problem is considered. Necessary and sufficient conditions for the existence of a unique control are found, and the control itself is obtained in an explicit analytical form.

We define radial multipliers using solutions of the Helmholtz equation, which depend on the radial coordinate, and we find the recurrence relations between them in the space of any dimension , in which the Helmholtz operator is defined. It is shown that the procedure of differentiation of these multipliers leads to a system of solutions of the Helmholtz equation, represented as products of the radial multipliers and harmonic polynomials. Theorems about the properties of radial multipliers and the structure of harmonic polynomials in the solutions of Helmholtz equation are given. These solutions constructed using radial multipliers and harmonic polynomials are proposed to be used in gradient elasticity for multi-layered domains with spherical and cylindrical boundaries, since they allow to present boundary conditions in explicit algebraic form.

In this paper we apply spectral method to the Gellerstedt problem for Lavrent’ev-Bitsadze equation in a half-strip. We consider Frankl matching condition at the type change line. Using the Darboux solution formulae for mixed type equationinthe hyperbolic subdomainwe reduce problem to an auxiliary problem for the Laplace operator.

Two boundary value problems in which one of the conditions is nonlocal and contains a real parameter are studied for an equation of mixed type in a half-strip. Sufficient conditions for the unique solvability of these problems are obtained under some restrictions on the parameter.

The Gellerstedt eigenvalue problem with homogeneous boundary conditions on interior characteristics is studied. We prove that the eigenfunction system of this problem is a Riesz basis in the L2 space in the elliptic domain.

We study the Gellerstedt problem for the Lavrent’ev–Bitsadze equation with boundary conditions on parallel characteristics in the hyperbolic domain of the equation. Three distinct types of conditions on the type change lines are considered, and existence and uniqueness theorems for the corresponding problems are proved.

We study a boundary value problem for an equation of mixed type with the Lavrent’ev–Bitsadze operator in the leading part and with variable deviation of the argument in lower-order terms. The general solution of the equation is constructed. We prove a uniqueness theorem without any conditions on the value of the deviation. The problem is uniquely solvable. We derive integral representations of the solutions in closed form in the elliptic and hyperbolic domains.

We study analogs of classical mixed-type problems (the Tricomi and Frankl problems) for the Lavrent'ev-Bitsadze equation in three-dimensional domains. There arise difficulties related to the well-posed statement of these problems. For each problem, the solutions are written out in closed form as function series, and the uniqueness and regularity of the constructed solutions are proved.