We study the solvability of new boundary problems for a special class of degenerate second-order hyperbolic equations. These problems have two features. The first is the presence of two variables in the equation, each of which can be considered as a time variable. This means that problems with fundamentally different boundary conditions can be correct for these equations. The second feature is the presence of degeneracy. This means that the boundary problems formulation can change depending on the nature of degeneration.
For the problems under study,we prove theorems of existence and uniqueness for regular solutions are proven, i.e. solutions that have all generalized according to Sobolev derivatives included in the equation.

We consider a singularly perturbed system of ordinary differential equations with a small parameter, in which different-scale variables are involved. The necessary information about the method of integral manifolds is presented, such as slow surface (ε = 0), sheets of slow surfaces, integral manifold (ε ̸= 0), its sheets, and asymptotic expansion of a slow integral manifold in powers of ε. As an example, a qualitative analysis of one singularly perturbed system with a small parameter is carried out in the work.

A brief overview of the authors’ research results on non-classical boundaryvalue problems for linear systems of partial differential equations is given. Some new results in this direction are presented.

The solvability of the initial-boundary value problem for linear integrodifferential equations with a condition on the lateral boundary that connects the values of the solution or the conormal derivative of the solution with the values of some integral operator of the solution is investigated. Theorems of existence and uniqueness of regular solutions are proved. The systematic study of nonlocal boundary value problems – the problems of finding periodic solutions to elliptic equations – was initiated by Bitsadze and Samarskii in 1969. Note also the studies for pseudoparabolic and pseudohyperbolic thirdorder equations with an integral condition on the lateral boundary. Great contributions to the development of the theory of nonlocal problems for differential equations of various classes were made by the monographs of Skubachevsky in 1997, Nakhushev in 2006 and 2012, and Kozhanov in 2024.

Sufficient conditions for unique solvability in the classical and generalized sense of the inverse problem for a nonlinear equation in a Banach space resolved with respect to the highest fractional derivative of Dzhrbashyan– Nersesyan are obtained. The overdetermination condition of the inverse problem is given by the Stieltjes integral; the lower derivatives are present in the equation non-linearly. The operator by the unknown function in the linear part of the equation is assumed to be bounded or generating an analytical resolving family of the corresponding linear homogeneous equation. Using our previous results for the direct problem for a linear inhomogeneous equation we obtain the main results here by the method of contraction mappings. An example of an inverse problem for a partial differential equation for which the conditions of an abstract theorem are fulfilled is given.

We provide new results on the Bergman type projections in products of tubular domains over symmetric cones extending some known classical assertions. Similar results with the same proof may be valid in the Siegel domains and bounded strongly pseudoconvex domains with smooth boundary. Our new Bergman projection theorems may have various interesting applications in function theory in tubular domains over symmetric cones.

A direct dynamic problem of the theory of elasticity is considered, which models the formation of seismic wave fields from earthquakes that occur during tectonic processes in the lower layers of Earth’s crust. The numerical solution to the stated problem is based on the method of complexing the analytical Laguerre transform and the finite difference method. A series of numerical calculations for a test model of media has been carried out.

An algorithm for the numerical solution of the equilibrium problem of a two-dimensional elastic body containing two thin elastic inclusions is developed. The inclusions are modeled within the framework of the theory of Timoshenko beams and intersect at right angle at an internal point of one of them, forming a T-shaped structure in an elastic body. One of the inclusions delaminates from the elastic matrix, forming a crack. On the crack faces, as part of the domain boundary, boundary conditions of the inequality form are specified. The presence of this type of boundary conditions leads to nonlinearity of the problem and formulation in the form of a variational inequality. To develop an algorithm for the numerical solution of the problem, an approximate problem of finding the saddle point of the Lagrangian is formulated. The convergence of solutions of the approximate problem to the solution of the original problem is proven. An iterative Uzawa-type algorithm is constructed and its convergence is shown. Examples of numerical implementation are given.