The classical solution of the first mixed problem for the wave equation in the cylindrical area in an odd-dimensional space is considered. The solution is constructed using spherical averaging operators which allow the reduction of the equation to the first mixed problem for the one-dimensional wave equation. Using the method of characteristics the explicit solution is obtained and necessary and sufficient matching conditions for the existence of a unique classical solution are proved.

We consider a stochastic differential–difference hybrid system with a feedback control and with a delay–difference equation. For the case of continuous coefficients of the system, a theorem on the existence of weak solutions is proved.

We study the basis properties of root functions of a $$2\times 2 $$ Dirac operator with integrable complex-valued potential and irregular boundary conditions. Under certain conditions on the spectrum of the operator, we prove that the root function system of this operator is incomplete in the space $$L_2(0,\pi )\oplus L_2(0,\pi )$$ but forms an unconditional basis in the closed linear span of itself.

We study how the asymptotic behavior of the fundamental solution system of linear differential equations generated by a symmetric two-term odd-order differential expression for large values of the independent variable depends on the coefficients of the highest derivative and the free term.

The paper studies the existence of a classical solution of an initial value problem with incomplete initial data on the boundary of a strip for a hyperbolic differential-difference equation. The equation contains a superposition of a differential operator and a translation operator with respect to the spatial variable, which varies along the entire real axis. Using the Gelfand–Shilov operational scheme, we obtain the solution of this problem in closed form.

We consider the spectral Dirichlet problem for the Laplace operator in a thin three-dimensional domain of variable thickness which admits a maximum value on a smooth closed contour either inside the longitudinal cross-section or at its boundary. We find asymptotic expansions of the eigenvalues which involve the eigenvalues of the harmonic oscillator on the line or half-line as well as of a certain second-order ordinary differential equation on the contour. The eigenfuctions are localized in a neighborhood of the contour.

We study the sensitivity of solutions of Riccati equations to asymptotically small perturbations of their coefficients. An upper bound on the difference between the solutions of algebraic and differential Riccati equations is derived. The result is applied to study optimality in a stochastic linear-quadratic control problem over an infinite time horizon for an asymptotically autonomous system. We also study the performance of an invariant control strategy.

We study the solvability of a boundary value problem for a system of five nonlinear second-order partial differential equations under given nonlinear boundary conditions, which describes the equilibrium state of elastic nonshallow inhomogeneous isotropic shells with free edges in the framen of the Timoshenko shear model in isometric coordinates. The boundary value problem is reduced to a nonlinear operator equation for the generalized displacements in a Sobolev space, the solvability of which is established using the contraction mapping principle.

For feedback systems governed by fractional semilinear differential inclusions and a sweeping process in a Hilbert space, controllability conditions are found. For the proof, topological methods of nonlinear analysis for multivalued condensing maps are used.

When solving localization problems numerically, the main task is to construct a universal section corresponding to a given localizing function. The paper proposes two methods for solving this problem that use estimates of the first- and second-order derivatives. A comparative analysis of these methods with the method based on the use of all nodes of a regular grid is carried out. A comparative analysis shows that the proposed methods are superior both in terms of computational complexity and in the quality of the resulting approximation of the universal section.