Vol 61, No 4 (2025)

In the works of I.A. Lappo-Danilevsky, in particular, the solutions of a system of linear ordinary differential equations in the vicinity of an isolated pole of arbitrary finite order were investigated. For the fundamental matrix of solutions of such a system, a series was obtained that absolutely converges in the punctured (annular) neighborhood of the pole. At the same time, recurrent relations of a rather complex type were found for the numerical coefficients of the specified series, which do not depend on the type of the system of equations. In this paper, explicit formulas for these coefficients are obtained for the first time. As an example, the results obtained are used to find the analytical formula for the trace of the monodromy matrix of an arbitrary regular singular point of the specified system of equations.

In a Hilbert space 𝐿2[0,+∞) the Sturm–Liouville operator generated by a differential expression of a special type containing the Dirac delta function with zero boundary condition is investigated. We prove that the eigenvalues 𝜆𝑛 of this operator satisfy the certain inequalities. The problem on the location of the first eigenvalue 𝜆1 depending on the parameters of the differential expression is solved. In particular, we obtain conditions under which 𝜆1 is negative.

The boundary value problem for stationary equations of magnetohydrodynamics of viscous heatconducting fluid with variable coefficients considered under the Dirichlet condition for velocity and mixed boundary conditions for electromagnetic field and temperature is studied. Sufficient conditions on the initial data, providing global solvability of the mentioned problem and local stability of its solution, are established.

For the hyperbolic biwave equation with nonlinear lower terms given in the first quadrant of Euclidean space, we consider a mixed problem in which the Cauchy conditions are specified on the spatial semi-axis, and Dirichlet and Wentzel conditions are specified on the temporal semi-axis. The solution is constructed by the method of characteristics in an implicit form as a solution of some integrodifferential equations. The solvability of these equations, as well as the dependence on the initial data and the smoothness of their solutions, is studied using the parameter continuation method and a priori estimates. For the problem under consideration, the uniqueness of the solution is proved and conditions under which there exists a classical solution are established. If the matching conditions are not met, then a problem with conjugation conditions is constructed, and if the data is not smooth enough, then a mild solution is constructed.

The article discusses the issues of studying stability and bifurcations in the reaction–diffusion system in a bounded domain with homogeneous Neumann boundary conditions. The main results of the article concern the study of problems of local bifurcations in the vicinity of spatially homogeneous equilibrium positions. A general scheme is proposed that allows us to obtain new formulas for studying the main characteristics of multiple equilibrium bifurcation and bifurcation Andronov–Hopf: sufficient signs of bifurcations, their type, approximate construction of solutions, stability analysis. The proposed approaches do not require complex and cumbersome transformations, the results obtained are brought to calculation formulas and algorithms. Some applications in problems of diffusion instability and corresponding bifurcations in reaction–diffusion systems are also discussed. The distributed model of the “brusselator” is considered as the main illustrative example.

For a hypersingular integral operator of the Calderon–Zygmund type, related to peridynamics problems, a lower bound is obtained. Thus, it is established that the previously found upper bound is exact.