Vol 217 (2022)

In this paper, we obtain necessary and sufficient conditions for the existence of a unique solution of the Showalter-Sidorov-Dirichlet problem for a second-order, semilinear Sobolev-type equation. For the initial-boundary-value problem considered, using the Galerkin method, we construct an approximate solution as an expansion in the system of eigenfunctions of the homogeneous Dirichlet problem for the Laplace operator. The proof of the *-weak convergence of the Galerkin approximations to the exact solution is based on a priori estimates, embedding theorems, and the Gronwall lemma.

This paper is a review of applications of the method of angular boundary functions to nonlinear equations. We consider the first boundary-value problem for the following singularly perturbed parabolic equation in a rectangle: ${\epsilon }^2\left(a^2\frac{{д}^{2}u}{дx^2}-\frac{дu}{дt}\right)=F(u, x, t, \epsilon )$ where the function F is nonlinear with respect to the variable и. We consider the case where the function F is quadratic or cubic in the variable и at the corner points of the rectangle and examine the possibility of constructing a complete asymptotic expansion of the solution of the problem as e ^ 0.

In this paper, we study a mathematical model of macroeconomics known as “demand-supply” or “market model.” The classical version of this model has no cycles. We show that the introduction of a delay may lead to the appearance of periodic solutions, including stable solutions, and find the minimum value of such a delay. Our analysis is based on methods of the theory of dynamical systems with infinite-dimensional spaces of initial conditions. For periodic solutions detected, we obtain asymptotic formulas.

On a bounded time interval, we consider the Cauchy problem for the of evolutionary Hamilton-Jacobi equation in the case where the dimension of the phase variable is equal to one.

The Hamiltonian depends on the phase and momentum variables and the dependence on the momentum variable is exponential. The domain in which the equation is considered is divided into three subdomains. Inside each of the three subdomains, the Hamiltonian is continuous, while at the boundaries of these subdomains it is discontinuous with respect to the phase variable. Based on the minimax/viscosity approach, we introduce the notion of a continuous generalized solution of the problem and prove its existence. The generalized solution is unique if the problem is considered in a domain bounded with respect to the phase variable.

This paper is devoted to the study of the phase dynamics of fractional linear systems with control. Two-dimensional systems with concentrated parameters are considered in most detail in the cases where the fractional differentiation operators in the governing equations are understood in the Caputo-Fabrizio sense. Systems modeled by equations with Atangana-Baleano and Prabhakara operators are also considered. We obtain and examine analytic solutions and boundary trajectories of systems, which determine domains of admissible values of the phase coordinates. The statement of the moment l-problem for the systems considered and its solvability are analyzed. An example of solving this problem in the case where the control is an essentially bounded function on a interval is given.

This paper is the final part of a review of results concerning the quantization approach
to the some classical aspects of noncommutative algebras. The first part is: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 213 (2022), pp. 110–144. The second part is: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 214 (2022), pp. 107–126. The third part is: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 215 (2022), pp. 95–128. The fourth part is: Itogi Nauki Tekhn. Sovr. Mat. Prilozh. Temat. Obzory, 216 (2022), pp. 153–171.