Том 239, № 6 (2019)

We consider the first mixed problem for the Vlasov–Poisson equations in infinite cylinder. This problem describes evolution of the distribution density for ions and electrons in a high-temperature plasma in the presence of an outer magnetic field. We construct stationary solutions of the Vlasov–Poisson system of equations with the trivial potential of the self-consistent electric field describing a two-component plasma in an infinite cylinder such that their supports are located at a distance from the boundary of the domain.

We investigate the well-posedness of initial-value problems for abstract integrodifferential equations with unbounded operator coefficients in a Hilbert space and provide the spectral analysis of operator-functions describing symbols of such equations. These equations are an abstract form of linear partial integrodifferential equations arising in the viscoelasticity theory and other important applications. We establish the localization and the spectrum structure of operator-functions describing symbols of these equations.

Boundary-value problems in bounded domains are studied for differential-difference equations with incommensurable translations of independent variables in principal terms. Conditions of the uniform (with respect to translations of independent variables) strong ellipticity of such equations are obtained.

This paper is devoted to differential-difference equations with degeneration in a bounded domain Q ⊂ ℝⁿ. We consider differential-difference operators that cannot be expressed as a composition of a strongly elliptic differential operator and a degenerated difference operator. Instead of this, the operators under consideration contain several degenerate difference operators corresponding to differential operators. Generalized solutions of such equations may not belong even to the Sobolev space \( {W}_2^1(Q) \).

Earlier, under certain conditions on the difference and differential operators, we obtained a priori estimates and proved that, instead of the whole domain, the orthogonal projection of the generalized solution to the image of the difference operator preserves certain smoothness inside some subdomains \( {Q}_r\subset Q\left(\underset{r}{\mathrm{U}}{\overline{Q}}_r=\overline{Q}\right) \).

In this paper, we prove necessary and sufficient conditions in algebraic form for the existence of traces on parts of boundaries of subdomains Q_r.

In this paper we study the convergence of the continuous Newton method for solving nonlinear equations with holomorphic mappings in complex Banach spaces. Our contribution is based on recent progress in the geometric theory of spiral-like functions. We prove convergence theorems and illustrate them by numerical simulations.