


卷 53, 编号 2 (2017)
- 年: 2017
- 文章: 14
- URL: https://journal-vniispk.ru/0012-2661/issue/view/9299
Ordinary Differential Equations
Solvability problems for a linear homogeneous functional-differential equation of the pointwise type
摘要
The Cauchy problem for a linear homogeneous functional-differential equation of the pointwise type defined on a straight line is considered. Theorems on the existence and uniqueness of the solution in the class of functions with a given growth are formulated for the case of the one-dimensional equation. The study is performed using the group peculiarities of these equations and is based on the description of spectral properties of an operator that is induced by the right-hand side of the equation and acts in the scale of spaces of infinite sequences.



Stability and attraction of solutions of nonlinear stochastic differential equations with standard and fractional Brownian motions
摘要
Conditions have been found that provide the stability and attraction of solutions of nonlinear stochastic differential systems, with a linear deterministic part, with standard and fractional Brownian motions.



Dulac–Cherkas criterion for exact estimation of the number of limit cycles of autonomous systems on a plane
摘要
The problem of exact nonlocal estimation of the number of limit cycles surrounding one point of rest in a simply connected domain of the real phase space is considered for autonomous systems of differential equations with continuously differentiable right-hand sides. Three approaches to solving this problem are proposed that are based on sequential two-step usage of the Dulac–Cherkas criterion, which makes it possible to find closed transversal curves dividing the connected domain in doubly connected subdomains that surround the point of rest, with the system having precisely one limit cycle in each of them. The effectiveness of these approaches is exemplified with polynomial Liènard systems, a generalized van der Pol system, and a perturbed Hamiltonian system. For some systems, the derived estimate holds true in the entire phase space.



Uniform, on the entire axis, convergence of the spectral expansion for Schrödinger operator with a potential-distribution
摘要
A uniform, on ℝ, estimate for the increment of the spectral function θ(λ; x, y) at x = y is proved for the self-adjoint Schrödinger operator A defined on the entire axis ℝ by the differential operation (−d/dx)2 + q(x) with a potential-distribution q(x) that uniformly locally belongs to the space W2−1. As a consequence, it is shown that for any function f(x) from the domain of power Aα of the operator with α > 1/4, the spectral expansion of the function that corresponds to the operator A is convergent absolutely and uniformly on the entire axis ℝ.



The case of a constant absolute invariant for the Lienard system
摘要
A criterion is suggested for defining such properties of the right-hand sides in the Lienard polynomial system that guarantee its first absolute invariant turning to a constant.



Partial Differential Equations
Nonlocal time-multipoint problem for a certain class of evolutionary pseudodifferential equations with variable symbols: II
摘要
The properties of the fundamental solution to a nonlocal time-multipoint problem for an evolutionary equation with a pseudodifferential operator constructed by a variable symbol are studied. The solvability of the above problem in the class of continuous bounded on R functions is established and a representation of the solution is derived.



On the weak solvability of the problem of viscoelasticity with memory
摘要
The existence of a weak solution is proved for a certain Oldroyd model of motion of a viscoelastic medium that allows for the memory of the system. The proof uses the theory of regular Lagrange flows and a topological approximation method that reduces the posed problem to an operator equation, its ε-regularization in smoother spaces, the use of a priori estimates and a topological degree for the proof of the solvability of the ε-regularized equations, and the passage to the limit as ε → 0.



Integral Equations
Regularized asymptotic solutions of singularly perturbed integral equations with a higher-order diagonal degeneration of the kernel and the initialization problem
摘要
An algorithm of the regularization method is developed for singularly perturbed integral equations with a higher-order diagonal degeneration of the kernel. The leading term of the asymptotics is analyzed to solve the problem of initialization (that is, extraction of the class of right-hand sides and kernels of the integral operator for which the exact solution of the original equation tends to some limit function as ε → +0 on the entire time interval, including the boundary-layer zone).



Convergence of a numerical method for solving a hypersingular integral equation on a segment with the use of piecewise linear approximations on a nonuniform grid
摘要
A numerical scheme has been constructed for solving a linear hypersingular integral equation on a segment with the integral treated in the sense of the Hadamard principle value by the method of piecewise linear approximations on an arbitrary nonuniform grid, with the hypersingular integral being regularized by approximating the unknown function with a constant in a small neighborhood of the singular point. The radius of the neighborhood can be chosen independently of the grid pitch, the latter understood as the maximum distance between the nodes. The uniform convergence of the obtained numerical solutions to the exact solution is proved as the grid pitch and the radius of the neighborhood in which the regularization is performed simultaneously tend to zero.



Control Theory
Optimal processes in the model of two-sector economy with an integral utility function
摘要
An infinite-horizon two-sector economy model with a Cobb–Douglas production function is studied for different depreciation rates, the utility function being an integral functional with discounting and a logarithmic integrand. The application of the Pontryagin maximum principle leads to a boundary value problem with special conditions at infinity. The presence of singular modes in the optimal solution complicates the search for a solution to the boundary value problem of the maximum principle. To construct the solution to the boundary value problem, the singular modes are written in an analytical form; in addition, a special version of the sweep algorithm in continuous form is proposed. The optimality of the extremal solution is proved.



Constructing an asymptotic solution of optimal observation problem for a quasilinear differential-algebraic system
摘要
The known results on the problem of optimal observation for ordinary quasilinear systems are generalized to differential-algebraic systems.



Short Communications
An addition to the cases of solvability of the Goursat problem in quadratures
摘要
Six new versions of solvability conditions for the Goursat problem in quadratures are formulated in terms of the coefficients of a linear inhomogeneous hyperbolic equation.



On solutions in the form of product of powers for second-order differential equations of the Fuchs type
摘要
Solutions in the form of the product of powers are found for second-order differential equations of the Fuchs type. Necessary and sufficient conditions are formulated for the existence of such solutions.



Problem with conditions similar to Neumann conditions for φ-Laplacian
摘要
A boundary value problem with functional boundary conditions, with the Neumann problem as its special case, is considered for the ϕ-Laplacian.


