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Vol 53, No 2 (2017)
- Year: 2017
- Articles: 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
Abstract
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.
145-156
157-170
Dulac–Cherkas criterion for exact estimation of the number of limit cycles of autonomous systems on a plane
Abstract
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.
171-179
Uniform, on the entire axis, convergence of the spectral expansion for Schrödinger operator with a potential-distribution
Abstract
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.
180-191
192-199
Partial Differential Equations
Nonlocal time-multipoint problem for a certain class of evolutionary pseudodifferential equations with variable symbols: II
Abstract
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.
200-211
On the weak solvability of the problem of viscoelasticity with memory
Abstract
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.
212-217
Integral Equations
Regularized asymptotic solutions of singularly perturbed integral equations with a higher-order diagonal degeneration of the kernel and the initialization problem
Abstract
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).
218-233
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
Abstract
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.
234-247
Control Theory
Optimal processes in the model of two-sector economy with an integral utility function
Abstract
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.
248-262
263-269
Short Communications
270-272
273-276
277-279