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Vol 52, No 2 (2016)
- Year: 2016
- Articles: 12
- URL: https://journals.rcsi.science/0012-2661/issue/view/9214
Ordinary Differential Equations
139-148
On the eigenvalues of a nonlinear spectral problem
Abstract
We consider a nonlinear eigenvalue problem of the Sturm–Liouville type with conditions of the third kind, which describes the propagation of polarized electromagnetic waves in a plane dielectric waveguide. The equation is nonlinear in the unknown function, and the boundary conditions depend on the spectral parameter nonlinearly. We obtain an equation for the spectral parameter and formulas for the zeros of the eigenfunctions and show that the problem has at most finitely many isolated eigenvalues.
149-156
Periodic two-cluster synchronization modes in completely connected genetic networks
Abstract
We consider nonlinear systems of differential-difference equations with delays that provide mathematical models for artificial complete genetic networks. We study problems of the existence and stability of special periodic motions referred to as two-cluster synchronization modes in these systems.
157-176
Estimates for solutions of linear and quasilinear systems in the nonautonomous case
Abstract
By using the freezing method, we obtain upper and lower estimates for the higher and lower characteristic exponents, respectively, of homogeneous n-dimensional linear differential and difference systems with coefficient matrix A(t) satisfying the condition ||A(t)−A(s)|| ≤ δ|t − s|α, δ > 0, α > 0, t, s ≥ 0. We also prove analogs of these estimates for quasilinear differential and difference systems.
177-185
Step-like contrast structure for a quasilinear system of singularly perturbed differential equations with a zero characteristic number
Abstract
We consider a system of singularly perturbed first-order differential equations with a zero characteristic number. The solution of such a problem is characterized by the presence of a contrast structure, that is, of an internal transition layer on a given interval. We prove the existence of an exact solution with a step-like contrast structure and construct its uniform asymptotic expansion. An example is given.
186-196
Partial Differential Equations
Simple layer potential and the first boundary value problem for a parabolic system on the plane
Abstract
By the method of boundary integral equations, we construct a classical solution of the first initial–boundary value problem for a one-dimensional (with respect to x) parabolic system in a domain with nonsmooth lateral boundary for the case in which the right-hand sides of the boundary conditions only have continuous derivatives of order 1/2. We study the smoothness of the solution.
197-209
Boundary value problems for a nonstrictly hyperbolic equation of the third order
Abstract
We study classical solutions of boundary value problems for a nonstrictly hyperbolic third-order equation. The equation is posed in a half-strip and a quadrant of the plane of two independent variables. The Cauchy conditions are posed on the lower boundary of the domain, and the Dirichlet conditions are posed on the lateral boundaries. By using the method of characteristics, we find the analytic form of the solution of considered problems. The uniqueness of the solutions is proved.
210-219
Inverse problem with nonlocal observation of finding the coefficient multiplying ut in the parabolic equation
Abstract
We study the inverse problem of the reconstruction of the coefficient ϱ(x, t) = ϱ0(x, t) + r(x) multiplying ut in a nonstationary parabolic equation. Here ϱ0(x, t) ≥ ϱ0 > 0 is a given function, and r(x) ≥ 0 is an unknown function of the class L∞(Ω). In addition to the initial and boundary conditions (the data of the direct problem), we pose the problem of nonlocal observation in the form ∫0Tu(x, t) dμ(t) = χ(x) with a known measure dμ(t) and a function χ(x). We separately consider the case dμ(t) = ω(t)dt of integral observation with a smooth function ω(t). We obtain sufficient conditions for the existence and uniqueness of the solution of the inverse problem, which have the form of ready-to-verify inequalities. We suggest an iterative procedure for finding the solution and prove its convergence. Examples of particular inverse problems for which the assumptions of our theorems hold are presented.
220-239
Gellerstedt problem with a generalized Frankl matching condition on the type change line with data on external characteristics
Abstract
We study the solvability of the Gellerstedt problem for the Lavrent’ev–Bitsadze equation with nonclassical matching conditions for the gradient of the solution (in the sense of Frankl) on the type change line of the equation. We prove that the inhomogeneous Gellerstedt problem with data on the external characteristics of the equation is solvable either uniquely or modulo a nontrivial solution of the homogeneous problem. We obtain integral representations of the solution of the problem in both the elliptic and the hyperbolic parts of the domain. The solution proves to be regular.
240-247
Periodic solutions of the wave equation with nonconstant coefficients and with homogeneous Dirichlet and Neumann boundary conditions
Abstract
We prove theorems on the existence and regularization of periodic solutions of the wave equation with variable coefficients on an interval with homogeneous Dirichlet and Neumann boundary conditions. The nonlinear term has a power-law growth or satisfies the nonresonance condition at infinity.
248-257
Short Communications
258-259
Smoothness of solutions of the Dirichlet problem for the biharmonic equation in nonsmooth 2D domains
Abstract
We study the smoothness of a generalized solution of the Dirichlet problem for the biharmonic equation in a two-dimensional domain. We introduce a weighted test function and derive an estimate for the absolute value of the solution in a neighborhood of an irregular boundary point.
260-264