Volume 59, Nº 9 (2023)
Articles
Estimates of Integrally Bounded Solutions of Linear Differential Inequalities
Resumo
We study integrally bounded solutions of the differential equation A(x)=z, where A is a linear differential operator of order l defined on functions x:R→H (R=(−∞,∞), H () and H is a finite-dimensional Euclidean space). The right-hand side z is an integrally bounded function on R ranging in H and satisfying the inequality (ψ(t),z(t))≥δ|z(t)|, t∈R, δ>0. Conditions are given on the operator A and the function ψ:R→H that guarantee an inverse inequality of the following form for the solutions x under consideration: ∫τ+1τ|x(l)(t)|dt≤c∫τ+2τ−1|x(t)|dt, where the constant is independent of the choice of a real number t and function x.
Construction of Polynomial Eigenfunctions of a Second-Order Linear Differential Equation
Resumo
A system of third-order recurrence relations for the coefficients of polynomial eigenfunctions (PEFs) of a differential equation is solved. A recurrence relation for three consecutive PEFs and a formula for differentiating PEFs are obtained. We consider differential equations one of which generalizes the Hermite and Laguerre differential equations and the other is a generalization of the Jacobi differential equation. For these equations, we construct functions bringing them to self-adjoint form and find conditions under which these functions become weight functions. Examples are given where the PEFs for nonweight functions do not have real zeros.
On the Existence of a Solution of a Boundary Value Problem on a Graph for a Nonlinear Equation of the Fourth Order
Resumo
A fourth-order nonlinear differential equation on a network that is a model of a system of Euler–Bernoulli rods is considered. Based on the monotone iteration method, the existence of a solution of a boundary value problem on a graph for this equation is established using the positiveness of the Green’s function and the maximum principle for the corresponding linear differential equation. An example is given to illustrate the results.
Approximation to the Sturm–Liouville Problem with a Discontinuous Nonlinearity
Resumo
We consider a continuous approximation to the Sturm–Liouville problem with a nonlinearity discontinuous in the phase variable. The approximating problem is obtained from the original one by small perturbations of the spectral parameter and by approximating the nonlinearity by Carathéodory functions. The variational method is used to prove the theorem on the proximity of solutions of the approximating and original problems. The resulting theorem is applied to the one-dimensional Gol’dshtik and Lavrent’ev models of separated flows
On the Existence of an Infinite Spectrum of Damped Leaky TE-Polarized Waves in an Open Inhomogeneous Cylindrical Metal–Dielectric Waveguide Coated with a Graphene Layer
Resumo
We consider the problem of leaky waves in an inhomogeneous waveguide structure covered with a layer of graphene, which is reduced to a boundary value problem for the longitudinal components of the electromagnetic field in Sobolev spaces. A variational statement of the problem is used to determine the solution. The variational problem is reduced to the study of an operator function. The properties of the operator function necessary for the analysis of its spectral properties are investigated. Theorems on the discreteness of the spectrum and on the distribution of the characteristic numbers of the operator function on the complex plane are proved.
Regularity of the Pressure Function for Weak Solutions of the Nonstationary Navier–Stokes Equations
Resumo
We study the nonstationary system of Navier–Stokes equations for an incompressible fluid. Based on a regularized problem that takes into account the relaxation of the velocity field into a solenoidal field, the existence of a pressure function almost everywhere in the domain under consideration for solutions in the Hopf class is substantiated. Using the proposed regularization, we prove the existence of more regular weak solutions of the original problem without smallness restrictions on the original data. A uniqueness theorem is proven in the two-dimensional case
Classical Solution of the Second Mixed Problem for the Telegraph Equation with a Nonlinear Potential
Resumo
For the telegraph equation with a nonlinear potential, we consider a mixed problem in the first quadrant in which the Cauchy conditions are specified on the spatial semiaxis and the Neumann condition is set on the temporal semiaxis. The solution is constructed by the method of characteristics in an implicit analytical form as a solution of some integral equations. The solvability of these equations is studied, as well as the dependence of the solutions on the smoothness of the initial data. For the problem under consideration, the uniqueness of the solution is proved and conditions are established under which a classical solution exists. If the matching conditions are not met, then a problem with conjugation conditions is constructed, and if the data is not smooth enough, then a mild solution is constructed.
Construction of Integral Representations of Fields in Problems of Diffraction by Penetrable Bodies of Revolution
Resumo
Based on integral representations with densities distributed along a segment of the symmetry axis, a representation of the solution of the boundary value problem of plane wave diffraction by a local penetrable body of revolution with smooth surface is constructed and justified. The resulting integral representation allows one to avoid resonances of the interior domain when analyzing the scattering frequency characteristics.
Integral Equations of Volterra Typewith Two Boundary and One Interior Singular Point
Resumo
Explicit solutions of model and nonmodel integral equations of Volterra type with two boundary and one interior singular point are obtained, and the properties of the resulting solutions are studied. The well-posed statement of problems with conditions specified on singular manifolds is found in the case where the solution of the model equation contains an arbitrary constant.
Fredholm Integral Equation for Problems of Acoustic Scattering by Three-Dimensional Transparent Structures
Resumo
We consider the differential and integral statements of problems of acoustic scattering by three-dimensional bounded transparent structures described by an integral equation. The results of the numerical solution of the integral equation describing the class of problems under consideration are presented. A theorem on existence and uniqueness of the solution is proved.
On the Optimality Conditions in the Weight Minimization Problem for a Shell of Revolution at a Given Vibration Frequency
Resumo
We consider shallow elastic shells with a given circular boundary and seek an axisymmetric shell shape minimizing the weight at a given fundamental frequency of shell vibrations. Using the resulting formula for the linear part of the increment of the frequency functional, the multiplicity of the minimum natural frequency of vibrations of the shell is estimated. The Fréchet differentiability of the frequency functional is also established, and optimal conditions for minimizing the weight of the shell at a given fundamental vibration frequency are obtained.
On a Control Problem for a System of Implicit Differential Equations
Resumo
We consider the differential inclusion F(t,x,x˙)∋0 with the constraint x˙(t)∈B(t), t∈[a,b], on the derivative of the unknown function, where F and B are set-valued mappings, F:[a,b]×Rn×Rn×Rm⇉ is superpositionally measurable, and B:[a,b]⇉Rn is measurable. In terms of the properties of ordered covering and the monotonicity of set-valued mappings acting in finite-dimensional spaces, for the Cauchy problem we obtain conditions for the existence and estimates of solutions as well as conditions for the existence of a solution with the smallest derivative. Based on these results, we study a control system of the form f(t,x,x˙,u)=0, x˙(t)∈B(t), u(t)∈U(t,x,x˙), t∈[a,b].