Vol 59, No 1 (2023)
Articles
Evgeniy Ivanovich Moiseev (07.03.1948-25.12.2022)
Abstract
Solution Uniqueness Criteria in a Time-Nonlocal Problem for the Operator Differential Equation l(.)-A with the Tricomi Operator A
Abstract
We study the uniqueness of the solution of a time-regular problem for the operator-differential equation with the Tricomi operator. The order of the differential expression is considered to be an arbitrary positive integer, and the regular boundary conditions are given with respect to the time variable. The operator is generated by the Tricomi equation. The boundary conditions for the Tricomi operator are given by the Dirichlet condition on the elliptic part and by the fractional derivative traces of the solution along the characteristics. It is indicated that this operator is a self-adjoint operator in the space. The self-adjointness of the operator guarantees the existence of a complete system of eigenfunctions orthonormal in if is a domain bounded by a Lyapunov curve and by characteristics of the wave equation.
Properties of Degenerate Systems of Linear Integro-Differential Equations and Initial Value Problems for These Equations
Abstract
We consider systems of higher order integro-differential equations in which the matrix multiplying the highest derivative of the unknown vector function is identically singular in the domain where the system is considered. We give criteria for the solvability of such systems and initial value problems for these systems as well as examples illustrating the theoretical results.
Equivalence of Entropy and Renormalized Solutions of a Nonlinear Elliptic Problem in Musielak–Orlicz Spaces
Abstract
We consider second-order elliptic equations with nonlinearities determined by Musielak–Orlicz functions and with right-hand side in the space. In the Musielak–Orlicz–Sobolev spaces, we establish some properties and uniqueness of both entropy and renormalized solutions of the Dirichlet problem in domains with Lipschitz boundary. In addition, the equivalence and sign-definiteness of the entropy and renormalized solutions is proved.
Solution Blow-Up and Global Solvability of the Cauchy Problem for a Model Third-Order Partial Differential Equation
Abstract
We obtain conditions for the existence of a global solution and the blow-up of the solution of the Cauchy problem on a finite time interval for a nonlinear third-order partial differential equation generalizing the equation of torsion vibrations of a cylindrical rod with allowance for internal and external damping modeling the propagation of longitudinal stress waves along a one-dimensional viscoelastic rod whose the material obeys Voigt–Kelvin medium deformation law.
Projector Approach to Constructing the Asymptotics of Solution of Initial Value Problems for Weakly Nonlinear Discrete Systems with Small Step in the Critical Case
Abstract
An algorithm for constructing an asymptotic solution containing boundary functions for an initial value problem for a weakly nonlinear system of discrete equations with small step in the critical case under certain conditions is given in the article by V.F. Butuzov and A.B. Vasil’eva in Differ. Uravn., 1970, vol. 6, no. 4, pp. 650–664. In the present paper, orthogonal projectors are used to construct the asymptotics of the solution of this problem. This projector approach greatly simplifies the understanding of the algorithm for constructing the asymptotics and permits explicitly writing the problems from which one can find the terms of any order in the asymptotics of the solution.
Asymptotics of Solutions of Linear Singularly Perturbed Optimal Control Problems with a Convex Integral Performance Index and a Cheap Control
Abstract
We consider an optimal control problem for a linear system with constant coefficients with an integral convex performance index containing a small parameter multiplying the integral term in the class of piecewise continuous controls with smooth geometric constraints. Such problems are called cheap control problems. It is shown that the limit problem will be a problem with a terminal performance index. It is established that if the terminal term of the performance index is a convex (strictly convex) and continuously differentiable function, then the performance functional in the limit problem has similar properties. It is proved that, in the general case, convergence with respect to the performance functional is valid, and under the condition of strict convexity of the terminal term of the performance index in the original problem, convergence to the minimum point of the terminal summand of the performance index in the limit problem is valid. The limit of the defining vector in the original problem is found as the small parameter tends to zero. In particular, it is shown that the first component of the defining vector in the original problem converges to the defining vector in the limit problem. The problems of controlling a point of low mass in a medium with and without resistance with a terminal part depending on both slow and fast variables are considered in detail, and complete asymptotic expansions of the defining vectors in these problems are constructed.
On Linearization of Single-Input Nonlinear Control Systems Based on Time Scaling and a One-Fold Prolongation
Abstract
We prove a necessary and sufficient condition for the linearizability of single-input nonlinear control systems in the class of transformations containing time scaling and preserving the state manifold. A description is given for systems that are obtained by a 1-fold prolongation of a single-input nonlinear control system and are
-orbitally linearizable. It is proved that the
-orbital linearizability of the system obtained by a 1-fold prolongation of a single-input affine control system implies the
-orbital linearizability of the original system as well. It is shown that if the system obtained by a fold prolongation of a single-input nonlinear control system, where, is orbitally linearizable, then the system obtained from the original system by its 1-fold prolongation is
-orbitally linearizable as well.
Numerical Construction of the Transform of the Kernel of the Integral Representation of the Poincaré–Steklov Operator for an Elastic Strip
Abstract
For an isotropic stratified elastic strip we consider the Poincaré–Steklov operator that maps normal stresses into normal displacements on part of the boundary. A new approach is proposed for constructing the transform of the kernel of the integral representation of this operator. A variational formulation of the boundary value problem for the transforms of displacements is obtained. A definition is given and the existence and uniqueness are proved for a generalized solution of the problem. An iteration method for solving variational equations is constructed, and conditions for its convergence are obtained based on the contraction mapping principle. The variational equations are approximated by the finite element method. As a result, at each step of the iteration method, it is required to solve two independent systems of linear algebraic equations, which are solved using the tridiagonal matrix algorithm. A heuristic algorithm is proposed for choosing the sequence of parameters of the iteration method that ensures its convergence. Verification of the developed computational algorithm is carried out, and recommendations on the use of adaptive finite element grids are given
A Continuous Method for Finding a Generalized Fixed Point of a Nonexpansive Mapping on a Set in a Hilbert Space
Abstract
We introduce the concept of a generalized fixed point of a nonexpansive operator on a convex closed set in a Hilbert space. To find this point, we construct a regularizing algorithm in the form of the Cauchy problem for a first-order differential equation and establish sufficient conditions for the strong convergence of the resulting approximations to a normal generalized fixed point under approximate specification of the nonexpansive operator and the convex closed set on which the desired generalized fixed point of the operator is located. Examples of parametric functions are given that ensure the convergence of the approximations in the norm of the Hilbert space to a normal generalized fixed point of the operator on the convex closed set in this space.
On a Nonlinear Second-Order Ordinary Differential Equation
Abstract
We consider a nonlinear second-order ordinary differential equation of a special form whose particular case arises when constructing exact solutions of the nonlinear heat equation with a power-law coefficient. Conditions are obtained for the parameters under which the equation admits a single integration. A number of examples of constructing exact solutions expressed in terms of elementary functions or in terms of the Lambert function are given.