We consider an ordinary differential operator of order 2m with integral conditions containing derivatives of the unknown function. We obtain an a priori estimate for the solutions of the problem for sufficiently large values of the spectral parameter. We prove the discreteness of the spectrum and its sectorial structure as well as the Fredholm property of the corresponding problem.
We consider various classes of abstract singular equations generalizing the Euler–Poisson–Darboux equation and containing the squared generator of an exponentially decaying semigroup. Sufficient conditions for the unique solvability of the corresponding incomplete problems are obtained.
This paper investigates the step-like contrast structure in a class of higher-order quasilinear singularly perturbed boundary value problems with the same type of singularity. To overcome the difficulty that the conditions for determining the transition point are overdetermined in the real domain, we extend the problem to the complex domain and employ complex-analytic methods. By utilizing the boundary layer method and the smooth matching technique, we not only construct the asymptotic expansion of the step-like solution with an interior layer but also rigorously prove the uniform validity of this smooth solution together with remainder estimates. The results show that, under appropriate analyticity and sign conditions, such quasilinear systems can achieve a balance between the number of equations and the number of unknowns for determining the transition point in the complex domain, thus yielding a solution with a step-like structure in the real domain. This work provides new theoretical tools and analytical approaches for the asymptotic analysis of higher-dimensional quasilinear singular perturbation systems.
Unlike solutions of ordinary differential equations, the smoothness of generalized solutions of differential–difference equations may be violated at interior points of the interval. Boundary value problems for such equations arise in many applied fields, particularly in the damping control problem for systems with aftereffect. The present paper studies the regularity of generalized eigenfunctions of differential–difference operators with Dirichlet type boundary conditions.
For an isotropic stratified elastic layer, we consider the Poincaré–Steklov operator that maps normal stresses into normal displacements on part of the boundary. To construct the transfer function of this operator, the variational formulation of the boundary value problem for the transforms of displacements is used. The positive definiteness of the bilinear form of the variational problem and the existence and uniqueness of a generalized solution are proved. Some properties of the transfer function are established, allowing for a significant reduction in computation costs. The variational problem is approximated by the finite element method. The developed computational algorithm is verified and recommendations for the use of adaptive finite element meshes are given.
We construct the asymptotics of a solution of the Neumann problem in a planar domain crossed from one edge to the other by a periodic row of small holes. Boundary layers of two types, namely the exponential and power-law ones, described by solutions of problems in a strip with a single hole and in a half-plane with a periodic family of holes of unit size, give rise to a new effect of appearance of singularities in higher-order asymptotic terms of “smooth” type at the point where the perforation line crosses the boundary. We derive estimates for asymptotic remainders.
The article discusses two approaches to describing the properties of solutions of ordinary differential equations, one by J. Yorke and the other by V.V. Filippov. It is shown that both approaches describe the same objects.
We study an equation that occurs in geophysical hydrodynamics and contains the second time derivative and a quadratic nonlinearity of the Monge–Ampère type in two spatial variables. A group analysis of this strongly nonlinear partial differential equation with three independent variables is carried out. A fourteen-parameter transformation that preserves the form of the equation is found. Some one-dimensional reductions permitting one to obtain self-similar and other invariant solutions that satisfy ordinary differential equations are described. New noninvariant solutions with additive, multiplicative, generalized, and functional separation of variables are obtained. Two-dimensional symmetry and nonsymmetry reductions leading to simpler partial differential equations with two independent variables are considered (including stationary Monge–Ampère type equations and linear wave equations).
The paper considers a spectral problem on a graph equipped with a differential operator. We consider a star-graph consisting of three edges, with a Sturm–Liouville operator defined on each edge. We consider Dirichlet boundary conditions at the free endpoints and the continuity and Kirchhoff conditions at the common vertex. The potential in the Sturm–Liouville problem is assumed to be complex–valued and singular: it is the generalized derivative of a square-summable function. For the case of identical potentials on all edges of the graph, the Riesz basis property for the system of eigenfunctions and associated functions is proved. In the general case, the Riesz basis property for their two-dimensional subspaces is established.
The global existence of a weak solution of the boundary value problem for a stationary model of magnetohydrodynamics of a viscous heat-conducting fluid, generalizing the Boussinesq approximation, is proved. A maximum-and-minimum principle for the temperature is established.
We consider the properties of a special class of boundary value problems on an interval with Dirichlet type boundary conditions for a linear system of ordinary differential equations with variable coefficients. These boundary value problems arise when calculating the plasma output parameters in stationary plasma thrusters (SPT), which are currently widely used to correct spacecraft orbits. In the approximation of incompressible plasma in SPT, with dissipative factors (conductivity; hydrodynamic viscosity and thermal conductivity of electrons and ions; and temperature relaxation) taken into account, the system in question is of the eighth order. In physically important cases, theorems on the existence and uniqueness of a solution are proved for the boundary value problems posed, Green’s functions are constructed, and two successive approximation methods converging to the solution of the boundary value problem are proposed. Finally, we construct efficient computational algorithms based on the finite difference method, which permit practically finding solutions of the boundary value problems in question.
We consider problems of local bifurcations in neighborhoods of equilibria of nonlinear continuous-discrete-time (hybrid) systems depending on a scalar parameter μ . The main attention is paid to the discussion of bifurcations of codimension 2; along with the parameter μ , the value h characterizing the discretization step of the system is considered as a bifurcation parameter. We discuss conditions on the coefficients of the system that lead to necessary and sufficient tests for the emergence of h -periodic solutions (bifurcation of a multiple equilibrium) and 2h -periodic solutions (period-doubling bifurcation) in the hybrid system. To study bifurcation problems, we make a transition from the hybrid system to a discrete-time system equivalent in the natural sense. The dynamics of the discrete-time system is analyzed, which permits one to study the properties of bifurcations in the original hybrid system. The results are illustrated by examples.
Based on a constructive regularization method, we establish sufficient conditions for the unique solvability of the Vallée-Poussin boundary value problem for a generalized nonlinear second-order matrix Lyapunov equation. An iterative classical algorithm for constructing the solution is developed.
The problem of reconstructing an unknown disturbance in a nonlinear system consisting of a combination of differential and algebraic equations is considered. Two cases are discussed. In the first case, the disturbance occurs in the system linearly, and in the second case, nonlinearly. In the case of linearity, the problem has two specific features. First, it is assumed that only part of phase coordinates of the system (namely, the coordinates described by the differential equation) is inaccurately measured at discrete times. Second, it is only known about the disturbance acting on the system that it is an element of the space of square integrable functions; i.e., it can be unbounded. These assumptions imply the impossibility of exact reconstruction. Taking into account this peculiarity, we construct a solving algorithm, which is stable with respect to information noises and computational errors. This algorithm is based on a combination of elements of the theory of ill-posed problems and the extremal shift method well known in the theory of positional differential games. A similar algorithm is designed for the general case in which the disturbance occurs in the system nonlinearly.
A new boundary value problem for the stationary heat and mass transfer equations with variable coefficients is considered. It is assumed that the leading viscosity, thermal conductivity, and diffusion coefficients, as well as the buoyancy force, occurring in the equations depend on the temperature and the concentration of the substance dissolved in the base medium. A mathematical technique based on a variational approach is developed to study this boundary value problem. This technique is used to prove the global existence of a weak solution of the boundary value problem and establish sufficient conditions on the problem data ensuring the local uniqueness of a weak solution under the additional condition that the temperature and concentration are smooth.
The article studies the problem of approximate online reconstruction of an unknown disturbance acting on a system described by ordinary differential equations. To solve the problem under the assumption that some of the system coordinates are measured inaccurately, we propose an algorithm based on a combination of feedback control methods and methods of the theory of ill-posed problems. The convergence of the approximations constructed by the algorithm to the exact disturbance is established under an appropriate matching of the measurement error and computational grids selected accordingly.
The article is devoted to solving the problem of controlling a system defined by ordinary differential equations with a nonlinear right-hand side, which has the property of quasi-monotonicity with respect to offdiagonal entries. The equations also contain control parameters and uncertainties (errors), whose possible values must satisfy some pointwise constraints. The problem of control over a finite time interval is considered in order to transfer the state of the system to a given target set. The current state of the system is unknown, and for the formation of a control strategy, only a priori estimates of the initial state are available, as well as the results of incomplete and inaccurate measurement results received online. To solve the problem, a well-known general scheme is used, according to which it is necessary to consistently solve three subproblems: the approximate construction of information sets of the system, solvability sets, and, finally, the control synthesis problem. In the present paper, this general scheme is successfully implemented for the special class of nonlinear systems under consideration. Theorems on external interval estimates of information sets, internal estimates of solvability sets, and on sufficient conditions for the solvability of the control problem are proved. Formulas for feedback control are obtained, depending on the so-called generalized position, formed on the basis of available information about the system and measurement results. The possibility of applying the theoretical results obtained to solve specific control problems is confirmed by the model example analyzed in the paper.
The time evolution of a system of small perturbations imposed on a triaxial homogeneous spreading–drain in an infinite three-dimensional space of a Newtonian incompressible fluid is investigated. In the first part of the paper, it is assumed that the main motion is stationary and the velocity field is defined by only two constants. In this case, the linearized problem for the velocity and pressure perturbations is reduced to a spectral problem in which the real part of the spectral parameter is related to the nature of the exponential decay or growth of the initial perturbations. Based on the method of integral relations for quadratic functionals, an upper bound for this parameter is obtained. In the second part of the paper, a more general case of unsteady triaxial spreading–drain is considered. For the perturbation growth, we derive an upper integral bound that includes a time function completely determined by the velocity field of the main fluid flow.
In this paper we give a simple proof of the lipschitz-regularity of the p-harmonic functions using the De Giorgi-Moser iterative techniques.
Asymptotic observer design problem is considered for an uncertain dynamical system. A method permitting one to apply a previously proposed cascading observer to a wider class of dynamical systems is developed. More specifically, the requirements of controllability of the system matrices and stability of the zero dynamics are lifted.
