Variational formulations of nonlinear boundary-value problems with a free boundary in the theory of interaction of surface waves with acoustic fields

Variational problems equivalent to nonlinear evolutionary boundary-value problems with a free boundary are formulated. These problems arise in the theory of interaction of limited volumes of liquid, gas, and their interface with acoustic fields. It is proved that the principle of separation of motions can be applied to these variational problems. The problem of a capillary-acoustic equilibrium form is given in a variational formulation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 12, pp. 1642–1652, December, 1993.     

The equilibrium boundary element method in boundary-value problems of elasticity theory

An equilibrium boundary element method is proposed for solving boundary-value problems in the theory of elasticity, thermo-elasticity, the dynamical theory of elasticity, bar torsion calculations, and the bending of a plate. The idea is to use simultaneously the method of constructing bundles of functions which exactly satisfy the equilibrium equations, the boundary variational equations of mechanics, and the methods of discrete finite-element approximation. The variational method of constructing the resolving boundary equations ensures that the linear system is symmetric and easily coupled to the finite-element method. Since volume integrals are eliminated the dimensions of the problem are reduced by one, but, unlike the boundary element method, there is no need to know the fundamental solutions. The solution of some bar torsion and plate bending problems confirms the high numerical efficiency of the method.     

A class of boundary-value problems for a system of nonlinear equations with a free boundary

The method of fixed domains is applied to derive conditions which ensure unique solvability of boundary-value problems for a system of nonlinear evolution equations with a partially free boundary.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 65, pp. 83–92, 1988.     

Analytic solution of boundary-value problems for nonstationary model kinetic equations

A theory for constructing the solutions of boundary-value problems for nonstationary model kinetic equations is constructed. This theory was incorrectly presented in the recent well-known monographs of Cercignani [1,2]. After application of a Laplace transformation to the studied equation, separation of the variables is used, this leading to a characteristic equation. Eigenfunctions are found in the space of generalized functions, and the eigenvalue spectrum is investigated. An existence and uniqueness theorem for the expansion of the Laplace transform of the solution with respect to the eigenfunctions is proved. The proof is constructive and gives explicit expressions for the expansion coefficients. An application to the Rayleight problem is obtained, and the corresponding result of Cercignani is corrected.Moscow Pedagogical University. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 92, No. 1, pp. 127–138, July, 1992.     

To the theory of boundary-value problems for elliptic equations with superposition operators in the boundary condition. I

We study the existence, uniqueness, and constant sign property of classical solutions to a nonlocal boundary-value problem for a second-order elliptic equation in a bounded domain of the Euclidean space. Using the system of maps that define superposition operators, we construct some subset of the domain boundary and establish the connection between the solvability of the problem under consideration and the solvability of the boundary value equation on the constructed subset.     

Time-space boundary element formulation for boundary-value problems

A time-space boundary element formulation is presented for the boundary-value problems with a governing equation expressed in terms of a certain type of linear operator. The boundary as well as the time domain are divided into a finite series of time-space elements, and the interpolation functions in time and space are introduced in order to construct a final discretized equation for the assembled system. The solving scheme is discussed, and the relation to the practical engineering problems is shown.     

Regularity up to the boundary of solutions to boundary-value problems of the theory of generalized Newtonian liquids

Boundary-value problems describing the stationary flow of a generalized Newtonian liquid are considered. The regularity of solutions to such problems is studied near the boundary. The W ₂ ² -estimate for a solution and the partial regularity of the strain velocity tensor are established. In the two-dimensional case, the complete regularity of the strain velocity tensor is also proved. Bibliography: 12 titles. Translated fromProblemy Matematicheskogo Analiza, No. 16. 1997, pp. 239–265.     

Inverse problems of finding the boundary condition in the theory of propagation of nonstationary waves. II

The problem of finding an unknown coefficient in the boundary condition of the third kind on the basis of certain information on Green's function is considered. Some problems more complicated than those in paper I are solved.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 51, pp. 129–133, 1975.     

Fully discrete approximations of parabolic boundary-value problems with nonsmooth boundary data

We study a numerical scheme for the approximation of parabolic boundary-value problems with nonsmooth boundary data. This fully discrete scheme requires no boundary constraints on the approximating elements. Our principal result is the derivation of optimal convergence estimates in L_p[0,T; L₂()] norms for boundary data in L_p[0, T; L₂()], 1p . For the same algorithms, we also show that the convergence remains optimal even in higher norms. The techniques employed are based on the theory of analytic semigroups combined with singular integrals.This paper was written in 1990, when the author was in the Department of Mathematical Sciences, University of Cincinnati. A preliminary version of this research was presented at the SIAM Annual Meeting in July 1989.     

Limit theorems in the theory of multipoint boundary-value problems

We present a reduction of a countable system of differential equations with countably-point boundary conditions to the case of a finite-dimensional multipoint boundary-value problem. We separately consider the case of a linear system. Kamenets-Podol'sk Pedagogic University, Kamenets-Podol'sk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 4, pp. 519–531, April, 1999.     

Application of the collocation method to multipoint boundary-value problems with integral boundary conditions

An algebraic collocation method for approximating solutions of systems of nonlinear ordinary differential equations is shown to be applicable in the case of linear multipoint boundary conditions containing definite integrals.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 11, pp. 1548–1555, November, 1992.     

A version of the asymptotic theory of the unsteady free interaction of a boundary layer with a transonic flow

The non-linear theory of strongly perturbed flows, with a structure of the velocity fields which is characteristic of domains of so-called free interaction of the boundary layer with an external potential flow, is considered. The specific details of the transonic flow show up not only in the estimates of the amplitudes and lengths of the perturbation waves in the asymptotic analysis of the problem but, also, in the fact that the motion may turn out to be simultaneously unsteady in the part of the boundary layer close to the wall and in the external potential flow. This mechanism of the evolution of the perturbations can be described by a single integro-differential equation which, assuming that the structure of the fluctuating fields is a quadrideck structure, is derived using the Fourier-Laplace method. Examples of its non-linear solutions are given in the form of solitary or periodic waves.     

The constructive investigation of boundary-value problems with approximate satisfaction of boundary conditions

We consider linear boundary-value problems for systems of functional differential equations when the number of boundary conditions is greater than the dimension of the system. We allow the boundary conditions to be fulfilled approximately. We propose an approach based on theorems whose conditions allow the verification by special reliable computing procedures.     

A time-variant version of the commutant lifting theorem and nonstationary interpolation problems

This paper contains a generalization of the commutant lifting theorem to a time-variant setting. The main result, which is called the three chains completion theorem, is used to solve various nonstationary norm constrained interpolation problems.     

Fully discrete Galerkin approximations of parabolic boundary-value problems with nonsmooth boundary data

Summary In this paper we study the convergence properties of a fully discrete Galerkin approximation with a backwark Euler time discretization scheme. An approach based on semigroup theory is used to deal with the nonsmooth Dirichlet boundary data which cannot be handled by standard techniques. This approach gives rise to optimal rates of convergence inL _p[O,T;L ₂()] norms for boundary conditions inL _p[O,T;L ₂()], 1p.     

Solvability of some boundary-value problems for polyharmonic equation with Hadamard-Marchaud boundary operator

We investigate solvability conditions for certain non-classical boundary-value problems for the polyharmonic equation. As the boundary operators we consider fractional differential operators in the sense of Hadamard-Marchaud. The considered problems generalize well-known Dirichlet and von Neumann boundary-value problems for boundary operators of fractional type.