On the convergence of solutions and eigenelements of a boundary value problem in a domain perforated along the boundary

We consider a boundary value problem in a model domain periodically perforated along the boundary. We assume that the homogeneous Neumann condition is posed on the external boundary and the homogeneous Dirichlet condition is posed on the boundary of the cavities. A limit (homogenized) problem is obtained. We prove the convergence of the solutions, eigenvalues, and eigenfunctions of the original problem to the solutions, eigenvalues, and eigenfunctions, respectively, of the limit problem.     

Existence and uniqueness of solutions of hyperbolic equations in divergence form with various boundary conditions on various parts of the boundary

We prove the existence and uniqueness of an energy class solution of an initial–boundary value problem for a semilinear equation in divergence form. We consider the case in which an inhomogeneous third boundary condition is posed on one part of the lateral surface of the cylinder in which the equation is studied and the homogeneous Dirichlet boundary condition is posed on the other part of the lateral surface.     

On an initial‐boundary value problem for a wide‐angle parabolic equation in a waveguide with a variable bottom

We consider the third‐order Claerbout‐type wide‐angle parabolic equation (PE) of underwater acoustics in a cylindrically symmetric medium consisting of water over a soft bottom B of range‐dependent topography. There is strong indication that the initial‐boundary value problem for this equation with just a homogeneous Dirichlet boundary condition posed on B may not be well‐posed, for example when B is downsloping. We impose, in addition to the above, another homogeneous, second‐order boundary condition, derived by assuming that the standard (narrow‐angle) PE holds on B, and establish a priori H² estimates for the solution of the resulting initial‐boundary value problem for any bottom topography. After a change of the depth variable that makes B horizontal, we discretize the transformed problem by a second‐order accurate finite difference scheme and show, in the case of upsloping and downsloping wedge‐type domains, that the new model gives stable and accurate results. We also present an alternative set of boundary conditions that make the problem exactly energy conserving; one of these conditions may be viewed as a generalization of the Abrahamsson–Kreiss boundary condition in the wide‐angle case. Copyright © 2008 John Wiley & Sons, Ltd.     

Optimal boundary displacement control at one end of a string with a medium exerting resistance at the other end

We study a problem of optimal boundary control of vibrations of a one-dimensional elastic string, the objective being to bring the string from an arbitrary initial state into an arbitrary terminal state. The control is by the displacement at one end of the string, and a homogeneous boundary condition containing the time derivative is posed at the other end. We study the corresponding initial-boundary value problem in the sense of a generalized solution in the Sobolev space and prove existence and uniqueness theorems for the solution. An optimal boundary control in the sense of minimization of the boundary energy is constructed in closed analytic form.     

Solvability of an inhomogeneous boundary value problem for the stationary magnetohydrodynamic equations for a viscous incompressible fluid

We study an inhomogeneous boundary value problem for the stationary magnetohydrodynamic equations for a viscous incompressible fluid corresponding to the case in which the tangential component of the magnetic field is specified on the boundary and the Dirichlet condition is posed for the velocity. We derive sufficient conditions on the input data for the global solvability of the problem and the local uniqueness of the solution.     

On the nonuniqueness of the solution of a mixed boundary value problem for the Laplace equation

We study the solvability of the mixed boundary value problem for the Laplace equation with three distinct boundary conditions, two of which include two directional derivatives with distinct tilt angles and the remaining one is the first boundary condition. An example of a nontrivial solution of the homogeneous problem is given, and conditions under which the problem has a unique solution are established. The solvability of the problem with a nonhomogeneous first boundary condition is studied.     

Determination of the number of an eigenvalue of a singular nonlinear self-adjoint spectral problem for a linear Hamiltonian system of differential equations

We suggest a method for determining the number of an eigenvalue of a self-adjoint spectral problem nonlinear with respect to the spectral parameter, for some class of Hamiltonian systems of ordinary differential equations on the half-line. The standard boundary conditions are posed at zero, and the solution boundedness condition is posed at infinity. We assume that the matrix of the system is monotone with respect to the spectral parameter. The number of an eigenvalue is determined by the properties of the corresponding nontrivially solvable homogeneous boundary value problem. For the considered class of systems, it becomes possible to compute the numbers of eigenvalues lying in a given range of the spectral parameter without finding the eigenvalues themselves.     

Quadratic functionals and nondegeneracy of boundary value problems on a geometric graph

Quadratic functionals defined on the space of functions differentiable on a geometric graph are considered. Analogs of the Lagrange and Dubois–Raymond lemmas are proved. Necessary extremum conditions for these quadratic functionals are obtained. A boundary value problem with conditions posed locally at the vertices of a geometric graph is shown to be selfadjoint if and only if it is generated by a quadratic functional. A subclass of quadratic energy functionals is singled out. The space of solutions of the homogeneous boundary value problem generated by a quadratic energy functional is described, and nondegeneracy criteria for such boundary value problems are derived.     

Stability estimates for solutions of control problems for the Maxwell equations with mixed boundary conditions

We consider extremal problems for the time-harmonic Maxwell equations with mixed boundary conditions for the electric field. Namely, the tangential component of the electric field is given on one part of the boundary, and an impedance boundary condition is posed on the other part. We prove the solvability of the original mixed boundary value problem and the extremal problem. We obtain sufficient conditions on the input data ensuring the stability of solutions of specific extremal problems under certain perturbations of both the performance functional and some functions occurring in the boundary value problem.     

On the solvability of a boundary value problem for the Laplace equation on a screen with a boundary condition for a directional derivative

We consider a three-dimensional boundary value problem for the Laplace equation on a thin plane screen with boundary conditions for the “directional derivative”: boundary conditions for the derivative of the unknown function in the directions of vector fields defined on the screen surface are posed on each side of the screen. We study the case in which the direction of these vector fields is close to the direction of the normal to the screen surface. This problem can be reduced to a system of two boundary integral equations with singular and hypersingular integrals treated in the sense of the Hadamard finite value. The resulting integral equations are characterized by the presence of integral-free terms that contain the surface gradient of one of the unknown functions. We prove the unique solvability of this system of integral equations and the existence of a solution of the considered boundary value problem and its uniqueness under certain assumptions.     

Convergence of solutions and eigenelements of Steklov type boundary value problems with boundary conditions of rapidly varying type

We consider boundary value problems for the Laplace operator in a domain with boundary conditions of rapidly varying type: the Dirichlet homogeneous condition and the third (Fourier) boundary condition or a Steklov type condition. We construct the limit (homogenized) problem and prove that solutions, eigenvalues, and eigenfunctions of the original problem converge respectively to solutions, eigenvalues, and eigenfunctions of the limit problem. Bibliography: 47 titles. Illustrations: 2 figures.     

A finite difference method for the wide‐angle “parabolic” equation in a waveguide with downsloping bottom

We consider the third‐order wide‐angle “parabolic” equation of underwater acoustics in a cylindrically symmetric fluid medium over a bottom of range‐dependent bathymetry. It is known that the initial‐boundary‐value problem for this equation may not be well posed in the case of (smooth) bottom profiles of arbitrary shape, if it is just posed e.g. with a homogeneous Dirichlet bottom boundary condition. In this article, we concentrate on downsloping bottom profiles and propose an additional boundary condition that yields a well‐posed problem, in fact making it L² ‐conservative in the case of appropriate real parameters. We solve the problem numerically by a Crank–Nicolson‐type finite difference scheme, which is proved to be unconditionally stable and second‐order accurate and simulates accurately realistic underwater acoustic problems. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Bifurcation approach to a logistic elliptic equation with a homogeneous incoming flux boundary condition

In this paper, we consider a semilinear elliptic boundary value problem in a smooth bounded domain, having the so-called logistic nonlinearity that originates from population dynamics, with a nonlinear boundary condition. Although the logistic nonlinearity has an absorption effect in the problem, the nonlinear boundary condition is induced by the homogeneous incoming flux on the boundary. The objective of our study is to analyze the existence of a bifurcation component of positive solutions from trivial solutions and its asymptotic behavior and stability. We perform this analysis using the method developed by Lyapunov and Schmidt, based on a scaling argument.     

On a boundary value problem of the theory of oscillations with parameter-dependent boundary conditions

A homogeneous second order differential equation with homogeneous boundary conditions dependent on the parameter, is investigated. Such an equation is obtained in the course of solution of the problem of characteristic oscillations of an ideal incompressible fluid in an elastic vessel, when the method of separation of variables is used. We prove the completeness of the system of eigenfunctions of our boundary value problem and we derive the expansion of an arbitrary, piecewise-continuous function into a series in terms of these eigenfunctions.     

Well-posedness and approximation of a measure-valued mass evolution problem with flux boundary conditions

This Note deals with imposing a flux boundary condition on a non-conservative measure-valued mass evolution problem posed on a bounded interval. To establish the well-posedness of the problem, we exploit particle system approximations of the mass accumulation in a thin layer near the active boundary. We derive the convergence rate for the approximation procedure as well as the structure of the flux boundary condition in the limit problem.     

On the rate of convergence for perforated plates with a small interior Dirichlet zone

The aim of the paper is to compare the asymptotic behavior of solutions of two boundary value problems for an elliptic equation posed in a thin periodically perforated plate. In the first problem, we impose homogeneous Dirichlet boundary condition only at the exterior lateral boundary of the plate, while at the remaining part of the boundary Neumann condition is assigned. In the second problem, Dirichlet condition is also imposed at the surface of one of the holes. Although in these two cases, the homogenized problem is the same, the asymptotic behavior of solutions is rather different. In particular, the presence of perturbation in the boundary condition in the second problem results in logarithmic rate of convergence, while for non-perturbed problem the rate of convergence is of power-law type.     

On the rate of convergence for perforated plates with a small interior Dirichlet zone

The aim of the paper is to compare the asymptotic behavior of solutions of two boundary value problems for an elliptic equation posed in a thin periodically perforated plate. In the first problem, we impose homogeneous Dirichlet boundary condition only at the exterior lateral boundary of the plate, while at the remaining part of the boundary Neumann condition is assigned. In the second problem, Dirichlet condition is also imposed at the surface of one of the holes. Although in these two cases, the homogenized problem is the same, the asymptotic behavior of solutions is rather different. In particular, the presence of perturbation in the boundary condition in the second problem results in logarithmic rate of convergence, while for non-perturbed problem the rate of convergence is of power-law type.     

Problem for the diffusion equation outside cuts on the plane with the Dirichlet condition and an oblique derivative condition on opposite sides of the cuts

We consider a boundary value problem for the stationary diffusion equation outside cuts on the plane. The Dirichlet condition is posed on one side of each cut, and an oblique derivative condition is posed on the other side. We prove existence and uniqueness theorems for the solution of the boundary value problem. We obtain an integral representation of a solution in the form of potentials. The densities of these potentials are found from a system of Fredholm integral equations of the second kind, which is uniquely solvable. We obtain closed asymptotic formulas for the gradient of the solution of the boundary value problem at the endpoints of the cuts.     

On the combination of some semi‐discretization methods and boundary integral equations for the numerical solution of initial boundary value problems

We consider initial boundary value problems for the homogeneous differential equation of hyperbolic or parabolic type in the unbounded two‐ or three‐dimensional spatial domain with the homogeneous initial conditions and with Dirichlet or Neumann boundary condition. The numerical solution is realized in two steps. At first using the Laguerre transformation or Rothe's method with respect to the time variable the non‐stationary problem is reduced to the sequence of boundary value problems for the non‐homogeneous Helmholtz equation. Further we construct the special integral representation for solutions and obtain the sequence of boundary integral equations (without volume integrals). For the full‐discretization of integral equations we propose some projection methods.     

A problem for the Helmholtz equation outside cuts on the plane with the dirichlet condition and the oblique derivative condition on opposite sides of the cuts

We consider a boundary value problem for the Helmholtz equation outside cuts on the plane. The Dirichlet condition is posed on one side of each cut, and an oblique derivative condition with pure imaginary coefficient of the tangential derivative is posed on the other side. We prove the uniqueness of the solution. The solvability of the problem is proved for the case in which the above-mentioned pure imaginary coefficient is less than unity in absolute value. In this case, we obtain an integral representation of the solution of the problem in the form of potentials. The densities of the potentials are found from a system of Fredholm integral equations of the second kind, which is uniquely solvable. The boundary value problem considered here generalizes the mixed Dirichlet-Neumann problem.