Boundary-value problems for an elliptic equation with complex coefficients and a certain moment problem

Elliptic systems of two second-order equations, which can be written as a single equation with complex coefficients and a homogeneous operator, are studied. The necessary and sufficient conditions for the connection of traces of a solution are obtained for an arbitrary bounded domain with a smooth boundary. These conditions are formulated in the form of a certain moment problem on the boundary of a domain; they are applied to the study of boundary-value problems. In particular, it is shown that the Dirichlet problem and the Neumann problem are solvable only together. In the case where the domain is a disk, the indicated moment problem is solved together with the Dirichlet problem and the Neumann problem. The third boundary-value problem in a disk is also investigated.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 11, pp. 1476–1483, November, 1993.     

The Augmented Scattering Matrix and Exponentially Decaying Solutions of an Elliptic Problem in a Cylindrical Domain

The self-adjoint elliptic boundary-value problem in a domain with cylindrical outlets to infinity is considered. The notion of an augmented scattering matrix is introduced on the basis of artificial radiation conditions. Properties of the augmented scattering matrix are studied, and the relationship with the classical scattering matrix is demonstrated. The central point is the possibility of calculating the number of linearly independent solutions of a homogeneous problem with fixed rate of decrease at infinity by analyzing the spectrum of the augmented scattering matrix. This property is applied to the problem on diffraction on a periodic boundary as an example. Bibliography: 21 titles.     

A mapping finite difference model for infinite elastic media

A simple mapping finite difference model is presented for the solution of boundary-value problems in the theory of time-harmonic elastic vibrations. The finite problem domain is condensed by mapping into a smaller finite domain using a suitable coordinate transformation. The field equations and the boundary conditions are also appropriately transformed. The radiation condition at infinity is satisfied through a change of the dependent variable. Finite difference forms of the transformed equations are then solved in the mapped domain, subject to the transformed boundary conditions.     

Some relationships between Green's functions and invariant imbedding

The method of invariant imbedding has been used to resolve the solution of linear two-point boundary-value problems into contributions associated with the homogeneous equation with homogeneous boundary conditions, with inhomogeneous boundary conditions, and with an inhomogeneous source term in the equation. The relationship between the Green's function and the invariant imbedding equations is described, and it is shown that the Green's function can be determined from an initial-value problem. Several numerical examples are given which illustrate the efficacy of the initial-value algorithm.This work was supported by the US Atomic Energy Commission.     

On general boundary-value problems for nonlinear elliptic equations of second order in a multiply connected plane domain

This paper deals with some general irregular oblique derivative problems for nonlinear uniformly elliptic equations of second order in a multiply connected plane domain. Firstly, we state the well-posedness of a new set of modified boundary conditions. Secondly, we verify the existence of solutions of the modified boundary-value problem for harmonic functions, and then prove the solvability of the modified problem for nonlinear elliptic equations, which includes the original boundary-value problem (i.e. boundary conditions without involving undertermined functions data). Here, mainly, the location of the zeros of analytic functions, a priori estimates for solutions and the continuity method are used in deriving all these results. Furthermore, the present approach and setting seems to be new and different from what has been employed before.The research was partially supported by a UPGC Grant of Hong Kong.     

On the uniqueness of solutions to some boundary-value problems for differential equations in a domain with algebraic boundary

Classes of differential equations with constant coefficients admitting unique solutions of Dirichlet and Cauchy boundary-value problems are considered in a bounded domain with algebraic boundary. For the Dirichlet problem in a ball, the necessary and sufficient conditions for the uniqueness of the solution are obtained in the form of a countable sequence of inequalities polynomial in the coefficients of the equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 7, pp. 898–906, July, 1993.     

The method of homogeneous solutions in an axisymmetric problem of statics for a finite cylinder with homogeneous conditions on the ends

Using the property of Papkovich generalized orthogonality of eigenfunctions, we develop a method of satisfying the boundary conditions on the lateral surface of a cylinder. The stresses and displacements in a finite cylinder with homogeneous conditions on the ends are represented in terms of the axial displacement. The solution is constructed as an expansion in a series of eigenfunctions of the corresponding homogeneous boundary-value problem. We find a class of boundary conditions that admits a solution of the problem without reduction to an infinite system of algebraic equations. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 135–139.     

A modification of an approach to the solution of the Hilbert boundary-value problem for an analytic function in a multiconnected circular domain

We propose a modification of the approach proposed by us in Russian Mathematics (Iz. VUZ) 44 (2), 58–62 (2000) for the solution of the Hilbert boundary-value problem for an analytic function in a multiconnected circular domain. This approach implies the solution of the corresponding homogeneous problem including the determination of an analytic function from the known boundary values of its argument in a circular domain.     

On the Global-in-Time Existence of a Generalized Solution to a Free-Boundary Problem

A problem with free (unknown) boundary for a one-dimensional diffusion-convection equation is considered. The unknown boundary is found from an additional condition on the free boundary. By the extension of the variables, the problem in an unknown domain is reduced to an initial boundary-value problem for a strictly parabolic equation with unknown coefficients in a known domain. These coefficients are found from an additional boundary condition that enables the construction of a nonlinear operator whose fixed points determine a solution of the original problem.

     

Numerical solution of the nonlinear seepage problem in a domain with an unknown boundary

A finite-element algorithm is developed for the problem of headless steady nonlinear seepage (boundary-value problem for a nonlinear elliptic equation in a domain with an unknown boundary) in a multicomponent medium with a piecewise-linear boundary. Numerical solution results are reported for a number of problems. The effects of the form of the nonlinearity on the characteristics of the seepage process are considered.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 61, pp. 83–90, 1987.     

Sinc-Galerkin method for solving linear sixth-order boundary-value problems

There are few techniques available to numerically solve sixth-order boundary-value problems with two-point boundary conditions. In this paper we show that the Sinc-Galerkin method is a very effective tool in numerically solving such problems. The method is then tested on examples with homogeneous and nonhomogeneous boundary conditions and a comparison with the modified decomposition method is made. It is shown that the Sinc-Galerkin method yields better results.

     

Necessary conditions for a domain optimization problem with various constraints

In this paper, a scheme for obtaining the necessary conditions in a wide class of domain optimization problems is given. In the case of a linear boundary-value problem of the Dirichlet type, necessary conditions are given.     

Perturbation of a Two-Point Problem

We investigate the problem of the effect of integral terms in boundary conditions on the well-posedness of nonlocal boundary-value problems for partial differential equations.     

Obtaining starting values for the shooting method solution of a class of two-point boundary-value problems

A method based on matching a zero of the right-hand side of the differential equations, in a two-point boundary-value problem, to the boundary conditions is suggested. Effectiveness of the procedure is tested on three nonlinear, two-point boundary-value problems.     

A robin-neumann problem in bounded domains with splitting boundary

We consider a mixed boundary-value problem for the homogeneous Laplace equation in a bounded domain which boundary splits up into two disjoint smooth components. On the one boundary component we pose a homogeneous Robin condition and an inhomogeneous Neumann condition on the other. We give a weak formulation, interpret this problem as a generalized spectral (eigenvalue) problem in the sense of F.Stummel (cf.[12]) and investigate existence, uniqueness and regularity of weak solutions. This problem is a cut-off version of a basic problem in water-wave theory (cf.Ramm [8], pp.394-395, Simon/Ursell [10] Stoker [11])     

Conditions of regularity of a general differential boundary-value problem for improperly elliptic equations

The Fredholm property and well-posedness of a general differential boundary-value problem for a general improperly elliptic equation are analyzed in a two-dimensional bounded domain with smooth boundary.     

The construction of boundary analogues of variational methods to approximate weak solutions of boundary-value problems in the theory of elasticity

Methods of approximating weak solutions of certain boundary-value problems in the theory of elasticity are proposed based on expanding the approximate solution in a finite series in basis functions which identically satisfy a homogeneous differential equation in the domain. The coefficients of the expansion are found by constructing a boundary analogue of the method of least squares (BAMLS). It is proved that the approximate solution thus obtained converges to a weak solution of the problem. Sufficient conditions for the stability of the BAMLS, easily verifiable by computational means, are derived. The construction of a boundary analogue of the collocation method (BACM) is proposed on the basis of the BAMLS, combined with discretization of the scalar product by quadrature formulae. The BACM obtained is convergent and stable and possesses better computational properties than the BAMLS.     

Multipoint Boundary-Value Solution of Two-Point Boundary-Value Problems

An iterative scheme, in which two-point boundary-value problems (TPBVP) are solved as multipoint boundary-value problems (MPBVP), which are independent TPBVPs in each iteration and on each subdomain, is derived for second-order ordinary differential equations. Several equations are solved for illustration. In particular, the algorithm is described in detail for the first boundary-value problem (FBVP) and second boundary-value problem (SBVP). A possible extension to higher-order BVPs is discussed briefly. The procedure may be used when the original TPBVP cannot be solved (does not converge) in a single long domain. It is suitable for implementation on computers with parallel processing. However, that issue is beyond the scope of this paper. The long domain is cut into a large number of subdomains and, based on assumed boundary conditions at the interface points, the resulting local BVPs are solved by any convenient conventional method. The local solutions are then patched by using simple matching formulas, which are derived below, rather than solving large systems of algebraic equations, as it is done in similar existing methods. Assuming that the local solutions are obtained by the most efficient methods, the overall convergence speed depends on the speed of matching. The proposed matching algorithm is based on a fixed-point iteration and has only a linear convergence rate. The rate can be made quadratic by applying standard accelerating schemes, which is beyond the scope of this article.     

Solutions in Hölder spaces of boundary-value problems for parabolic equations with nonconjugate initial and boundary data

The first and second one-dimensional boundary-value problems for parabolic equations are investigated in the case where the conjugation conditions for all required orders are not satisfied. The existence and uniqueness are proved. Estimates of solutions in classical and weighted Hölder spaces are obtained. We prove that the violation of conjugation for the given functions on the boundary of the domain at the initial-time moment causes the appearance of singular solutions. The order of singularity (as a power of t) is found for the singular solutions for t = 0.     

A problem of the bending of a plate for a doubly connected domain with a partially unknown boundary

The problem of the bending of an isotropic elastic plate, bounded by two rectangles with vertices lying on the same half-line, drawn from the common centre, is considered. The vertices of the inner rectangle are cut by convex smooth arcs (we will call the set of these arcs the unknown part of the boundary). It is assumed that normal bending moments act on each rectilinear section of the boundary contours in such a way that the angle of rotation of the midsurface of the plate is a piecewise-constant function. The unknown part of the boundary is free from external forces. The problem consists of determining the bending of the midsurface of the plate and the analytic form of the unknown part of the boundary when the tangential normal moment acting on it takes a constant value, while the shearing force and the normal bending moments and torques are equal to zero. The problem is solved by the methods of the theory of boundary-value problems of analytical functions.