Scalar Boundary Value Problems Research Articles

Stable Asymptotics for Elliptic Systems on Plane Domains with Corners

We consider boundary value problems for elliptic systems in the sense of Agmon-Douglis-Nirenberg on plane domains with corners, where the domain, the coefficients of the operators and the right hand sides all depend on a parameter. We construct corner singularities in such a way that the corresponding decomposition of the solution into regular and singular parts is stable, i. e. the regular part and the coefficient of the singular functions depend smoothly on the parameter. The construction of these singular functions continues the paper [3] and generalizes results known for second order scalar boundary value problems — see [4,5] [12].

Saddle point theorem and nonlinear scalar Neumann boundary value problems

AbstractWe are concerned with existence results for nonlinear scalar Neumann boundary value problems u″ + g(x, u) = 0, u′(0) = u′(π) = 0 where g(x, u) satisfies Carathéodory conditions and is (possibly) unbounded. On the one hand we only assume that the function (sgn u)g(x, u) is bounded either from above or from below in some function space, and we impose conditions which relate the asymptotic behavior of the function (for¦u¦large) with the first two eigenvalues of the corresponding linear problem (here G(x, u) = is the potential generated by g). On the other hand we consider cases where the function (sgn u)g(x, u) is unbounded. The potential G(x, u) is not necessarily required to satisfy a convexity condition. Our method of proof is variational, we make use of the Saddle Point Theorem.

Numerical calculations of the optical propagation properties of the interconnections of symmetric multimode slab waveguides

The implementation of a program to compute the optical properties of the interconnection region of two multimode waveguides is discussed. The interconnection problem is described by a scalar boundary value problem defined for the TE polarized component of the electric field. Because of the particular design of the boundary value problem, the conditions at the open boundaries of the incoming and outgoing waveguides are relatively simple. A discretization is obtained using a hybrid system of finite element and boundary element methods. The method is illustrated by calculations on two interconnection configurations. Estimates of the accuracy of the results are given.

On the reduction of the factorization of matrix functions of daniele-khrapkov type to a scalar boundary value problem on a riemann surface

A survey of methods to construct the factorization of a class of non-rational 2 × 2-matrix functions isgiven This paper presents the reduction of those problems to a scalar Riemann boundary value probleem a Riemann surface.

A contribution to the scalar free boundary value problem of physical geodesy

A contribution to the scalar free boundary value problem of physical geodesy

On the subsonic stationary motion of stamps and flexible cover-plates on the boundary of an elastic half-plane and a composite plane

A mixed dynamic problem for an elastic half-plane on different sections of whose boundary shear and normal stresses and displacements are given simultaneously in four fundamental combinations is considered. It is assumed that all the sections move at an identical constant subsonic velocity along the half-plane boundary and their number and mutual arrangement are arbitrary. An analogous problem on the interaction of two half-planes of different materials (a composite plane) is examined under the formulation of six kinds of contact conditions simultaneously in two modifications. The solutions are constructed in quadratures on the basis of new representations of the complex Galin potentials /1/. The first problem is reduced to a scalar combined Hilbert-Riemann boundary value problem /2/ for a plane with slots, and the second to unrelated Hilbert-Riemann and Hubert problems for the same domain. Both problems of the theory of analytic functions are solved by a new method different from /2/. The problem of the wedging of a composite plane by a finite stamp moving at a sub-Rayleigh velocity /3/, and the problem of the motion of a stamp and a flexible cover plate over a half-plane boundary at subsonic velocity are examined as examples. The exact solutions of stationary contact problems for a half-plane with two kinds of boundary conditions were first obtained by Galin /1/. The problem was formulated for a composite plane with three kinds of boundary conditions, whose solution is obtained in quadratures in the case of one slipping section /4/. However, as shown in /3, 5/, the method described in /4/ does not result in an exact solution for a large number of sections.

Singular perturbation of boundary value problem for a vector fourth order nonlinear differential equation

We study the vector boundary value problem with boundary perturbations: $$\begin{gathered} \varepsilon ^2 y^{(4)} = f(x,y,y'',\varepsilon ,\mu ) (\mu< x< 1 - \mu ) \hfill \\ y(x,\varepsilon ,\mu )\left| {_{x = \mu } = A_1 (\varepsilon ,\mu ), y(x,\varepsilon ,\mu )} \right|_{x = 1 - \mu } = B_1 (\varepsilon ,\mu ) \hfill \\ y''(x,\varepsilon ,\mu )\left| {_{x = \mu } = A_2 (\varepsilon ,\mu ), y''(x,\varepsilon ,\mu )} \right|_{x = 1 - \mu } = B_2 (\varepsilon ,\mu ) \hfill \\ \end{gathered}$$ where yf, Aj andB j (j=1,2) are n-dimensional vector functions and e, μ are two small positive parameters. This vector boundary value problem does not appear to have been studied, although the scalar boundary value problem has been treated. Under appropriate assumptions, using the method of differential inequalities we find a solution of the vector boundary value problem and obtain the uniformly valid asymptotic expansions.

An approach to the scalar boundary value problem of physical geodesy by means of Nash-Hörmander theorem

An approach to the scalar boundary value problem of physical geodesy by means of Nash-Hörmander theorem

SOME ESTIMATIONS OF SOLUTION OF NONLINEAR BOUNDARY VALUE PROBLEM FOR SECOND ORDER SYSTEMS

We study the nonlinear boundary value problem: ey″=f(x, ɛ, y, y′), y(0, ɛ)=A(ɛ), y(1, ɛ)=B(ɛ)where ɛ>0 is a small parameter, y, f, A and B are n-dimensional vector functions. This vector nonlinear boundary value problem does not appear to have been studied, although the scalar boundary value problem has been treated extensively. Under appropriate assumptions, using method of differential inequality we can find a solution and estimate it as well.

The Numerical Solution of Second-Order Boundary Value Problems on Nonuniform Meshes

In this paper, we examine the solution of second-order, scalar boundary value problems on nonuniform meshes. We show that certain commonly used difference schemes yield second-order accurate solutions despite the fact that their truncation error is of lower order. This result illuminates a limitation of the standard stability, consistency proof of convergence for difference schemes defined on nonuniform meshes. A technique of reducing centered-difference approximations of first-order systems to discretizations of the underlying scalar equation is developed. We treat both vertex-centered and cell-centered difference schemes and indicate how these results apply to partial differential equations on Cartesian product grids.

A monotone method for constructing extremal solutions of second order scalar boundary value problems

A monotone iterative method is given for the construction of extremal solutions to second order scalar boundary value problems, starting from a pair of upper and lower solutions. The fully nonlinear case is considered.

The scalar boundary value problem of physical geodesy

The scalar boundary value problem of physical geodesy

The numerical solution of second-order boundary value problems on nonuniform meshes

In this paper, we examine the solution of second-order, scalar boundary value problems on nonuniform meshes. We show that certain commonly used difference schemes yield second-order accurate solutions despite the fact that their truncation error is of lower order. This result illuminates a limitation of the standard stability, consistency proof of convergence for difference schemes defined on nonuniform meshes. A technique of reducing centered-difference approximations of first-order systems to discretizations of the underlying scalar equation is developed. We treat both vertex-centered and cell-centered difference schemes and indicate how these results apply to partial differential equations on Cartesian product grids.

Boundary and Angular Layer Behavior in Singularly Perturbed Semilinear Systems

AbstractSome authors have employed the method and technique of differential inequalities to obtain fairly general results concerning the existence and asymptotic behavior, as ∊ → 0+, of the solutions of scalar boundary value problems∊y" = h(t,y), a &lt; t &lt; b,y(a,∊) = A, y(b,∊) = B.In this paper, we extend these results to vector boundary value problems, under analogous stability conditions on the solution u = u(t) of the reduced equation 0 = h(t,u).Two types of asymptotic behavior are studied, depending on whether the reduced solution u(t) has or does not have a continuous first derivative in (a,b), leading to the phenomena of boundary and angular layers.

Asymptotic structures in nonlinear dissipative and dispersive systems

Recent results in the asymptotic theory of ordinary differential equations are applied to certain scalar and vector boundary value problems that model various nonlinear dissipative and dispersive wave phenomena.

Practical stability test for finite elements with reduced integration

AbstractAn easy to use stability test is presented for finite elements which are under‐ or selectively‐integrated. This test is reminiscent of the patch test in that it must be applied only to one prototype element, and it is a sufficient test for the nonsingularity of the global coefficient matrix. The minimum number of integration points which can be used with each of several common one‐, two‐ and three‐dimensional elements for a variety of steady‐state and time‐dependent, scalar boundary value problems are given.

A semilinear parabolic system arising in the theory of superconductivity

arises in the theory of superconductivity of liquids proposed by Abrahams and Tsuneto in [ 11. System (1) has also been briefly treated as an example by Chueh, Conley and Smaller in (61. Throughout this paper we shall assume that D is a bounded domain in R” (n = 1,2 or 3) with smooth boundary X!. A denotes the Laplacian; all the results of the paper which hold for n > 1 also hold when A is replaced by any second-order symmetric elliptic differential expression with positive spectrum. Because of the very symmetric nature of (l), it is reasonable to expect solutions of (1) to have nice properties. We show in this paper that this is in fact the case. In Section 2 we give a complete description of steady state solutions of (1) satisfying Dirichlet or Neumann boundary conditions; we show that often all steady states are of the form u = u, t‘ = au for some constant a where u is the solution of a single differential equation. It is possible, using standard techniques for scalar nonlinear boundary value problems, to obtain lots of information about the solutions of this single equation and so about solutions of (1). In [6 1 it is shown that all solutions of (1) are attracted towards the set {(u, v): u* + v* < 1). In Section 3 we show that (1) has a Lyapunov function and use this fact and the standard theory of w-limit sets to prove that any solution of (1) tends, as t + 03, towards the family of all steady state solutions of (1). Often, in the case of scalar equations, it can be shown that

A Shooting Algorithm for the Best Least Squares Solution of Two-Point Boundary Value Problems

A shooting algorithm is presented for the best least squares solution (BLSS) of the linear two-point boundary value problem ${\bf y}' + A(t){\bf y} = {\bf g}(t)$, $M{\bf y}(a) + N{\bf y}(b) = {\bf d}$, in the noninvertible case. The initial vector of the BLSS is given by an explicit formula involving the generalized inverse of a characteristic matrix. The BLSS is obtained by solving $n + 1$ initial value problems, where n is the dimension of the system. The nth order scalar boundary value problem is also solved in general. In the invertible case, this method reduces to the Goodman–Lance method of adjoints, which gives the classical solution. The method is illustrated by an application to a noninvertible example.

Some disconjugacy criteria for self-adjoint linear differential equations

1. In this paper we obtain disconjugacy criteria for 2nth-order selfadjoint scalar differential equations by methods based on the relationship which exists between disconjugacy of self-adjoint equations and certain features of eigenvalue problems. This relationship was first mentioned and used in [8] to study the disconjugacy of second-order equations. Its use was later extended to the particular case of the fourth-order equation [3] and to second-order vector-matrix equations [12]. In Section 2 of the present paper, this relationship is established for the equations with which we are concerned, and it is then used in Section 3 in conjunction with the classical minimum properties of eigenvalues to obtain disconjugacy criteria involving arbitrary functions (of certain classes). By appropriate choice of these functions these criteria lead to explicit disconjugacy conditions. Although conditions in the literature [4, 5, IO, 1 I] are similar to some of these, the present conditions are new results or strengthened versions of known results. It is of interest to note that, in spite of the equivalence of the scalar boundary value problems considered in this paper to a system of two first-order matrix-matrix equations [4], the criteria which we shall obtain are not special cases of results in [I21 concerning disconjugacy problems for second-order vector-matrix differential equations. Indeed, the transformation of such scalar systems into vector-matrix form leads to boundary value problems which are quite different from those considered in [12].