Showing papers on "Free boundary problem published in 1968"

Inverse problems in partial differential equations.

Abstract: : A procedure for identification in partial differential equations is described and illustrated by the Laplace equation and the unsteady heat conduction equation. The procedure for solution involves the substitution of difference operators for the partial derivatives with respect to all but one of the independent variables. The linear boundary value problem is solved by superposition of particular solutions. For non-linear boundary value problems which arise from the original form of the equation or from the identification procedure, a Newton-Raphson-Kantorovich expansion in function space is used to reduce the solution to an iterative procedure of solving linear boundary value problems. For the problems considered, this procedure has proven to be effective and results in a reasonable approximation to the solution of the boundary value problem in partial differential equations. For the identification problem, it is shown that the constant parameters are identified to the same accuracy as the supplementary data used in the identification procedure. (Author)

Boundary Control Systems

Abstract: Boundary conditions control systems described by partial differential equation in domain of Euclidean space analyzed for optimal controllability

Stability theory for difference approximations of mixed initial boundary value problems. I

Abstract: are given, u1 and u11 are defined according to the partition of A, i.e. u1 = (wU), • • -, um)', ulL = (w

Effect of Boundary Conditions on the Onset of Convection

Abstract: The effect of various boundary conditions on the onset of convection in a semi‐infinite fluid which has a time‐dependent temperature profile is investigated. The linearized hydrodynamic and thermodynamic equations are solved by expanding the variables in a series of orthonormal functions satisfying the boundary conditions and integrating the resulting set of coupled ordinary differential equations numerically. For a semi‐infinite fluid the initial behavior of the system is found to depend mainly on the Prandtl number and the manner in which the fluid is cooled or heated. The various combinations of rigid, free, conducting, and insulating boundary conditions only slightly affect the time of onset of manifest convective behavior and the horizontal wavenumber of the disturbances which are amplified the most at this time.

Adjoint and self-adjoint boundary value problems with interface conditions.

Abstract: In this paper the class of adjoint and in particular self-adjoint boundary value problems associated with ordinary linear differential equations is extended to include problems which may have discontinuities in the solution or some of its derivatives at a finite number of interior points.

One dimensional Stefan problems with nonmonotone free boundary

Abstract: problem The weak solution of [3] is a classical solution of the appropriate parabolic equation in every open set where u> 0 and a(u) >0 (or where uO and a(u) ? - a) However the set W where u = 0 and - a 1) W stays away from the fixed boundaries It is not known, for instance, whether or not W has a positive measure or interior points For the one-phase problem, however, it was proved in [3] that W is contained in the boundary of measurable point and, consequently, has no interior points Physically, Wrepresents the "weak" free boundary between the solid and the liquid of the problem The first purpose of the present paper is to prove that, in the case n = 1, W is actually a continuous curve x=s(t) (which is then the free boundary in the usual sense) and that u(x, t) = 0 at all the points where x = s(t) This result is stated in Theorem 3 below, and is proved in ??2, 3 The second purpose of the paper is to extend the results from two-phase problems to k-phase problems for any k This is done in ??5-7 The lines t=a where some of the phases degenerate cause some difficulty There may occur perhaps some intervals on t = ai where u(x, ai)_ 0 (see ?6) and regions lying above such intervals where u 0 (see ?7) But in any case we prove that W consists of a finite number of curves x = x(t) which are continuous in the intervals (at, ai + 1) As far as we know, global existence theorems of classical solutions for the twophase Stefan problem were considered only by Rubinstein [4] However, there is a fundamental gap in his proof of existence In ?8 we explain the nature of this gap Thus it is not known, at this point, whether classical solutions do in fact exist globally

Mathematical problems of boundary layer theory

Abstract: The equations of boundary layer theory were derived by Prandtl in 1904. These equations are the basis of boundary layer theory, which has been intensively developed for more than half a century and is one of the important branches of contemporary hydromechanics. Boundary layer theory is the subject of numerous monographs (see [3], [4] and others), written in the USSR and abroad, and also of a large collection of articles setting out the results of theoretical and experimental investigations. In connection with engineering problems (in particular, the important problem of determining the resistance of a medium to a body moving in it), the most important methods are numerical methods for solving Prandtl's system. Problems of the approximate solution of systems of boundary layer equations were investigated by Karman, Polhausen, Kochin, Dorodnitsyn, Loitsyanskii, and many others. The present paper deals with mathematical problems in boundary layer theory. One of these is the problem of conditions for which a solution of Prandtl's system exists. This mathematical problem is important, since in the boundary layer there may occur the so-called boundary layer separation, which leads to the appearance of singularities in the solutions of Prandtl's system and to their noncontinuability. In the paper we also discuss the problem of uniqueness and stability of solutions of Prandtl's system, the problem of unsteady flow in the boundary layer resulting in steady conditions as , we construct approximate solutions of Prandtl's system and we prove their convergence. The treatment of all these problems in this paper follows that of [7], [8]. Solutions of all principal two-dimensional problems in boundary layer theory are constructed here for steady and unsteady flow of an incompressible fluid. The corresponding uniqueness and stability theorems are proved for these solutions.

Solution of 2-dimensional field problems by boundary relaxation

Abstract: A numerical method is described for solution of 2-dimensional electric- and magnetic-field problems of the exterior type. Such problems are temporarily converted into interior problems by defining an arbitrary closed boundary, and then improving field values on the boundary iteratively, until a solution valid both within and without the artificial boundary is obtained. Within the boundary, the solution is found at a finite number of points, by any of the well known finite-difference methods. The final result is independent of the choice of artificial boundary, and corresponds exactly to the solution that would be obtained by applying finite-difference techniques to an infinite array of points. An empirical study of the convergence properties of this process is described, and typical computing speeds are indicated. Use of this method is illustrated by a variety of simple problems.

On the convergence of difference schemes approximating the second and third boundary value problems for elliptic equations

Abstract: THE present paper considers difference schemes approximating the second and third boundary value problems for a self-adjoint elliptic equation without mixed derivatives in a rectangular region, and represents a direct continuation of [1].

Boundary value problems in thin shallow shells of arbitrary plan form

Abstract: This paper is concerned with a general method of analysis of boundary-value problems in thin shallow shells of arbitrary plan form. Two specific shell configurations are considered. General solutions to the governing partial differential equations are obtained in complex form, containing a sufficient number of arbitrary elements to satisfy the four boundary conditions permitted by classical thin shell theory. An algorithm for the determination of these arbitrary elements from a general form of boundary condition is presented. The method of solution is based on I.N.Vekua's theory of elliptic partial differential equations.

Approximate solution of initial-boundary wave equation problems by boundary-value techniques

Abstract: A new boundary-value technique is proposed for the treatment of initial-boundary-value problems for linear and mildly non-linear wave equations. Several illustrative examples are offered to demonstrate the ease with which the method can be applied.