A Riemann-Hilbert boundary value problem in neutron transport theory

Summary A Riemann-Hilbert boundary value problem, occurring in neutron transport theory, is presented and solved in this note for the case of an infinite straight line and zero index.

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Sommario Viene presentato e risolto in questa nota un problema al contorno di Riemann-Hilbert di indice zero e relativo ad un intervallo infinito. Esso ha origine da un problema di diffusione di neutroni, studiato nell'ambito della teoria del trasporto.

     

A boundary value problem associated with the second painlevé transcendent and the Korteweg-de Vries equation

The differential equation considered is \(y'' - xy = y|y|^\alpha \) . For general positive α this equation arises in plasma physics, in work of De Boer & Ludford. For α=2, it yields similarity solutions to the well-known Korteweg-de Vries equation. Solutions are sought which satisfy the boundary conditions (1) y(∞)=0 (2) (1) $$y{\text{(}}\infty {\text{)}} = {\text{0}}$$ (2) $$y{\text{(}}x{\text{) \~( - }}\tfrac{{\text{1}}}{{\text{2}}}x{\text{)}}^{{{\text{1}} \mathord{\left/ {\vphantom {{\text{1}} \alpha }} \right. \kern-\nulldelimiterspace} \alpha }} {\text{ as }}x \to - \infty $$ It is shown that there is a unique such solution, and that it is, in a certain sense, the boundary between solutions which exist on the whole real line and solutions which, while tending to zero at plus infinity, blow up at a finite x. More precisely, any solution satisfying (1) is asymptotic at plus infinity to some multiple kA i(x) of Airy's function. We show that there is a unique k^*(α) such that when k=k^*(α) the condition (2) is also satisfied. If 0 ^*, the solution exists for all x and tends to zero as x→-∞, while if k>k ^* then the solution blows up at a finite x. For the special case α=2 the differential equation is classical, having been studied by Painlevé around the turn of the century. In this case, using an integral equation derived by inverse scattering techniques by Ablowitz & Segur, we are able to show that k^*=1, confirming previous numerical estimates.     

On the first boundary value problem for the classical theory of elasticity in a three-dimensional domain with a singular boundary

An existence and uniqueness theorem is proved for the first boundary value problem of classical elasticity, relating to a broad class of three-dimensional domains whose boundaries may have edges and vertexes.The qualitative properties, up to the boundary, of the solution are investigated.

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Résumé On démontre un théorème d'existence et unicité pour le premier problème de valeurs au contour de l'élasticité classique, concernant une large classe de domaines de l'éspace à trois dimensions, avec des frontières présentant des arètes et des sommets.On étudie les propriétés qualitatives, jusque'au contour, de la solution.

     

On the boundary value problem for an elastic cylinder with homogenized fuzzy-cracks

Summary An idea of homogenized fuzzy-cracks (HFC) as a model of the classical cracks in a solid body is introduced. A boundary value problem for an elastic cylinder with HFC is presented and analyzed under the assumptions of axisymmetry and generalized plane strain. Among others, the convergence of an iterative algorithm to the solution of the boundary value problem is shown.

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Zum Bandwertproblem für einen elastischen Zylinder mit homogenisierten Fuzzy-Cracks

Übersicht Der Begriff der homogenisierten Fuzzy-Cracks (HFC) wird als Modell für die klassischen Risse in einem Festkörper eingeführt. Ein Randwertproblem für einen elastischen Zylinder mit Fuzzy-Cracks wird dargestellt und gelöst, wobei Axialsymmetrie und verallgemeinerter ebener Verzerrungszustand angenommen werden. Unter anderem wird die Konvergenz eines Iterationsverfahrens für die Lösung des Randwertproblems gezeigt.

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Former with: Department of Fuel Element Analysis Nuclear Research Institute CS-25068 Rez near Prague Czechoslovakia     

The boundary value problem for the free oscillations of an ideal rotating liquid

We consider the problem, in its linear formulation, of the free oscillations of an ideal incompressible liquid, partially or completely filling a rotating cavity in a field of body forces. The unperturbed motion is the rotation of the liquid-cavity system at constant angular velocity as a solid body. The problem of determining the frequency spectrum is reduced to the solution of a boundary value problem for the eigenvalues, which can be solved analytically or numerically. Particular cavity shapes and types of motion are discussed for which exact and approximate solutions of the boundary value problem are obtained. The results of numerical computations are given.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 4, pp. 81–88, July–August, 1973.In conclusion, the author wishes to thank G. S. Narimanov, B. I. Rabinovich, and V. M. Rogovoi for valuable advice, and also I. A. Artemov who gave great help with the calculations.     

On the initial-boundary value problem for the Boltzmann equation

A theorem on the traces of the solutions of initial-boundary value problems for the Boltzmann equation is proved. This result makes it possible to extend a recent theorem of existence proved by Hamdache to more realistic situations.     

On a boundary layer problem for the nonlinear Boltzmann equation

This article deals with a boundary-layer problem arising in the kinetic theory of gases when the mean free path of molecules tends to zero. The model considered here is the stationary, nonlinear Boltzmann equation in one dimension with a slightly perturbed reflection boundary condition. We restrict our attention to the case of hard spheres collisions, with Grad's cutoff assumption. Existence, uniqueness and asymptotic behavior are derived by means of energy estimates.     

A variational formulation for a boundary value problem considering an electro-sensitive elastomer interacting with two bodies

We present a variational formulation for an electro-elastic body in contact with two semi-infinite rigid bodies, which are electric conductors and have a distribution of free charge. These three bodies are surrounded by free space, where far away we have a given electric displacement and an electric potential on disjoint surfaces.