A steady-state electrochemical machining problem with a threshold current

A critical steady-state electrochemical machining problem in a periodic domain with a threshold current is considered. The existence and uniqueness of the classical solution and the regularity of the free boundary to the problem are proved.     

An optimization problem in hydrodynamic lubrication theory

If an arbitrary film profile is given, then we show how to find a film profile which will support more weight than the given profile. We do this for a discrete problem-our methods may be used on a digital computer-and we establish convergence results. The problem of finding a film profile which will support more weight than any other profile is also discussed.     

A scheme for the electrochemical machining of metals by a cathode tool with a curvilinear part of the boundary

The solution of a non-linear plane problem in the theory of the electrochemical machining of metals, associated with the determination of the shape of a surface (the anode) during its treatment with a cathode tool with a curvilinear part of the boundary, is obtained by methods developed for problems of jet flow past curvilinear obstacles. A condition is obtained which is identical to the well known Brillouin-Villat condition in fluid dynamics for smooth separation, the use of which enables one to determine the position of the transition point from the zone of steady treatment conditions into a region where the dissolution of the metal does not occur. The fixed shapes of the anode boundary are found for two cathode configurations.     

One inverse problem in the theory of liquid metals

The possibility of direct determination of the Fourier component of the model potential of the electron-ion interaction, as well as the polarization and structural characteristics of a degenerate electron liquid, based on the condition of the availability of experimental information about partial structural factors, is demonstrated using the model of a one-component plasma as an initial approximation. The possibility for self-consistent calculations of the volume and surface properties of liquid metals is also shown.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 105, No. 3, pp. 462–470, December, 1995.     

The Cauchy–Neumann problem for a multi‐dimensional isentropic hydrodynamic model for semiconductors

We investigate a multi‐dimensional isentropic hydrodynamic (Euler–Poisson) model for semiconductors, where the energy equation is replaced by the pressure–density relation p(n) . We establish the global existence of smooth solutions for the Cauchy–Neumann problem with small perturbed initial data and homogeneous Neumann boundary conditions. We show that, as t→+∞, the solutions converge to the non‐constant stationary solutions of the corresponding drift–diffusion equations. Moreover, we also investigate the existence and uniqueness of the stationary solutions for the corresponding drift–diffusion equations. Copyright © 2005 John Wiley & Sons, Ltd.     

A problem in elasticity theory

The problem of determining the dimensions of the transverse cross-sections of a beam from the given frequencies of its natural vibrations is examined. Frequency spectra are indicated that determine the dimenions of the transverse cross-sections of the beam uniquely, an effective procedure is presented for solving the inverse problem, and a uniqueness theorem is proved. The method of standard models /1/ is used to solve the inverse problem.     

A problem in prediction theory

Summary A new kind of ? prediction problem ? is defined and studied, first for certain special distributions and then generally. The general results are connected with conjugate trigonometric series. To Giovanni Sansone on his 70^th birth day. The authors acknowledge the support of theAlfred P. Sloan Foundation and the National Science Foundation, respectively.     

A problem in the theory of mixing layers

The classical problem of the free steady mixing layer which is formed as the result of the interaction between two parallel homogeneous flows which move with different velocities and come into contact in a certain section is considered. Subject to the additional condition that the first derivative of the solution in a class of self-similar functions is positive, a boundary-value problem is studied, for values of the self-similarity index m > 0, which describes the mixing of two viscous streams of the same fluid for m = 1 [1] and for m = 2 [2]. The method of investigation used [3–5] enables the third-order non-linear equation to be reduced to a first-order equation and enables the corresponding solutions (Gz) to be constructed in a parametric form as a function of the values of m. A knowledge of the behaviour of the velocity profile of the main stream can be used to investigate the flow stability. The results obtained form the basis of the subsequent construction of the solution of Lock's problem [6] and the investigation of the uniqueness of the solutions obtained.     

A theory for optimal regularization in the finite dimensional case

Consider the matrix problem $\text{Ax = y + ε =}\text{y}\text{?}$ in the case where A is known precisely, the problem is ill conditioned, and ε is a random noise vector. Compute regularized “ridge” estimates,$\text{x}\text{?}{}_{\mathrm{\lambda }}\text{= (A}{}^1\text{A + λI)}{}^{-1}\text{A}{}^1\text{y}\text{?}$,where ¹ denotes matrix transpose. Of great concern is the determination of the value of λ for which x?_λ “best” approximates $\text{x}{}_{\mathrm{<mtext>0</mtext>}}\text{= A}{}^{\mathrm{+}}\text{y}$. Let $\text{Q = 6}\text{x}\text{?}{}_{\mathrm{\lambda }}\text{? x}{}_{\mathrm{<mtext>0</mtext>}}\text{6}{}^2$,and define λ₀ to be the value of λ for which Q is a minimum. We look for λ₀ among solutions of dQ/dλ = 0. Though Q is not computable (since ε is unknown), we can use this approach to study the behavior of λ₀ as a function of y and ε. Theorems involving “noise to signal ratios” determine when λ₀ exists and define the cases λ₀ > 0 and λ₀ = ∞. Estimates for λ₀ and the minimum square error Q₀ = Q(λ₀) are derived.     

Opposing flows in a one dimensional convection-diffusion problem

In this paper, we examine a particular class of singularly perturbed convection-diffusion problems with a discontinuous coefficient of the convective term. The presence of a discontinuous convective coefficient generates a solution which mimics flow moving in opposing directions either side of some flow source. A particular transmission condition is imposed to ensure that the differential operator is stable. A piecewise-uniform Shishkin mesh is combined with a monotone finite difference operator to construct a parameter-uniform numerical method for this class of singularly perturbed problems.     

On a class of atmospheres occurring in stellar hydrodynamic theory

Summary A derivation is given of the basic equation governing the behaviour of small disturbances in a stratified compressible atmosphere (with a locally unstable region bounded by semi-infinite stable regions) in which the time of radiative relaxation is constant.By means of a new independent variable that is a function of the equilibrium entropy stratification, various properties of such media can be established in a straightforward manner. A general expression for the wave energy flux is derived, and integral properties arising from this quantity are used to obtain in particular, results pertaining to the eigenfunctions and eigenvalues for strongly radiating systems. In addition, a model of an atmosphere with a strongly-radiating region embedded in a convective zone, and bounded by semi-infinite radiation-free layers is discussed in some detail. The paper is an approach towards providing some preliminary but unified results in the theory of stellar hydrodynamics.

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Résumé Nous donnons une dérivation de l'équation de base qui détermine la propagation des petites perturbations dans une atmosphère compressible stratifiée (présentant une zone d'instabilité locale entre des zones de stabilité semi-infinies) dont le temps de relaxation radiative est constant.Au moyen d'une nouvelle variable indépendante, fonction de la stratification de l'entropie en équilibre, diverses propriétés de tels milieux sont déterminées simplement. On donne une formulation du flux de l'énergie ondulative et on emploie les propriétés intégrales qui en résultent pour obtenir en particulier des résultats concernant les eigenfonctions et les eigenvalues pour des systèmes à forte radiation énergétique. En outre, on étudie en détail le modèle d'une amtosphère caractérisée par une zone à forte radiation insérée dans une zone convective entre des couches semi-infinies sans radiation.Ce travail présente quelques conclusions préliminaires mais cohérentes dans le domaine de la théorie de l'hydrodynamique stellaire.

     

A problem of steady-state electrochemical shaping with a non-Schlicht velocity hodograph

We solve the problem of the steady-state electrochemical shaping by two semi-infinite cathode plates oriented and located arbitrarily with respect to the direction of the feed motion. A characteristic feature of this problem is a non-schlicht velocity hodograph.     

A contact problem of the theory of consolidation for a strip

A plane mixed boundary-value problem of the linear theory of inertialess two-phase consolidation is considered [1]. A strip lying on a smooth undeformable foundation, impermeable to liquid, is under the pressure of a semi-infinite permeable. The material of the solid phase and the liquid are compressible. Using Laplace transformations with respect to time and the space coordinate, the problem is reduced to a Wiener-Hopf equation. The general features of the distribution of the roots of the characteristic equations, corresponding to different homogeneous conditions on the faces of the strip, are investigated. An effective solution is constructed in multiple integrals which converge exponentially with respect to all the variables. The temporal processes of the settling of the punch and the extrusion of the liquid are investigated.