Heat transfer process in an elliptical channel

In the case of n functions of a single variable, the vanishing of the wronskian is the most familiar criterion for their linear dependence. The wronskian also plays an important part in connection with the theory of a single ordinary homogeneous linear differential equation of the nth order, even in questions which do not concern the linear dependence of solutions of the equation. It is the purpose of the present paper to generalize the fundamental facts connected with these topics, to the case of functions of several variables. The characteristic property of an ordinary homogeneous linear differential equation is that any solution is expressible linearly, with constant coefficients, in terms of a fundamental set of solutions. The natural generalization to several independent variables is afforded by the completely integrable system of homogeneous linear partial differential equations, in one dependent variable, any solution of which is likewise linearly dependent on a finite number of solutions of the system. It is such systems which are discussed in the latter part of this paper; the subject of the linear dependence of functions of several variables is developed in the first part, chiefly with a view to its subsequent application to completely integrable systems. In our discussion of linear dependence, we consider throughout n functions Yl, Y2, Y n,, of p independent variables u1, u2, * up . Functions and variables may be either real or complex; if any of the variables be complex, we suppose the functions to be analytic in those variables. All of the main theorems, however, are stated and proved under the supposition that the independent variables are real; the modifications usually simplifications and omissions-are easily supplied for cases in which one or more of the independent variables is complex. The n functions, then, of the real variables

We treat the master equation in one dimension for the Rayleigh-piston problem of hard rods, i.e., a single heavy tagged particle with finite mass ${m}_{A}$ moving in an ensemble of light bath particles with mass ${m}_{B}$ and a time-independent velocity distribution. The backward form of the master equation is used to obtain a transport equation for a conditional average of a time-dependent physical quantity. It is shown how this integro-differential equation can be solved successively by transforming it into a closed set of first-order linear partial differential equations. The solution of the $l$th linear partial differential equation is completely determined by the solutions of the $l\ensuremath{-}1$ linear partial differential equations, meaning that higher approximations do not change lower ones. It is shown how the motion of the tagged particle can be separated into a deterministic (nonfluctuating) part and into fluctuating contributions. It turns out that the $l=0$ term in the successive approximation scheme satisfies a homogeneous linear partial differential equation and describes the nonfluctuating motion, whereas the higher approximations ($l\ensuremath{\ge}1$), which are solutions of nonhomogeneous linear partial differential equations, describe the fluctuating contributions. The calculations for arbitrary time-dependent conditional averages are performed explicitly up to order 2 in the expansion parameter ${\ensuremath{\Omega}}^{\ensuremath{-}1}=\frac{{m}_{B}}{({m}_{A}+{m}_{B})}$. This new method is employed to calculate the conditional averages for the time-dependent mean velocity, the mean-square velocity, the velocity-autocorrelation function, and the self-diffusion coefficient. In addition, the results show that conditional averages calculated via a Gaussian distribution function deviate considerably from the exact results obtained for mass ratios of order 1.

Within the framework of the kinetic approach, the authors have solved the problem on rarefied-gas flow in a cylindrical duct in the presence of the longitudinal temperature gradient. The Williams kinetic equation was used as the basic equation, and the diffuse-reflection model, as the boundary condition on the duct wall. This enabled the authors to consider the solution of the problem in linearized form. To find a linear correction to the locally equilibrium distribution function, the problem was reduced to solution of a linear homogeneous partial differential equation of first order. A solution to the latter was constructed with the method of characteristics. With account taken of the obtained solution and on the basis of the statistical meaning of the distribution function of gas molecules by coordinates and velocities, the authors constructed the profile of the heat-flux vector in the duct and computed the heat flux through the duct cross section. A numerical analysis of final expressions was made. A comparison with analogous results obtained with the discrete-ordinates method has shown that the solution procedure proposed in the work leads to correct results in a wide range of values of the duct radius.

A note on non-separable solutions of linear partial differential equations

Operator products in 2-dimensional critical theories

Mayer variational problems in which the control variable appears linearly are considered and a canonical form sought for the system equations which is somewhat analogous to that adopted by Wonham and Johnson for linear constant coefficient systems with cost functional quadratic in the state variables. A means of synthesizing a transformation to the canonical form in terms of the mutually independent solutions of a first order linear homogeneous partial differential equation is described. It is then shown how the Legendre-Clebsch necessary condition applied in the transformed system of variables may be employed to obtain information on the singular extremals of the problem and the possible appearance of singular subarcs in the solution. Two examples are employed for illustration, one a simple servomechanism problem and the other Goddard's problem of optimal thrust programming for a sounding rocket.

Frictional-cohesive solids such as soil, ores, chemicals, sugar, flour are regarded as plastic and represented by the Jenike-Shield yield function [1] during steady flow. The stress-strain rate relations are based on isotropy, continuity, and a one-to-one dependence of density on the major pressure. In plane strain and in axial symmetry the stress field requires the solution of a system of two hyperbolic partial differential equations. The velocity field can then be computed by solving another system of two linear homogeneous partial differential equations of the hyperbolic type. In straight conical channels, a particular stress field called the “radial stress field” assumes a special importance because evidence has been presented elsewhere that all general fields tend to approach the radial stress fields in the vicinity of the vertex. Examples of numerical solutions of radial stress fields are given.

On Solutions of the Linear Homogeneous Partial Differential Equation in the Vicinity of the Singularity. II

The results concerning the linear dependence of functions of several variables were read on the latter date.

Parameter and State Estimation for a Diffusion Process

We derive a method of obtaining approximate numerical solution of linear variable-coefficient partial differential equations (PDEs) with two independent variables from Kida's optimum approximation theory. It is shown that a certain generalized filter bank implements linear PDEs. By applying generalized discrete orthogonality of Kida's optimum approximation to this filter bank, we prove that our approximate solution satisfies a given linear PDEs and the corresponding initial or boundary conditions at all given sample points, simultaneously.

The aim of this paper is to derive the dependence of the nature of a solution of a class of differential equations of n-th order with polynomial coefficients on the solutions of the corresponding characteristic algebraic equation of n-th degree.

The Lagrange Multiplier method has been applied in solving the regular Sturm Liouville (RSL) equation under a boundary condition of the first kind (Dirichlet boundary condition). This method is a very powerful tool for solving the RSL equation and it involves the solution of the RSL equation with the expansion of eigenfunctions into trigonometric series. The efficiency of this approach is emphasized by solving two examples of regular Sturm Liouville problem under homogenous Dirichlet boundary conditions. The methodology is effectively demonstrated, and the results show a high degree of accuracy of the solution in comparison with the exact solution and reasonably fast convergence.

Small perturbations in the type of boundary conditions for an elliptic operator

By the maximum principle the solution of the homogeneous heat equation with homogeneous Dirichlet boundary conditions is nonnegative for positive time if the initial values are nonnegative. In recent work it has been shown that this does not hold for the standard spatially discrete and fully discrete piecewise linear finite element methods. However, for the corresponding semidiscrete and Backward Euler Lumped Mass methods, nonnegativity of initial data is preserved, provided the underlying triangulation is of Delaunay type. In this paper, we study the corresponding problems where the homogeneous Dirichlet boundary conditions are replaced by Neumann and Robin boundary conditions, and show similar results, sometimes requiring more refined technical arguments.