Transformation Of Independent Variables Research Articles

Self-similarity in the vapour bubble dynamics

The covariance of equations governing the homogeneous vapour bubble dynamics with respect to the transformation of independent variables is stated. It leads to the self-similar time dependence of bubble radius in three cases: 1. (1) when liquid pressure is constant, 2. (2) when it varies with time and 3. (3) in an ultrasonic field. This is confirmed by numerical calculations.

A regularization of multiple encounters in gravitationalN-body problems

A method is introduced to regularize binary collisions between one of the bodies and any number of other bodies in the three-dimensional problem ofn-bodies. The coordinates are first transformed from an inertial system to a system relative to one of the bodies. The KS dependent variable transformation and a new independent variable transformation are introduced for the regularization.

Covariant properties and symmetry of the parabolic equation of second order

In this article the defining equations are constructed for the group of transformations of independent variables, with respect to which the normal parabolic equation of second order is invariant. This group is calculated for the time-dependent Schrodinger equation. The symmetry of the parabolic equation is investigated.

Repeated transformations to solve nonuniform transmission line problems

A new technique is introduced to solve nonuniform transmission line problems; namely, repeated transformations, which consist of a number of alternate transformations of both dependent and independent variables of the differential equation of the voltage of a line. The two forms of the resulting equation may then be used to derive new classes of solvable nonuniform lines. They have been used here to derive two theorems which generalize a class of solvable nonuniform line problems. The matrix parameters as also the equivalent circuits of the generalized lines are given.

An Exploratory Study of Taylor’s Tool-Life Equation by Power Transformations

Transformations of both dependent and independent variables are employed to investigate the linearization of Taylor’s tool-life equation. This exploratory study indicates that a logarithmic transformation, which is a special case of the general class of power transformations, gives the best fit for HSS tool-life data. However, the study does not show that a logarithmic transformation is the best for carbide tool-life data. For a wide cutting range, where Taylor’s tool-life equation does not hold, a linear equation instead of a second-order relationship for the prediction of tool life can be determined by the proper transformation.

Remarks on the problem of slow motions in a rotating fluid

This note is concerned with the problem of the slow motions of an inviscid, incompressible rotating fluid, and in particular with the motion of a sphere along the axis of rotation. This problem was studied recently by Stewartson (2), who overcomes the principal mathematical difficulty, viz. that of formulating the problem in a coordinate system in which the appropriate differential equation can be solved simply and in which the sphere is a coordinate surface, by a very elegant transformation of independent variables. Stewartson, however, uses inappropriate initial conditions. It is the purpose of this note to discuss the question of initial conditions in the light of results previously obtained by the writer (1).

Infra-red radiation from an engine cylinder

We speak about an exact solution when the original system of partial differential equations describing the beam can be reduced to a system of ordinary differential equations. The set of exact solutions also includes the solutions that can be expressed in terms of elementary and special functions.The most important function of the exact solutions is testing of the approximate and numerical models. In doing so, not just one but a specially chosen set of exact solutions should be used because no single solution contains all the peculiarities inherent to practical problems. These peculiarities may include the emission in ρ- and T- modes from a curvilinear surface with a non-homogeneous current density in the presence of an external magnetic field and compensating background, as well as relativistic and high-frequency effects, extended trajectories, high-compression and deep-deceleration modes, and so forth.A complete set of exact solutions can be constructed by investigating the group properties of the beam equations. The main goal of those investigations is to reveal all the transformations of independent and dependent variables that preserve the form of the beam equations. A general approach based on the theory of groups and the notion of invariant solution was developed by L. V. Ovsyannikov. This section also familiarizes readers with the theory of continuous groups as applied to the problems of intense beam optics.The group properties of 3D nonstationary relativistic flows are studied, and the tables of the H-solutions that can be described by the ordinary differential equations are constructed. Aside from the transformations of translation, scaling, and rotations, the transformations with arbitrary functions of time are revealed for the non-relativistic case.Examples of the solutions in elementary functions for the axisymmetric and 3D flows are given. Most of the known exact solutions turn out to be invariant; however, there exist the solutions, the relation of which to the group properties of the beam equation remains still unknown.