Showing papers in "Ima Journal of Applied Mathematics in 1980"

The Fractional Order Fourier Transform and its Application to Quantum Mechanics

Abstract: We introduce the concept of Fourier transforms of fractional order, the ordinary Fourier transform being a transform of order 1. The integral representation of this transform can be used to construct a table of fractional order Fourier transforms. A generalized operational calculus is developed, paralleling the familiar one for the ordinary transform. Its application provides a convenient technique for solving certain classes of ordinary and partial differential equations which arise in quantum mechanics from classical quadratic hamiltonians. The method of solution is first illustrated by its application to the free and to the forced quantum mechanical harmonic oscillator. The corresponding Green's functions are obtained in closed form. The new technique is then extended to three-dimensional problems and applied to the quantum mechanical description of the motion of electrons in a constant magnetic field. The stationary states, energy levels and the evolution of an initial wave packet are obtained by a systematic application of the rules of the generalized operational calculus. Finally, the method is applied to the second order partial differential equation with time-dependent coefficients describing the quantum mechanical dynamics of electrons in a time-varying magnetic field.

Three-dimensional Contact of a Rigid Roller Traversing a Viscoelastic Half Space

Abstract: In this paper, the authors consider a nontrivial three-dimensional viscoelastic contact problem which has some physical significance. (Although the subject of the analysis is an elliptical roller, it is only a small step onward to the consideration of a crowned cylindrical roller.) Generally, it is the intractability of the mathematics which hinders analytic solution of true three-dimensional problems in (visco)elasti city. The traditional method of surmounting this difficulty is to reduce the problem to two dimensions, either by choosing a suitable geometry, or by using an appropriate co-ordinate system. The elastic line integral theory represents another approach; certain approximations are used to simplify the governing equations, thus allowing the solution of the problem. After the development of a viscoelastic analogue of the Boussinesq equation valid for the solution of quasi-steady state viscoelastic contact problems, analysis proceeds making use of near field and extended line integral approximations. Results are generated showing the velocity dependence of several physical parameters, including the size and shape of the contact zone. One additional point of interest is uncovered, namely the presence of a pressure peak near the leading edge of the contact zone.

Variational Principles for Problems with Linear Constraints: Prescribed Jumps and Continuation Type Restrictions

Abstract: : A general theory of the problems subjected to linear constraints is developed. As applications, a problem for which the solutions are required to satisfy prescribed jumps and another one whose solutions are restricted to be such that they can be continued smoothly into solutions of given equations in neighboring regions, are formulated abstractly. General variational principles for these types of problem are reported. In addition it is shown that sets of functions that can be extended in the manner explained above constitute, generally, completely regular subspaces, here defined. These results have a bearing on boundary methods which are being developed for treating numerically partial differential equations associated with many problems of Science and Engineering. (Author)