Use Of Partial Differential Equations Research Articles

An effective Lagrangian approach to the problem of parametric interaction of Gaussian acoustic beams

A new technique for studying the nonlinear interaction of collinear Gaussian acoustic beams based on the use of Lagrangian field-theoretic methods is discussed. The technique is based on treating the standard beam parameters, i.e., spot size and on-axis pressure, of the Gaussians as dynamic variables, i.e., unknown functions of distance down the beam axis. An effective one-dimensional Lagrangian derived for these quantities leads to a set of Euler–Lagrange ordinary differential equations that can be solved rapidly on a computer. The method avoids the use of partial differential equations entirely, and gives results that are in good agreement with previous work both on second-harmonic and parametric generation of sound by Gaussian beams.

On Homothetic Functions

In economic analysis, the importance of the homotheticity of production functions (or utility functions), which is due to Shephard (1953), has been well recognized. Its important feature lies in the fact that every expansion path is a ray from the origin and the underlying production (or utility) function is homothetic. Although the proof of the part of this statement is easy, the proof of the only if part, at least as it appears in the literature, is not easy, requiring a few pages for the proof. Lau (1969) proved it by way of partial differential equations (cf. pp. 379-81), and Fare and Shephard (1977a, b) used a set-theoretic approach, whereas Sandler and Swimmer (1978) proved it for the two-input case. F0rsund (1975) provides a simple proof for a closely related proposition (cf. his Proposition 2), and yet his proof requires the use of partial differential equations. It would thus be desirable to obtain an elementary, short alternative proof of the above important proposition. The purpose of this note is to offer such a proof. In this paper we prove both the only if and the part simultaneously. Although our proof is carried out in terms of production theory, the same proof obviously applies to the theory of consumption. Let f(x) be the production function of a firm which produces a single output, where x is an n-dimensional input vector. The firm minimizes its

An eigenfunction expansion associated with a two-parameter system of differential equations. II

SynopsisWe continue with the work of an earlier paper concerning the use of partial differential equations to prove the uniform convergence of the eigenfunction expansion associated with a left definite two-parameter system of ordinary differential equations of the second order.