Showing papers in "Communications on Pure and Applied Mathematics in 1954"

On the exponential solution of differential equations for a linear operator

Abstract: The present investigation was stimulated by a recent paper of K. 0. Friedrichs 113, who arrived at some purely algebraic problems in connection with the theory of linear operators in quantum mechanics. In particuIar, Friedrichs used a theorem by which the Lie elements in a free associative ring can be characterized. This theorem is proved in Section I1 of the present paper together with some applications which concern the addition theorem of the exponential function for non-commuting variables, the so-called BakerHausdorff formula. Section I contains some algebraic preliminaries. It is of a purely expository character and so is part of Section 111. Otherwise, Section 111 deals with the following problem, also considered by Friedrichs: Let A( t ) be a linear operator depending on a real variable 1. Let Y(t) be a second operator satisfying the differential equation

Remarks on the theory of partial differential equations of mixed type and applications to the study of transonic flow

Abstract: I shall begin by recalling briefly some well known statements on problems arising in the theory of transonic flow. These, from the mathematical point of view, are related to a partial differential equation of the mixed type. But they are not the simplest basic problems of the theory of equations of this type. Although there are close similarities between the study of transonic flow and the mathematical study of mixed equations, one cannot say that the studies are strictly equivalent. The second part of this lecture is devoted to the formation and the study of elementary solutions of equations of this type; a few applications to certain boundary value problems are proposed.

Solutions of Second Order Hyperbolic Differential Equations with Constant Coefficients in a Domain with a Plane Boundary

Abstract: It is known that solutions of partial differential equations often have special regularity properties that set them apart from the “arbitrary” functions of any class C m or even C ∞. For example solutions of analytic elliptic equations are themselves analytic. Similarly, solutions of the heat equation are analytic in the space variables and of class C ∞ in the time. Even solutions of hyperbolic or ultra-hyperbolic equations have their special regularity properties, where however “regularity” does not always consist in possessing a large number of derivatives. This may find its expression in the fact that local integral transforms of the solution have many derivatives or even are analytic. Often the solutions form a family of functions, which share with the analytic functions the property of unique continuation, at least within certain limits.

Computation of one‐dimensional compressible flows including shocks

Abstract: The solutions of problems involving compressible flows have different functional forms in different regions, separated not only by discontinuities of the fluid itself but also by shock fronts and by characteristics starting from the origins of shock fronts and from points where such characteristics meet shocks. To obtain accuracy in numerical integration, small intervals and formulas using differences of low order are convenient immediately after crossing these lines, but much larger intervals, with differences of higher order, are sufficient elsewhere. The purpose of this paper is to explain a method of exploiting the economy made possible in this way. If the equations for the one-dimensional case are transformed to characteristic variables, they form a ‘canonical hyperbolic system’. Adams' method for ordinary differential equations extends to a variable interval by using divided differences of the derivative, and it then becomes practical to use it for starting and across discontinuities. Further, it applies directly to canonical hyperbolic systems. Numerical solution is then very direct over regions determined by two characteristics and is convenient where the flow crosses already existing shocks. Finally, the origin of a new shock in any interval can be detected and arrangements can be made for it.