Fritz John

Plane waves and spherical means : applied to partial differential equations

The author would like to acknowledge his obligation to all his (;Olleagues and friends at the Institute of Mathematical Sciences of New York University for their stimulation and criticism which have contributed to the writing of this tract. The author also wishes to thank Aughtum S. Howard for permission to include results from her unpublished dissertation, Larkin Joyner for drawing the figures, Interscience Publishers for their cooperation and support, and particularly Lipman Bers, who suggested the publication in its present form. New Rochelle FRITZ JOHN September, 1955 [v] CONTENTS Introduction...1 CHAPTER I Decomposition of an Arbitrary Function into Plane Waves Explanation of notation ...7 The spherical mean of a function of a single coordinate. 7 9 Representation of a function by its plane integrals . CHAPTER II Tbe Initial Value Problem for Hyperbolic Homogeneous Equations with Constant Coefficients Hyperbolic equations...15 Geometry of the normal surface for a strictly hyperbolic equation. 16 Solution of the Cauchy problem for a strictly hyperbolic equation . 20 Expression of the kernel by an integral over the normal surface. 23 The domain of dependence ...2 9 The wave equation ...32 The initial value problem for hyperbolic equations with a normal surface having multiple points ...36 CHAPTER III The Fundamental Solution of a Linear Elliptic Differential Equation witL Analytic Coefficients Definition of a fundamental solution ...43 The Cauchy problem ...45 Solution of the inhomogeneous equation with a plane wave function as right hand side ...49 The fundamental solution...

Partial Differential Equations.

Many physical problems involve quantities that depend on more than one variable. The temperature within a “large”1 solid body of conducting material varies with both time and location within the material. When such problems are modeled, what results is a differential equation involving partial derivatives, or a partial differential equation..

Lectures on advanced numerical analysis

Автор(ы): Fritz John Год изд.: 1966 Описание: A large number of mathematical books begin as lecture notes; but, since mathematicians are busy, and since the labor required to bring lecture notes up to the level of perfection which authors and the public demand of formally published books is very considerable, it follows that an even larger number of lecture notes make the transition to book form only after great delay or not at all. The present lecture note series aims to fill the resulting gap. It will consist of reprinted lecture notes, edited at least to a satisfactory level of completeness and intelligibility, though not necessarily to the perfection which is expected of a book. In addition to lecture notes, the series will include volumes of collected reprints of journal articles as current developments indicate, and mixed volumes including both notes and reprints. Оглавление:

Contributions to the Theory of Partial Differential Equations. (AM-33)

The description for this book, Contributions to the Theory of Partial Differential Equations. (AM-33), will be forthcoming.