Partial differential equations.

*(English)*Zbl 0209.40001
Applied Mathematical Sciences. Vol. 1. New York-Heidelberg-Berlin: Springer-Verlag, viii, 221 p. (1971).

This book is a new edition (minor corrections have been made in the text and an up-to-date bibliography has been added) of the classic work of the author formerly available only as bound lecture notes. Even though the original notes are nearly 20 years old this book still serves as an excellent introduction to partial differential equations.

The book is divided into four chapters listed below with a summary of their contents:

I. The single first order equation, linear and quasi-linear equations, general first order equation for a function of two variables, general first order equation for a function of \(n\) independent variables.

II. The Cauchy problem for higher order equations, analytic functions of several real variables, formulation of the Cauchy problem and the notion of characteristics, the Cauchy problem for the general nonlinear equation, the Cauchy-Kowalewsky theorem.

III. Second order equations with constant coefficients, equations in two independent variables, canonical forms, the one-dimensional wave equation, the wave equation in higher dimensions, method of spherical means, method of descent, Duhamel’s principle, the potential equation in two dimensions, the Dirichlet problem, the Green’s function and the fundamental solution, equations related to the potential equation, continuation of harmonic functions, the heat equation.

IV. The Cauchy problem for linear hyperbolic equations in general, Riemann’s method, higher order equations in two independent variables, the method of plane waves.

The book is divided into four chapters listed below with a summary of their contents:

I. The single first order equation, linear and quasi-linear equations, general first order equation for a function of two variables, general first order equation for a function of \(n\) independent variables.

II. The Cauchy problem for higher order equations, analytic functions of several real variables, formulation of the Cauchy problem and the notion of characteristics, the Cauchy problem for the general nonlinear equation, the Cauchy-Kowalewsky theorem.

III. Second order equations with constant coefficients, equations in two independent variables, canonical forms, the one-dimensional wave equation, the wave equation in higher dimensions, method of spherical means, method of descent, Duhamel’s principle, the potential equation in two dimensions, the Dirichlet problem, the Green’s function and the fundamental solution, equations related to the potential equation, continuation of harmonic functions, the heat equation.

IV. The Cauchy problem for linear hyperbolic equations in general, Riemann’s method, higher order equations in two independent variables, the method of plane waves.

Reviewer: Vincent G. Sigillito

##### MSC:

35-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations |