Introduction

Abstract

A partial differential equation for a function u(x,y,...) with partial derivatives u_x,u_y,u_xx,u_xy,... is a relation of the form

$$F\left( {x,y,...,u,{u_x}{u_y}{u_{xx}},...,} \right) = 0,$$

(1)

where F is a given function of the variables x,y,...,u,u_x,u_y,u_xx,... . Only a finite number of derivatives shall occur. Needless to say, a function u(x,y,...) is said to be a solution of (1), if in some region of the space of its independent variables, the function and its derivatives satisfy the equation identically in x,y,... .