Generalized Solutions and Singular Characteristics of First Order PDEs

Abstract

A classical solution of the non-linear first order PDE

$$ F\left( {x,u,{{\partial u} \mathord{\left/ {\vphantom {{\partial u} {\partial x}}} \right. \kern-\nulldelimiterspace}{\partial x}}} \right) = 0.x \in \Omega\in {\mathbb{R}^n} $$

(2.1)

is generally defined as a function of the class C¹Ω, i.e. a solution u(x) must have partial derivatives δ;u(x)/δx;,, i = 1,…, n, continuously depending upon x in the domain Ω. Formally, to veri fy whether a given continuous function u(x) is a solution or is not, one needs only the existence of the partial derivatives. There are functions for which these derivatives, being discontinuous, exist in one coordinate system and do not exist in another one. The continuity requirement makes the definition of the solution invariant with respect to some specific coordinate system.