# Generalized Solutions and Singular Characteristics of First Order PDEs

Chapter

## Abstract

A classical solution of the non-linear first order PDE
is generally defined as a function of the class C

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

(2.1)

^{1}Ω, 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.## Keywords

Viscosity Solution Dispersal Surface Initial Value Problem Regular Characteristic Singular Surface
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Copyright information

© Springer Science+Business Media New York 1998