First-Order, Quasi-linear Equations and Method of Characteristics

Abstract

Many problems in mathematical, physical, and engineering sciences deal with the formulation and the solution of first-order partial differential equations. From a mathematical point of view, first-order equations have the advantage of providing a conceptual basis that can be utilized for second-, third-, and higher-order equations.

This chapter is concerned with first-order, quasi-linear and linear partial differential equations and their solutions by using the Lagrange method of characteristics and its generalizations.

As long as a branch of knowledge offers an abundance of problems, it is full of vitality.

David Hilbert

The advance of analysis is, at this moment, stagnant along the entire front of nonlinear problems. That this phenomenon is not of a transient nature but that we are up against an important conceptual difficulty … yet no decisive progress has been made against them … which could be rated as important by the criteria that are applied in other, more successful (linear!) parts of mathematical physics.

It is important to avoid a misunderstanding at this point. One may be tempted to qualify these (shock wave and turbulence) problems as problems in physics, rather than in applied mathematics, or even pure mathematics. We wish to emphasize that it is our conviction that such an interpretation is wholly erroneous.

John Von Neumann