Hyperbolic Partial Differential Equations in More Than Two Independent Variables

Abstract

In Chapter 2 we gave a treatment of the partial differential equations (PDEs) of hyperbolic type in two independent variables, as a mathematical description of electromagnetic wave propagation in the (x, t) plane. We first discussed the one-dimensional wave equation, then the theory of quasilinear hyperbolic equations in two independent variables, and finally the theory of fully nonlinear equations in two variables. In accordance with our plan of going from simple to more complex situations, we extend the treatment of PDEs of hyperbolic type to more than two independent variables. This will essentially supply the mathematical description for wave propagation phenomena in electromagnetic media in more than one dimension vis-à-vis the Cauchy initial value problem and related problems. We shall show that the extension of characteristic curves to characteristic surfaces, and the rays along which disturbances are propagated, will play a central role in our generalization to more than two dimensions. We showed in Chapter 2 that we can transform from a single second-order to a pair of first-order PDEs.