Power Series Methods

Abstract

It takes a little time and a few basic examples to develop intuition. This is particularly true of the subject of partial differential equations which has an enormous variety of technique and phenomena within its confines. This section describes the simplest nontrivial partial differential equation

$${u_t}(t,x) + c{u_x}(t,x) = 0,t,x \in \mathbb{R},c \in \mathbb{C}$$

(1)

The equation is of first order, is linear with constant coefficients, and involves derivatives with respect to both variables. The unknown is a possibly complex valued function u of two real variables. This example reveals one of the fundamental dichotomies of the subject, the equation is hyperbolic if c E ll and elliptic otherwise. The equation is radically different in these two cases in spite of the similar appearance.