Introduction

Abstract

This part considers the most general class of PDEs treated in this book. It contains systems of $n + m$ linear coupled hyperbolic PDEs of which n equations convect information from $x = 1$ to $x = 0$, and m equations convect information in the opposite direction. Such systems are usually stated in the form (1.25), which we for the reader’s convenience restate here

$$u_t(x, t) + \Lambda ^+(x) u_x(x, t) = \Sigma ^{++}(x) u(x, t) + \Sigma ^{+-}(x) v(x, t)$$

$$v_t(x, t) - \Lambda ^-(x) v_x(x, t) = \Sigma ^{-+}(x) u(x, t) + \Sigma ^{--}(x) v(x, t)$$

$$u(0, t) = Q_0 v(0, t) $$

$$v(1, t) = C_1 u(1, t) + U(t)$$

$$u(x, 0) = u_0(x)$$

$$v(x, 0) = v_0(x)$$