Introduction

Abstract

We now proceed by investigating systems of coupled linear hyperbolic PDEs. The simplest type of such systems are referred to as $2 \times 2$ systems, and consists of two PDEs convecting in opposite directions.

$$u_t(x, t) + \lambda (x) u_x(x, t) = c_{11}(x) u(x, t) + c_{12}(x) v(x, t)$$

$$v_t(x, t) - \mu (x) v_x(x, t) = c_{21}(x) u(x, t) + c_{22}(x) v(x, t)$$

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

$$v(1, t) = \rho v(1, t) + U(t)$$

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

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