Introduction

Abstract

We now generalize the class of systems considered, and allow an arbitrary number of states convecting in one of the directions. They are referred to as $n + 1$ systems, where the phrasing “$n + 1$” refers to the number of variables, with u being a vector containing n components convecting from $x = 0$ to $x = 1$, and v is a scalar convecting in the opposite direction. They are typically stated in the following form

$$u_t(x, t) + \varLambda (x) u_x(x, t) = \varSigma (x) u(x, t) + \omega (x) v(x, t) $$

$$v_t(x, t) - \mu (x) v_x(x, t) = \varpi ^T(x) u(x, t) + \pi (x) v(x, t) $$

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

$$v(1, t) = c^T u(1, t) + k_1 U(t)$$

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

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

$$y_0(t) = k_2 v(0, t)$$

$$y_1(t) = k_3 u(1, t) $$