First-order linear equations and characteristics

The transport equation \( u_t + c\,u_x = 0 \) says that the derivative of \( u \) in the direction \( (1, c) \) of the \( (t, x) \) plane is zero. So \( u \) is constant along every line \( x - ct = \text{const} \), and the solution with initial value \( u(x, 0) = g(x) \) is \( u(x, t) = g(x - ct) \). The example checks this for \( g(x) = x^2 \), \( c = 3 \): \( u_t = -6(x - 3t) \) and \( 3u_x = 6(x - 3t) \).

The same idea handles variable coefficients. For \( a(x,t)\,u_t + b(x,t)\,u_x = 0 \), the curves with \( dx/dt = b/a \) are the characteristics, and along each one \( \frac{d}{dt}u(x(t), t) = u_t + \frac{dx}{dt}u_x = 0 \). For \( u_t + x\,u_x = 0 \) the characteristics are \( x = x_0 e^{t} \), so \( u(x, t) = g(x e^{-t}) \). With a source, \( u_t + c\,u_x = f \), the value changes along the characteristic at rate \( f \), and integrating gives Duhamel's formula \[ u(x,t) = g(x - ct) + \int_0^t f\big(x - c(t - s),\, s\big)\,ds. \]

Picture it: the graph of \( g \) slides to the right at speed \( c \) without changing shape. In the \( (x, t) \) plane, draw the family of parallel lines \( x - ct = x_0 \): each line carries the single value \( g(x_0) \) up from the axis.

Think it: the method converts one PDE into infinitely many ODEs, one per characteristic. It works when the initial curve is non-characteristic, crossing every characteristic exactly once. Data given along a characteristic cannot be arbitrary, because the equation already dictates how \( u \) varies there. In fluids the operator \( \partial_t + u\cdot\nabla \) is the material derivative, transport along particle paths: in two-dimensional ideal flow the vorticity is carried unchanged by exactly this mechanism.