First-Order PDEs: The Transport Equation

The transport (advection) equation is the simplest first-order PDE:

Its general solution is any function of the single combination $x - ct$:

This describes the initial profile $f(x) = u(x,0)$ rigidly sliding to the right at speed $c$, its shape unchanged. The lines $x - ct = \text{const}$ along which $u$ stays constant are called characteristics.

Worked example. Solve $u_t + 4u_x = 0$ with $u(x, 0) = x^2$. The solution is $u(x,t) = (x - 4t)^2$ — replace $x$ by $x - ct = x - 4t$. Check: $u_t = 2(x-4t)(-4) = -8(x-4t)$ and $4u_x = 4\cdot 2(x-4t) = 8(x-4t)$, so $u_t + 4u_x = 0$ ✓. At $x = 10, t = 2$: $u = (10 - 8)^2 = 4$.