 Method of characteristics

The method of characteristics converts a first-order partial differential equation into ordinary differential equations along characteristic curves. Forsolve $\dot{X}=b(t,X)$, $X(0)=\alpha$, and $\dot{U}=h(t,X,U)$, $U(0)=g(\alpha )$. The chain rule identifies $U(t,\alpha )=u(t,X(t,\alpha ))$. If the characteristic flow map is invertible, it reconstructs the solution by $u(t,x)=U(t,X_t^{-1}(x))$. This turns spatial propagation and the evolution of the carried value into separate calculations.^[ref]

For $u_t+t{u}_x=0$, the characteristic curves are $X(t,\alpha )=\alpha +t^2/2$ and $U=g(\alpha )$, so $u(t,x)=g(x-t^2/2)$. Substitution verifies both the partial differential equation and the initial condition. More generally, a quasilinear partial differential equation allows $b$ to depend on $U$, so the two ordinary differential equations become coupled. Noncharacteristic Cauchy data and a nonzero projection Jacobian determinant give local reconstruction; characteristic crossing can destroy it even while the lifted trajectories remain smooth.