Partial differential equations: method of characteristics
LMB — rotate. Wheel — zoom. RMB — pan. Equation a·u_x + b·u_y = c is solved by characteristics. Characteristics are curves in 3D (x, y, u).
How it works: PDE \( a\, u_x + b\, u_y = c \) with initial condition \( u(x_0(s), y_0(s)) = u_0(s) \) is solved by integrating the characteristic system: \( \frac{dx}{dt} = a,\; \frac{dy}{dt} = b,\; \frac{du}{dt} = c \). The solution surface is spanned by these curves.
Quick reference
A quasilinear first-order PDE:
\[ a\,u_x + b\,u_y = c \]Characteristics reduce it to ODEs in \((x,y,u)\). An initial curve generates an integral surface. Transport keeps the profile; inviscid Burgers can form a shock.
Method of characteristics for PDEs online
ODEPLOT solves first-order PDEs by Cauchy characteristics and builds a 3D integral surface from \(u=\varphi(s)\).
Compare linear transport with Burgers wave breaking. Change \(a,b,c\) and the initial data; download PNG.
What you can do on this page
On this page you can open the scene full screen, rotate and zoom the integral surface, animate equation parameters with the sliders, and save a PNG.