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).

Surface Characteristics Initial cond. Auto-rotate

Presets (examples with parameters)

How it works: PDE a·u_x + b·u_y = c with initial condition u(x₀(s), y₀(s)) = u₀(s) is solved by integrating the characteristic system: dx/dt = a, dy/dt = b, du/dt = c. The solution surface is spanned by these curves.

Equation a·u_x + b·u_y = c

a(x, y, u) =

b(x, y, u) =

c(x, y, u) =

Initial condition (parametric curve)

x₀(s) =

y₀(s) =

u₀(s) =

s range: … t:

Cauchy problem

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.