Hyperbolic Partial Differential Equation

A Partial Differential Equation of second-order, i.e., one of the form

$$Au_{xx} + 2B u_{xy} + C u_{yy} + Du_x + Eu_y + F = 0,$$ (1)

is called hyperbolic if the Matrix

$${\hbox{\sf Z}} \equiv \left[{\matrix{A & B\cr B & C\cr}}\right]$$ (2)

satisfies det $({\hbox{\sf Z}})<0$. The Wave Equation is an example of a hyperbolic partial differential equation. Initial-boundary conditions are used to give

$$u(x,y,t) = g(x,y,t) \quad\hbox{for }x \in \partial \Omega, t > 0$$ (3)

$$u(x,y,0) = v_0(x,y) \quad\hbox{in } \Omega$$ (4)

$$u_t(x,y,0)=v_1(x,y) \quad\hbox{in } \Omega,$$ (5)

where

$$u_{xy} = f(u_x,u_t,x,y)$$ (6)

holds in $\Omega$.

See also Elliptic Partial Differential Equation, Parabolic Partial Differential Equation, Partial Differential Equation