Parabolic 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 parabolic if the Matrix

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

satisfies det $({\hbox{\sf Z}})=0$. The Heat Conduction Equation and other diffusion equations are examples. Initial-boundary conditions are used to give

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

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

where

$$u_{xx} = f(u_x,u_y,u,x,y)$$ (5)

holds in $\Omega$.

See also Elliptic Partial Differential Equation, Hyperbolic Partial Differential Equation, Partial Differential Equation