Inhomogeneous wave equation

Hyperbolic | Linear

We now consider an inhomogeneous wave equation,

\[\pdd{u}{t}=\vnabla\cdot(f(x,y)\vnabla u),\]

with homogeneous Neumann (aka no-flux) boundary conditions. This equation can be solved numerically as long as $f(x,y)>0$ for all $x,y$ in the domain.

Damped waves and inhomogeneous boundaries

We next consider the damped wave equation,

\[\pdd{u}{t} +d\pd{u}{t}=D\nabla^2 u,\]

with inhomogeneous Dirichlet boundary conditions,

\[u|_{\partial \Omega} = \cos\left(\frac{m \pi x}{100}\right)\cos\left(\frac{m \pi y}{100}\right),\]

on a square domain.