Initial-boundary value problem

An initial-boundary value problem specifies a partial differential equation, initial conditions, and boundary conditions. For example, the half-line wave equation $u_{tt}=c^2{u}_{xx}$ requires an initial displacement and initial velocity for $x\ge 0$ together with an endpoint condition such as $u(0,t)=0$ for $t>0$. Its meaning also depends on the chosen solution class: a classical solution satisfies the equation pointwise, while a weak solution satisfies an integrated form. These data must obey the compatibility conditions needed by the chosen regularity.