Initial and boundary data for evolution equations

On $0<x<L$, $t>0$, an initial condition $u(x,0)=g(x)$ gives the starting profile. For the heat equation, Dirichlet boundary conditions $u(0,t)=a(t)$, $u(L,t)=b(t)$ can model ends held at prescribed temperatures. Alternatively, zero Neumann boundary conditions describe insulated ends in the constant-conductivity model. These choices can produce different evolutions from the same initial condition.

For a continuous classical solution reaching the corners, the Dirichlet boundary data must satisfy $g(0)=a(0)$ and $g(L)=b(0)$; higher smoothness can require further compatibility. In dimensionless variables, a complete worked heat equation problem isIts exact classical solution is $u(x,t)=x+e^{-{\pi }^{2}t}\sin (\pi x)$. The straight profile $x$ is the equilibrium; the extra bump decays. Substituting verifies the equation and every datum, and the energy estimate proves uniqueness among smooth solutions with these data.

For the transport equation with $c>0$ on the same interval, prescribe an inflow transport boundary condition only at $x=0$. Backtracking a characteristic curve gives $g(x-ct)$ if $x\ge ct$, and $a(t-x/c)$ otherwise. Values at $x=L$ are determined by this formula; independently prescribing them can make the problem inconsistent.^[ref]

On the unit square, these data determine only the zero solution of the heat equation $u_y=u_{xx}$. The squared $L^2$ norm decreases because its derivative is $-2\int u_x^{2}dx$, and its initial value is zero. For Laplace's equation, $\sin (\pi x/2)\sinh (\pi y/2)$ is a nonzero solution with the same three data; adding zero Dirichlet data on the top edge restores uniqueness by Green's first identity. For the wave equation $u_{yy}=u_{xx}$, $\sin (\pi x/2)\sin (\pi y/2)$ is a counterexample; the missing datum is initial velocity. Adding $u_y(x,0)=0$ restores uniqueness by conservation of $\frac{1}{2}\int (u_x^2+u_y^2)dx$. Thus the type of equation determines what information is needed on an unprescribed edge.

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.

A prescribed PDE source term changes the heat equation to $u_t=D{u}_{xx}+f(x,t)$, or the transport equation to $u_t+c{u}_x=f(x,t)$. If $u$ is a temperature, $f$ has units of temperature per time. The formulas below assume $f=0$; a nonzero PDE source term can change the spatial average even with periodic boundary conditions.