= 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 is
$$
u_t=u_{xx},\quad 0<x<1,\quad t>0,\qquad u(x,0)=x+\sin(\pi x),\quad u(0,t)=0,\quad u(1,t)=1.
$$
Its exact <classical solution> is $u(x,t)=x+e^{-\pi^2t}\sin(\pi x)$. 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 $x\geq ct$, and $a(t-x/c)$ otherwise. Values at $x=L$ are determined by this formula; independently prescribing them can make the problem inconsistent.https://www.math.utoronto.ca/ivrii/PDE-textbook/Chapter2/S2.1.html[Victor Ivrii, First order PDEs, initial-boundary value problems]{ref}
\Image[/pde-boundary-data.png]
{title=Heat flow uses initial and both endpoint data; rightward transport uses initial and inflow data}
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