A PDE on its own has far too many solutions — the heat equation is solved by a flat bar, a cooling bar, a bar heated at one end, and endlessly more. To pin down one physical solution we must say what happens at the edges of space (boundary conditions) and, for time-dependent problems, what the situation is at the start (initial conditions).

The three boundary conditions

On the spatial boundary, three kinds cover almost everything. Picture a rod occupying [0, L]:

• Dirichlet — fix the value: u(0, t) = T_0. The end is held at a known temperature (an ice bath).
• Neumann — fix the derivative: u_x(0, t) = 0. The flux is prescribed; zero means an insulated end (no heat flows through).
• Robin — fix a combination: a\,u(0, t) + b\,u_x(0, t) = g. Models convective cooling, where heat loss depends on the temperature difference with the surroundings.

How many conditions, and of what kind

The type of the equation decides what data it needs:

• Parabolic (heat): boundary conditions on the spatial edges plus one initial condition u(x, 0) = f(x) — the equation is first order in time.
• Hyperbolic (wave): boundary conditions plus two initial conditions, u(x, 0) and u_t(x, 0) — second order in time needs both the initial shape and the initial velocity.
• Elliptic (Laplace): no time at all — just boundary values around the entire edge of the region. It is a pure boundary-value problem.