08 / CHAPTER 10
Boundary and initial conditions
Make a differential equation into a physical problem.
ANCHOR EQUATION
Boundary condition
In symbols: u|∂Ω = g
A boundary condition tells the differential equation what the domain edge must do.
Boundary and initial conditions make the model physical
A differential equation describes local behavior but does not know what happens at walls, inlets, or the initial instant. Boundary conditions encode the environment; initial conditions encode preparation.
Count conditions from derivative order
A first-order time equation needs one initial field; a second-order time equation needs two, such as displacement and velocity. A second-order spatial equation on an interval typically needs two boundary conditions, such as fixed values or fluxes. They select a unique solution when compatibility conditions hold.
For a rod with steady temperature T'' = 0, boundary values T(0) = 100°C and T(1) = 20°C give T(x) = 100 − 80x. The slope is −80°C/m, and the heat flux points toward +x.
Over-specifying can contradict physics
Imposing both temperature and heat flux independently at the same boundary can be inconsistent unless they happen to match. At a fluid wall, no-slip sets velocity while pressure adjusts through the solution.
Work it: Solve T'' = 0 on 0 ≤ x ≤ 2 with T(0) = 50 and T'(2) = −10. Interpret the resulting slope and endpoint temperature.