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Chapter 5

Boundary and initial conditions

As you all know, solutions to ordinary differential equations are usually not unique (inte-
gration constants appear in many places). This is of course equally a problem for PDE’s.
PDE’s are usually specified through a set of boundary or initial conditions. A boundary
condition expresses the behaviour of a function on the boundary (border) of its area of def-
inition. An initial condition is like a boundary condition, but then for the time-direction.
Not all boundary conditions allow for solutions, but usually the physics suggests what
makes sense. Let us remind you of the situation for ordinary differential equations, one
you should all be familiar with, a particle under the influence of a constant force,

2
d x
= a.
dt 2
Which leads to
dx
= at + v 0 ,
dt
and
1
2
x = at + v 0 t + x 0 .
2
This contains two integration constants. Standard practice would be to specify ∂x (t =
∂t
0) = v 0 and x(t = 0) = x 0 . These are linear initial conditions (linear since they only
involve x and its derivatives linearly), which have at most a first derivative in them. This
one order difference between boundary condition and equation persists to PDE’s. It is kind
of obviously that since the equation already involves that derivative, we can not specify
the same derivative in a different equation.
The important difference between the arbitrariness of integration constants in PDE’s
and ODE’s is that whereas solutions of ODE’s these are really constants, solutions of
PDE’s contain arbitrary functions.
Let us give an example. Take u = yf(x) then ∂u = f(x). This can be used to
∂y
eliminate f from the first of the equations, giving u = y ∂u which has the general solution
∂y
u = yf(x).
One can construct more complicated examples. Consider

u(x, y) = f(x + y) + g(x − y)

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