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which gives on double differentiation
2
2
∂ u ∂ u
− = 0.
∂x 2 ∂y 2
The problem is that without additional conditions the arbitrariness in the solutions makes
it almost useless (if possible) to write down the general solution. We need additional
conditions, that reduce this freedom. In most physical problems these are boundary
conditions, that describes how the system behaves on its boundaries (for all times) and
initial conditions, that specify the state of the system for an initial time t = 0. In the
ODE problem discussed before we have two initial conditions (velocity and position at
time t = 0).
Because PDEs typically have many solutions, we single out one solution by imposing
auxiliary conditions. We attempt to formulate the conditions so as to specify a unique
solution. These conditions are motivated by the physics and they come in two varieties,
initial conditions and boundary conditions.
An initial condition specifies the physical state at a particular time t 0 . For the diffusion
equation the initial condition is
u(x, t 0 ) = φ(x), (5.1)

where φ(x) = φ(x, y, z) is a given function. For a diffusing substance, φ(x) is the
initial concentration. For heat flow, φ(x) is the initial temperature. For the Schr¨odinger
equation, too, (5.1) is the usual initial condition.
For the wave equation there is a pair of initial conditions

∂u
u(x, t 0 ) = φ(x) and (x, t 0 ) = ψ(x), (5.2)
∂t

where ψ(x) is the initial position and ψ(x) is the initial velocity. It is clear on physical
grounds that both of them must be specified in order to determine the position u(x, t) at
later times.
In each physical problem we have seen that there is a domain D in which the PDE
is valid. For the vibrating string, D is the interval 0 < x < l, so the boundary of D
consists only of the two points x = 0 and x = l. For the drumhead, the domain is a plane
region and its boundary is a closed curve. For the diffusing chemical substance, D is the
container holding the liquid, so its boundary is a surface S. For the hydrogen atom, the
domain is all of space, so it has no boundary.
It is clear, again from our physical intuition, that it is necessary to specify some
boundary condition if the solution is to be determined. The three most important kinds
of boundary conditions are:

(D) u is specified (”Dirichlet condition”);

(N) the normal derivative ∂u/∂n is specified (”Neumann condition”);

(R) ∂u/∂n + au is specified (”Robin condition”)

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