Di erential Equations
Separation of Variables and Heat
1. Initial Value Problems
Partial di erential equations generally have lots of solutions. To specify a unique one, we’ll need some additional conditions. These conditions are usually motivated by the physics and come in two varieties: initial conditions and boundary conditions. An initial value problem for a PDE consists of the equation, initial conditions, and boundary conditions.
An initial condition specifies the physical state at a given time t . For example, an initial condition for the heat equation would be the starting temperature distribution
u(x, 0) = f(x).
This is the only initial condition required because the heat equation is first order with respect to time. The wave equation, which we will look at later, is second order with respect to time, and so needs two initial conditions.
PDEs are also generally only valid on a certain domain. Boundary conditions specify how the solution is to behave on the boundary of this domain. These need to be specified, because the solution isn’t on the one side of the boundary, meaning we might have problems with di erentiabiltiy there.
Our heat equation was derived for a one-dimensional bar of length l, so the relevant domain in question can be taken to be the interval 0 < x < l and the boundary consists of the two points x = 0 and x = l. We could have derived a two-dimensional heat equation, for example, in which case the domain would be some region in the xy-plane with boundary some closed curve.
It will usually be clear from a physical description of the problem what the appropriate boundary conditions are. We might know that, at the endpoints x = 0 and x = L, the temperatures u(0, t) and u(l, t) are fixed. Boundary conditions that give the value of the solution are called Dirichlet conditions. Or we might insulate the ends, meaning there should be no heat flow out of the boundary; this would yield the boundary conditions ux(0, t) = ux(l, t) = 0. If the boundary conditions specify the derivative of the solution, they’re called Neumann conditions. We could also specify that we have one insulated end and at the other, we control the temperature; this is an example of a mixed boundary condition.
As we’ve seen, changing boundary conditions can significantly change the character of a prob- lem. Initially, to get a feel for our solution method, we’ll work with the homogeneous Dirichlet conditions u(0, t) = u(l, t) = 0, giving us the following initial value problem:
ut = kuxx
u(0, t) = u(l, t) = 0
u(x, 0) = f(x).
After we’ve seen the general method, we’ll see what happens with homogeneous Neumann conditions, leaving nonhomogeneous conditions for a later discussion.