1.4. EXAMPLE 2: THE HEAT EQUATION

has a nontrivial (i.e. a nonzero) solution is called an eigenvalue; w then is called a corresponding eigenfunction.

If λn is an eigenvalue of problem (1.24) with corresponding eigenfunction wn, then

e−λntwn(x)

is a solution of the first two equations in (1.21). It is known that equation (1.24) has infinitely many real eigenvalues 0 ≤ λ1 ≤ λ2 ≤ · · · , (λnt→∞−→ ∞). Multiple eigenvalues are counted according to their multiplicity. An arbitrary bounded piecewise continuous function can be represented as a linear combination of the eigenfunctions w1, w2, . . .. Therefore, the solution of (1.21) can be written in the form

The smallest eigenvalue of (1.24) is λ1 = 0 with w1 = 1 and λ2 > 0. Therefore we see from (1.26) that

t→∞

Thus, in the limit (i.e., as t goes to infinity), the temperature will be constant in the whole container. The convergence rate towards this equilibrium is determined by the smallest positive eigenvalue λ2 of (1.24):

n=2

Here we have assumed that the value of the constant function w1(x) is set to unity.

1.5Example 3: The wave equation

The air pressure u(x, t) in a volume with acoustically “hard” walls satisfies the equations

Sound propagates with speed − u, along the (negative) gradient from high to low pressure.

To solve the wave equation we proceed as with the heat equation in section 1.4: separation of u according to (1.23) leads again to equation (1.24) but now together with

We know this equation from the analysis of the vibrating string, see (1.11). From there we know that the general solution of the wave equation has the form

where the wk, k = 1, 2, . . ., are the eigenfunctions of the eigenvalue problem (1.24). The coe cients ak and bk are determined by (1.31) and (1.32).

If a harmonic oscillation is forced on the system, an inhomogeneous problem

is obtained. The boundary and initial conditions are taken from (1.29)–(1.32). This problem can be solved by setting

An and Bn are real constants that are determined by the initial conditions. If ω gets close

√ √

to λ1, then the last term can be very large. In the limit, if ω = λn, v˜n gets the form

In this case, v˜n is not bounded in time anymore. This phenomenon is called resonance. Often resonance is not desirable; it may, e.g., mean the blow up of some structure. In order to prevent resonances eigenvalues have to be known. Possible remedies are changing the domain (the structure).

Remark 1.1. Vibrating membranes satisfy the wave equation, too. In general the boundary conditions are di erent from (1.30). If the membrane (of a drum) is fixed at its boundary, the condition

is imposed. This boundary conditions is called Dirichlet boundary conditions. The boundary conditions in (1.21) and (1.30) are called Neumann boundary conditions. Combinations of these two can occur.