Page 37 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

THE FOURIER METHOD: SEPARATION OF VARIABLES 27
value problem is as follows:

2
y tt = a y xx (2.94)
y(t, 0) = 0
y(t, L) = 0 (2.95)

y(0, x) = f (x)
y t (0, x) = 0

2.2.1 Scales and Dimensionless Variables
The problem has the obvious length scale L. Hence let ξ = x/L. Now let τ = ta/L and the
equation becomes

(2.96)
y ττ = y ξξ
One could now nondimensionalize y, for example, by defining a new variable as
f (x)/f max , but it wouldn’t simplify things. The boundary conditions remain the same except t
and x are replaced by τ and ξ.

2.2.2 Separation of Variables
You know the dance. Let y = P(τ)Q(ξ). Differentiating and substituting into Eq. (2.96),

(2.97)
P ττ Q = PQ ξξ
Dividing by PQ and noting that P ττ /P and Q ξξ /Q cannot be equal to one another unless
they are both constants, we find

P ττ /P = Q ξξ /Q =±λ 2 (2.98)

It should be physically clear that we want the minus sign. Otherwise both solutions will be
hyperbolic functions. However if you choose the plus sign you will immediately find that
the boundary conditions on ξ cannot be satisfied. Refer back to (2.63) and the sentences
following.
The two ordinary differential equations and homogeneous boundary conditions are

2
P ττ + λ P = 0 (2.99)
P τ (0) = 0