Page 31 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

THE FOURIER METHOD: SEPARATION OF VARIABLES 21
condition.

W ξξ + W ηη = 0 (2.57)

W(0,η) = W(1,η) = W(ξ, 0) = 0 (2.58)
W(ξ, 1) = 1
V ξξ + V ηη = 0 (2.59)

V (0,η) = V (1,η) = V (ξ, 1) = 0 (2.60)
V (ξ, 0) = 1

(It should be clear that these two problems are identical if we put V = W(1 − η). We will
therefore only need to solve for W.)

2.1.14 Separation of Variables
Separate variables by letting W(ξ, η) = P(ξ)Q(η).

P ξξ Q ηη 2
=− =±λ (2.61)
P Q
2.1.15 Choosing the Sign of the Separation Constant
Once again it is not immediately clear whether to choose the plus sign or the minus sign. Let’s
see what happens if we choose the plus sign.

2
P ξξ = λ P (2.62)
The solution is exponentials or hyperbolic functions.

P = A sinh(λξ) + B cosh(λξ) (2.63)

Applying the boundary condition on ξ = 0, we find that B = 0. The boundary condition on
ξ = 1 requires that A sinh(λ) = 0, which can only be satisfied if A = 0or λ = 0, which yields
a trivial solution, W = 0, and is unacceptable. The only hope for a solution is thus choosing
the minus sign.
If we choose the minus sign in Eq. (2.61) then

2
P ξξ =−λ P (2.64)
2
Q ηη = λ Q (2.65)

with solutions

P = A sin(λξ) + B cos(λξ) (2.66)