Page 27 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

THE FOURIER METHOD: SEPARATION OF VARIABLES 17
To normalize temperature we choose

u − u 0
U = (2.22)
u 1 − u 0
The problem statement reduces to

a
2
U ξξ + U ηη = 0 (2.23)
b
U(0,ξ) = U(1,ξ) = U(η, 1) = 0
U(η, 0) = 1 (2.24)

2.1.7 Separation of Variables
As before, we assume a solution of the form U(ξ, n) = X(ξ)Y (η). We substitute this into the
differential equation and obtain

a
2
Y (η)X ξξ (ξ) =−X(ξ) Y ηη (η) (2.25)
b
Next we divide both sides by U(ξ, n)and obtain
a
2
X ξξ Y nn 2
=− =±λ (2.26)
X b Y
In order for the function only of ξ on the left-hand side of this equation to be equal to the
function only of η on the right-hand side, both must be constant.

2.1.8 Choosing the Sign of the Separation Constant
However in this case it is not as clear as the case of Example 1 what the sign of this constant
2
must be. Hence we have designated the constant as ±λ so that for real values of λ the ± sign
determines the sign of the constant. Let us proceed by choosing the negative sign and see where
this leads.
Thus
2
X ξξ =−λ X
Y (η)X(0) = 1
Y (η)X(1) = 0 (2.27)

or

X(0) = 1

X(1) = 0 (2.28)