Page 29 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

THE FOURIER METHOD: SEPARATION OF VARIABLES 19
Solution for X(ξ) yields hyperbolic functions.

X(ξ) = C cosh(λ n ξ) + D sinh(λ n ξ) (2.40)

The boundary condition at ξ = 1 requires that

0 = C cosh(λ n ) + D sinh(λ n ) (2.41)

or, solving for C in terms of D,

C =−D tanh(λ n ) (2.42)

One solution of our problem is therefore

U n (ξ, η) = K n sin(nπη)[sinh(anπξ/b) − cosh(anπξ/b)tanh(anπ/b)] (2.43)

2.1.9 Superposition
According to the superposition theorem (Theorem 2) we can now form a solution as

∞

U(ξ, η) = K n sin(nπη)[sinh(anπξ/b) − cosh(anπξ/b)tanh(anπ/b)] (2.44)
n=0

The final boundary condition (the nonhomogeneous one) can now be applied,

∞

1 =− K n sin(nπη)tanh(anπ/b) (2.45)
n=1

2.1.10 Orthogonality
We have already noted that the sine function is an orthogonal function as defined on (0, 1).
Thus, we multiply both sides of this equation by sin(mπη)dη and integrate over (0, 1), noting
that according to the orthogonality theorem (Theorem 3) the integral is zero unless n = m.
The result is
1 1

2
sin(nπη)dη =−K n sin (nπη)dη tanh(anπ/b) (2.46)
η=0 η=0
1 n 1
[1 − (−1) ] =−K n tanh(anπ/b) (2.47)
nπ 2
n
2[1 − (−1) ]
K n =− (2.48)
nπ tanh(anπ/b)