Page 32 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

22 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
and

Q = C sinh(λη) + D cosh(λη) (2.67)

respectively. Remembering to apply the homogeneous boundary conditions first, we find that for
W(0,η) = 0, B = 0and for W(1,η) = 0, sin(λ) = 0. Thus, λ = nπ, our eigenvalues cor-
responding to the eigenfunctions sin(nπξ). The last homogeneous boundary condition is
W(ξ, 0) = 0, which requires that D = 0. There are an infinite number of solutions of the form

PQ n = K n sinh(nπη)sin(nπξ) (2.68)

2.1.16 Superposition
Since our problem is linear we apply superposition.

∞

W = K n sinh(nπη)sin(nπξ) (2.69)
n=1
Applying the final boundary condition, W(ξ, 1) = 1

∞

1 = K n sinh(nπ)sin(nπξ). (2.70)
n=1

2.1.17 Orthogonality
Multiplying both sides of Eq. (2.70) by sin(mπξ) and integrating over the interval (0, 1)

1 1
∞

sin(mπξ)dξ = K n sinh(nπ) sin(nπξ)sin(mπξ)dξ (2.71)
n=0
0 0
The orthogonality property of the sine eigenfunction states that

1

0, m = n
sin(nπξ)sin(mπξ)dξ = (2.72)
1/2, m = n
0
Thus,

K n = 2/ sinh(nπ) (2.73)

and
∞
2

W = sinh(nπη)sin(nπξ) (2.74)
sinh(nπ)
n=0