Page 157 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

STURM–LIOUVILLE TRANSFORMS 147
Solution of this differential equation and applying the boundary conditions yields an
infinite number of functions (as any Sturm–Liouville problem)

(x,λ n ) = A cos(λ n x)

with
(2n − 1)
cos(λ n ) = 0 λ n = π
2
π
Thus, the appropriate kernel function is K(x,λ n ) = cos(λ n x)with λ n = (2n − 1) .
2
Using this kernel function in the original partial differential equation, we find
dY
2
=−λ Y
dt n

where C λ y(x, t) = Y (t,λ n ) is the cosine transform of y(t, x). The solution gives
Y (t,λ n ) = Be −λ 2 t

and applying the cosine transform of the initial condition
1

B = f (x)cos(λ n x)dx
x=0
According to Eq. (9.19) the solution is as follows:

1
∞
cos(λ n x)
2
−λ t
y(x, t) = f (x)cos(λ n x)dxe n
# # 2
cos(λ n x)
n=0 # #
x−0
Example 9.3 (The Hankel transform). Next consider the diffusion equation in cylindrical
coordinates.
1 d du
u t = r
r dr dr
Boundary and initial conditions are prescribed as
u r (t, 0) = 0
u(t, 1) = 0

u(0,r) = f (r)
First we find the proper kernel function

d d n 2
[ (r,λ n )] = r =−λ r
n
dr dr