Page 158 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

148 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
with boundary conditions

r (λ n , 0) = 0
(λ n , 1) = 0

The solution is the Bessel function J 0 (λ n r)with λ n given by J 0 (λ n ) = 0. Thus the
transform of u(t, r)isasfollows:
1

H λ u(t,r) = U(t,λ n ) = rJ 0 (λ n r)u(t,r)dr
r=0
This is called a Hankel transform. The appropriate differential equation for U(t,λ n )is

dU n 2
=−λ U n
n
dt
so that

2
U n (t,λ n ) = Be −λ t
n
Applying the initial condition, we find

1

B = rf (r)J 0 (λ n r)dr
r=0
and from Eq. (9.19)

∞ 1
rf (r)J 0 (λ n r)dr 2
r=0 −λ t
u(t,r) = J 0 (λ n r)e n
# # 2
J 0 (λ n r) #
n=0
#
Example 9.4 (The sine transform with a source). Next consider a one-dimensional transient
diffusion with a source term q(x):
u t = u xx + q(x)

y(0, x) = y(t, 0) = t(t,π) = 0

First we determine that the sine transform is appropriate. The operator is such that

= xx = λ