Page 144 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

134 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
Obtaining the homogeneous and particular solutions yields

1 √ √
U(r, s ) = + AJ 0 (i sr) + BY 0 (i sr)
s
The boundedness condition requires that B = 0, while the condition at r = 1

1
A =− √
sJ 0 (i s )
Thus
√
1 J 0 (i sr)
U(r, s ) = − √
s sJ 0 (i s )
The inverse transform is as follows:
√

st J 0 (i sr
u(r, t) = 1 − Residues of e √
sJ 0 (i s )
√ √
Poles of the function occur at s = 0and J 0 (i s ) = 0or i s = λ n , the roots of the Bessel
2
function of the first kind order are zero. Thus, they occur at s =−λ . The residues are
n
√
lim st J 0 (i sr)
e √ = 1
s → 0 J 0 (i s )
and
√ √
lim J 0 (i sr) lim J 0 (i sr) 2 J 0 (λ n r)
e st √ = e st √ √ = e −λ t
n
1

s →−λ 2 n sJ (i s ) s →−λ 2 n −J 1 (i s ) i/2 s − λ n J 1 (λ n )
0
2
The two unity residues cancel and the final solution is as follows:
∞
−λ 2t J 0 (λ n r)
u(r, t) = e n
λ n J 1 (λ n )
n=1
Problems
1. Consider a finite wall with initial temperature zero and the wall at x = 0 insulated.
Thewallat x = 1 is subjected to a temperature u(1, t) = f (t)for t > 0. Find u(x, t).

2. Consider a finite wall with initial temperature zero and with the temperature at x =
0 u(0, t) = 0. The temperature gradient at x = 1 suddenly becomes u x (1, t) = f (t)
for t > 0. Find the temperature when f (t) = 1 and for general f (t).
3. A cylinder is initially at temperature u = 1 and the surface is subject to a convective
boundary condition u r (t, 1) + Hu(t, 1) = 0. Find u(t, r).