Page 876 - Advanced_Engineering_Mathematics o'neil

856 Answers to Selected Problems

with
4ω sin(4ω) + cos(4ω) − 1 sin(4ω) − 4ω cos(4ω)
a ω = ,b ω =
πω 2 πω 2
By Fourier transform,
1 4
u(x,t) = √ ξe −(x−ξ) 2 /4kt dξ
2 πkt 0

ω
∞ −ω 2 kt
2
5. u(x,t) = e dω
π 0 α + ω 2
2

1 − cos(hω)
2 ∞ −ω 2 kt
7. u(x,t) = 0 sin(ωx)e dω
π ω
ω
∞ −ω 2 t −t 2 /2
4
9. u(x,t) = sin(ωx)e e dω
2 2
π 0 (1 + ω )
Section 17.4 Laplace Transform Techniques
(2n + 1)L − x (2n + 1)L + x

∞
1. u(x,t) = T 0 erfc √ − erfc √
n=0
2 kt 2 kt
√ x
kt

kt
3. u(x,t) = e kt−x − e ∗ L −1 e − s/kx = e kt−x − e ∗ √ e −x 2 /4kt
2 πt 3
Section 17.5 Heat Conduction in an Infinite Cylinder
1. Write
∞ 1
2
− j 2
2
U(r,t) = ξ J 0 ( j n ξ)dξ J 0 ( j n r)e n t.
[J 1 ( j n )] 2
n=1 0
Inserting the approximate values, we have
U(r,t) ≈0.8170J 0 (2.405r)e −5.785t − 1.1394J 0 (5/520r)e −30.47t
+ 0.7983J 0 (8.654r)e −74.89t − 0.747J 0 (11.792r)e −139.04t
+ 0.6315J 0 (14.931r)e −222.93t .
3.
∞ 1
2

2
n
U(r,t) = ξ(9 − ξ )J 0 ( j n ξ)dξ J 0 ( j n r/3)e − j 2 t/18
(J 1 ( j n )) 2
n=1 0
The fifth partial sum approximation is
U(r,t) ≈9.9722J 0 (2.405r/3)e −5.78t/18 − 1.258J 0 (5.520r/3)e −30.47t/18
+ 0.4093J 0 (8.654r/3)e −74.89t/18 − 0.1889J 0 (11.792r/3)e −139.04t/18
+ 0.1048J 0 (14.931r/3)e −222.93t/18
Section 17.6 Heat Conduction in a Rectangular Plate
1.
∞ ∞

u(x,t,t) = b nm sin(nπx/L)sin(mπy/K)e −β nm kt
n=1 n=1
where
n m
2 2
β nm = + π 2
L 2 K 2
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October 14, 2010 17:50 THM/NEIL Page-856 27410_25_Ans_p801-866