Page 877 - Advanced_Engineering_Mathematics o'neil

Answers to Selected Problems 857

and
4 K L
b nm = f (ξ,η)sin(nπξ/L)sin(mπη/K)dξ dη.
LK 0 0
8 m
3. u(x, y,t) = sin(x) ∞ sin(my)e −(1+m 2 )kt
m=1
π 4m − 1
2
CHAPTER EIGHTEEN THE POTENTIAL EQUATION
Section 18.1 Laplace’s Equation
1. ( f + g) xx + ( f + g) yy = ( f xx + f yy ) + (g xx + g yy ) = 0and (cf ) xx + (cf ) yy = c( f xx + g yy ) = 0
3
3
2
2
2. (a) (x − 3xy ) xx + (x − 3xy ) yy = 6x − 6x = 0
4
4
2
4
2
2
4
2
(c) (x − 6x y − y ) xx + (x − 6x y − y ) yy = 6y − 6y = 0
−y
−y
y
−y
y
−y
y
y
(e) (sin(x)(e + e )) xx + (sin(x)(e + e )) yy =−sin(x)(e + e ) + sin(x)(e + e ) = 0
2
2
2y − zx 2 2x − 2y 2
(g) f xx = , f yy =
2
2
2 2
2 2
(x + y ) (x + y )
Section 18.2 Dirichlet Problem for a Rectangle
1
1. u(x, y) = sin(πx)sinh(π(π − y))
2
sinh(π )
32 n(−1) n+1

∞
3. u(x, y) = sin(nπx)sinh(nπy)
n=1 π sinh(4nπ) (2n − 1) (2n + 1) 2
2
2
1
5. u(x, y) = sin(πx)sinh(πy)+
sinh(π )
2
n
16n[(−1) − 1]
∞

+ sin(nπx/2)sinh(nπy/2)
n=1,n
=2 π (n − 2) (n + 2) sinh(nπ /2)
2
2
2
2
7.
∞

u(x, y) = c n sin((2n − 1)πx/2a)sinh((2n − 1)πy/2a)
n=1
where
2 a
c n = f (ξ)sin((2n − 1)πξ/2a)dξ
a sinh((2n − 1)πb/2a) 0
n
−1 2 2(1 − (−1) )
9. u(x, y) = sinh(π(x − 4))sin(πy) + ∞ sinh(nπx)sin(nπy)
n=1
sinh(4π) sinh(4nπ) π n
3 3
Section 18.3 Dirichlet Problem for a Disk
1. u(r,θ) = 1
r
1 ∞ n 1
2
3. u(r,θ) = π + 2(−1) n [2cos(nθ) + n sin(nθ)]
3 n=1 2 n 2
5.
1
u(r,θ) = sinh(π)
π
∞ n (−1)
2 r n
+ [a n cos(nθ) + b n sin(nθ)],
2
π 4 n + 1
n=1
where
a n = sinh(π) and b n = n sinh(π).
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October 14, 2010 17:50 THM/NEIL Page-857 27410_25_Ans_p801-866