1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

472 Answers to Odd-Numbered Exercises

Membrane is attached to a frame that is flat on the left and right but has

the shape of the graph off(x)at top and bottom.

b. ∂

(^2) u
∂x^2

+∂

(^2) u
∂y^2
=0, 0 <x<a,0<y<b,
∂u
∂x(^0 ,y)=0, u(a,y)=0,^0 <y<b,
u(x, 0 )=0, u(x,b)=100, 0 <x<a.
The bar is insulated on the left; the temperature is fixed at 100 on the top,
at 0 on the other two sides.
c. ∂
(^2) u
∂x^2

+∂

(^2) u
∂y^2
=0, 0 <x<a,0<y<b,
u( 0 ,y)=0, u(a,y)=100, 0 <y<b,
∂u
∂y(x,^0 )=0,
∂u
∂y(x,b)=0,^0 <x<a.
The sheet is electrically insulated at top and bottom. The voltage is fixed
at 0 on the left and 100 on the right.
d. ∂
(^2) φ
∂x^2 +
∂^2 φ
∂y^2 =0,^0 <x<a,0<y<b,
∂φ
∂x
( 0 ,y)=0, ∂φ
∂x
(a,y)=−a,0<y<b,
∂φ
∂y(x,^0 )=0,
∂φ
∂y(x,b)=b,0<x<a.
The velocities, given byV=−∇φ,areVx=a,Vy=0ontheright,
Vx=0,Vy=−bonthetop;andwallsontheothertwosidesmakeve-
locities 0 there.
Section 4.2

1. Show by differentiating and substituting that both are solutions of the
   differential equation. The Wronskian of the two functions is
   ∣∣
   ∣∣ sinh(λy) sinh(λ(b−y))
   λcosh(λy) −λcosh(λ(b−y))

∣∣

∣∣=−λsinh(λb)= 0.

3. In the caseb=a, use two terms of the series:u(a/ 2 ,a/ 2 )= 0 .32.

5.u(x,y)=

∑∞

1

bnsin

(

nπx

a

)

sinh(nπy/a)

sinh(nπb/a)

,bn=^8

n^2 π^2

sin

(

nπ

2

)

.