1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

274 Chapter 4 The Potential Equation

that cos(a/ 2 )=0:

v(x,y)=

cos

(

x−^12 a

)

cos

( 1

2 a

) e−y.

What partial differential equation and boundary conditions are satisfied

byw(x,y)=u(x,y)−v(x,y)ifuis the function of Exercise 7?

9.Solve the potential equation in the slot 0<y<b,0<xfor each of the

following sets of boundary conditions:

a. u( 0 ,y)=0,u(x, 0 )=0,u(x,b)=f(x)=

{ 1 , 0 <x<a,

0 , a<x;

b. u( 0 ,y)=0,u(x, 0 )=e−x,u(x,b)=0.

10.Find product solutions of this potential problem in a strip:

∂^2 u

∂x^2 +

∂^2 u

∂y^2 =^0 ,^0 <x<a, −∞<y<∞,

subject to the boundedness conditionu(x,y)bounded asy→±∞.

11.Solve the potential problem consisting of the equation and boundedness

conditions from Exercise 10 and the boundary conditions

u( 0 ,y)= 0 , u(a,y)=e−|y|, −∞<y<∞.

12.Show how to solve the potential problem of Exercise 10 together with the

boundary conditions

u( 0 ,y)=g 1 (y), u(a,y)=g 2 (y), −∞<y<∞,

whereg 1 andg 2 are suitable functions.

13.Find product solutions of this potential problem in the quarter-plane:

∂^2 u

∂x^2

+∂

(^2) u
∂y^2
= 0 , 0 <x, 0 <y,
u( 0 ,y)= 0 , u(x, 0 )=f(x).
Note thatu(x,y)must remain bounded asx→∞and asy→∞.
14.Solve the potential equation in the quarter-plane,x>0,y>0, subject to
the boundary conditions
u( 0 ,y)=e−y, y> 0 ; u(x, 0 )=e−x, x> 0.