286 Chapter 4 The Potential Equation
5.Same as Exercise 3, but the boundary conditions are
u( 0 ,y)= 1 , u(a,y)= 1 , 0 <y<b,
∂u
∂y(x, 0 )= 0 , u(x,b)= 0 , 0 <x<a.6.Same as Exercise 3, but the boundary conditions are
u( 0 ,y)= 1 , u(a,y)= 0 , 0 <y<b,
u(x, 0 )= 1 , u(x,b)= 0 , 0 <x<a.7.Same as Exercise 3, but the region is a square(b=a)and the boundary
conditions are
u( 0 ,y)=f(y), u(a,y)= 0 , 0 <y<a,
u(x, 0 )=f(x), u(x,a)= 0 , 0 <x<a,
wherefis a function whose graph is an isosceles triangle of heighthand
widtha.
8.Solve the potential equation in the region 0<x<a,0<ywith the
boundary conditions
u(x, 0 )= 1 , 0 <x<a,
u( 0 ,y)= 0 , u(a,y)= 0 , 0 <y.9.Find the solution of the potential equation on the strip 0<y<b,
−∞<x<∞, subject to the conditions that follow. Supply bounded-
ness conditions as necessary.u(x, 0 )={
1 , −a<x<a,
0 , |x|>a,
u(x,b)= 0 , −∞<x<∞.10.Show that the functionu(x,y)=tan−^1 (y/x)is a solution of the potential
equation in the first quadrant. What conditions doesusatisfy along the
positivex-andy-axes?
11.Solve the potential problem in the upper half-plane,
∂^2 u
∂x^2+∂
(^2) u
∂y^2
= 0 , −∞<x<∞, 0 <y,
u(x, 0 )=f(x), −∞<x<∞,
takingf(x)=exp(−α|x|).