262 Chapter 4 The Potential Equation
(a) (b)Figure 2 (a) Level curves of the solutionu(x,y)of the example problem (see
Eq. (12)) for the caseb=a=1. Each curve is part of the locus of points that
satisfyu(x,y)=constant for constants 0 to 0.9 in steps of 0.1. For some constants,
the locus consists of more than one connected curve. (b) Perspective view of the
surfacez=u(x,y).
Now we have seen a solution of Dirichlet’s problem in a rectangle with ho-
mogeneous conditions on two parallel sides. In general, of course, the bound-
ary conditions will be nonhomogeneous on all four sides of the rectangle. But
this more general problem can be broken down into two problems like the one
we have solved.
Consider the problem
∇^2 u= 0 , 0 <x<a, 0 <y<b, (13)
u(x, 0 )=f 1 (x), 0 <x<a, (14)
u(x,b)=f 2 (x), 0 <x<a, (15)
u( 0 ,y)=g 1 (y), 0 <y<b, (16)
u(a,y)=g 2 (y), 0 <y<b. (17)
Letu(x,y)=u 1 (x,y)+u 2 (x,y). We will put conditions onu 1 andu 2 so that
they can readily be found, and from themucan be put together. The most
obvious conditions are the following:
∇^2 u 1 = 0 , ∇^2 u 2 = 0 ,
u 1 (x, 0 )=f 1 (x), u 2 (x, 0 )= 0 ,
u 1 (x,b)=f 2 (x), u 2 (x,b)= 0 ,
u 1 ( 0 ,y)= 0 , u 2 ( 0 ,y)=g 1 (y),
u 1 (a,y)= 0 , u 2 (a,y)=g 2 (y).