for in

6

6

¦ ¦ ¦

0

6

0 6

£ ¦ ¦©

where is a bounded, open domain in . Here, are the two space variables.

¤˜

£ ¥

6

¦

0

¦

„

¥ ¡¢

D T

£

Domain for Poisson™s equation.

The above equation is to be satis¬ed only for points that are located at the interior of

£

the domain . Equally important are the conditions that must be satis¬ed on the boundary

£¥of . These are termed boundary conditions, and they come in three common types:

¦

BV 1

B

"

¦ "

¦

Dirichlet condition

C ¦B % C

7

©§§

Neumann condition

¨‘

C "B V ¦B

— C B ¨%‘ §§ B

"

¦ ¦

¦

Cauchy condition §

C

C C

¤˜

The vector usually refers to a unit vector that is normal to and directed outwards. ¥

Note that the Neumann boundary conditions are a particular case of the Cauchy conditions

¤

6

7 0

with . For a given unit vector, with components and , the directional

§

¤

(

derivative is de¬ned by

V

C ¤ —

BV BV 3

¢ C B ¤

¦ "

¦

¡

V

¦

C

&¢

‘t ¥

V BV

— 0 C B¤ 0 C 6 ¨

&&

"

¦

¦

6©¦

¦

¨ ¢‘ ¥

t

&

V

” ” ”n

uu u ™

n

¢

¤§ ¥ §¢

¡

¡© £ £ ¥ £ ¤ § ¨¡ ¢

§¥ © ©

where is the gradient of ,

V V

%

§§ Y

t

%§ ¨ £ &¥

V

§ 6

3

and the dot in (2.3) indicates a dot product of two vectors in .

In reality, Poisson™s equation is often a limit case of a time-dependent problem. It can,

£

for example, represent the steady-state temperature distribution in a region when there is

¡

a heat source that is constant with respect to time. The boundary conditions should then

model heat loss across the boundary . ¥

¡ B 78

"

¦

The particular case where , i.e., the equation

C

z ¡

aV

7

to which boundary conditions must be added, is called the Laplace equation and its solu-

tions are called harmonic functions.

Many problems in physics have boundary conditions of mixed type, e.g., of Dirichlet

type in one part of the boundary and of Cauchy type in another. Another observation is that

the Neumann conditions do not de¬ne the solution uniquely. Indeed, if is a solution, then V

—w

so is for any constant .

V

The operator

6 6

¡ —