Mathematical Methods for Physicists: A concise introduction - Site Map

SOLUTIONS OF LAPLACE'S EQUATION
and
@u…x; t†
@t
ˆ g…x†; 0 x l; …10:46†
tˆ0
where f and g are given functions.
Assuming that the solution of Eq. (10.43) may be written as a product
u…x; t† ˆX…x†T…t†;
then substituting into Eq. (10.43) and dividing by XT we obtain
1 d 2 X
X dx 2 ˆ 1 d 2 T
v 2 T dt 2 :
…10:47†
Both sides of this last equation must be equal to a constant, say b 2 =v 2 . Then we
have two equations
1 d 2 X
X dx 2 ˆb2 v 2 ;
…10:48†
1 d 2 T
T dt 2 ˆb2 : …10:49†
The solutions of these equations are periodic, and it is more convenient to write
them in terms of trigonometric functions
X…x† ˆA sin bx v ‡ B cos bx v
; T…t† ˆC sin bt ‡ D cos bt; …10:50†
where A; B; C, and D are arbitrary constants, to be ®xed by the boundary conditions.
Equation (10.47) then becomes
u…x; t† ˆ A sin bx v ‡ B cos bx
…C sin bt ‡ D cos bt†: …10:51†
v
The boundary condition u…0; t† ˆ0…t > 0† gives
for all t, which implies
0 ˆ B…C sin bt ‡ D cos bt†
B ˆ 0:
…10:52†
Next, from the boundary condition u…l; t† ˆ0…t > 0† we have
0 ˆ A sin bl …C sin bt ‡ D cos bt†:
v
Note that B ˆ 0 would make u ˆ 0. However, the last equation can be satis®ed
for all t when
sin bl
v ˆ 0;
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