Mathematical Methods for Physicists: A concise introduction - Site Map

PARTIAL DIFFERENTIAL EQUATIONS
which implies
b ˆ nv ; n ˆ 1; 2; 3; ...: …10:53†
l
Note that n cannot be equal to zero, because it would make b ˆ 0, which in turn
would make u ˆ 0.
Substituting Eq. (10.53) into Eq. (10.51) we have
u n …x; t† ˆsin nx
l
C n sin nvt
l
‡ D n cos nvt
l
; n ˆ 1; 2; 3; ...: …10:54†
We see that there is an in®nite set of discrete values of b and that to each value of
b there corresponds a particular solution. Any linear combination of these particular
solutions is also a solution:
u n …x; t† ˆX1
nˆ1
sin nx
l
C n sin nvt
l
‡ D n cos nvt
l
: …10:55†
The constants C n and D n are ®xed by the boundary conditions (10.45) and
(10.46).
Application of boundary condition (10.45) yields
f …x† ˆX1
Similarly, application of boundary condition (10.46) gives
g…x† ˆv
l
nˆ1
X 1
nˆ1
D n sin nx : …10:56†
l
nC n sin nx : …10:57†
l
The coecients C n and D n may then be determined by the Fourier series method:
D n ˆ 2
l
Z l
0
f …x† sin nx dx; C
l
n ˆ 2
nv
Z l
0
g…x† sin nx dx:
l
…10:58†
We can use the method of separation of variable to solve the heat conduction
equation. We shall leave this as a home work problem.
In the following sections, we shall consider two more methods for the solution
of linear partial di€erential equations: the method of Green's functions, and the
method of the Laplace transformation which was used in Chapter 9 for the
solution of ordinary linear di€erential equations with constant coecients.
Solution of Poisson's equation. Green's functions
The Green's function approach to boundary-value problems is a very powerful
technique. The ®eld at a point caused by a source can be considered to be the total
e€ect due to each ``unit'' (or elementary portion) of the source. If G…x; x 0 † is the
404