8.7. THE LAPLACE EQUATION
8.7.2
117
Dirichlet problem for a circle
The Laplace equation is commonly written symbolically as
2 = 0,
(8.50)
where 2 is called the Laplacian, sometimes denoted as . The Laplacian can
be written in various coordinate systems, a
112
CHAPTER 8. PARTIAL DIFFERENTIAL EQUATIONS
Example: Determine the concentration of a dye in a pipe of length
, where the dye has unit mass and is initially concentrated at the
center of the pipe, and the ends of the pipe are sealed
Again we model the i
116
8.7.1
CHAPTER 8. PARTIAL DIFFERENTIAL EQUATIONS
Dirichlet problem for a rectangle
We consider the Laplace equation (8.49) for the interior of a rectangle 0 < < ,
0 < < , (see Fig. 8.4), with boundary conditions
(, 0) = 0,
(, ) = 0,
0 < < ;
(0, ) = 0,
8.5. SOLUTION OF THE DIFFUSION EQUATION
111
which can be seen as extending the formula obtained for eigenvalues and eigenvectors for positive given by (8.27) and (8.28) to = 0.
We now turn to the equation for (). The equation corresponding to eigenvalue ,
8.6. SOLUTION OF THE WAVE EQUATION
113
yielding the two ordinary differential equations
+ = 0,
+ 2 = 0.
(8.38)
We solve first the equation for (). The appropriate boundary conditions for
are given by
(0) = 0, () = 0,
(8.39)
and we have solved this equa
118
CHAPTER 8. PARTIAL DIFFERENTIAL EQUATIONS
so that the Laplacian may be found by multiplying both sides of (8.56) by its
complex conjugate, taking care with the computation of the derivatives on the
right-hand-side:
(
)
(
)
2
2
+
=
+
2
2
2
2
1
8.7. THE LAPLACE EQUATION
115
b
y
u=0
u=0
u = f(y)
u=0
0
0
a
x
Figure 8.4: Dirichlet problem for the Laplace equation in a rectangle.
8.6.3
General initial conditions
If the initial conditions on (, ) are generalized to
(, 0) = (),
(, 0) = (),
0 ,
(8.48)
8.5. SOLUTION OF THE DIFFUSION EQUATION
109
As , we assume that a stationary concentration distribution () will
attain, independent of . Since () must satisfy the diffusion equation, we have
() = 0,
0 ,
with general solution
() = + .
Since () must satisf
114
CHAPTER 8. PARTIAL DIFFERENTIAL EQUATIONS
Our solution to the wave equation with plucked string is thus given by (8.43)
and (8.44). Notice that the solution is time periodic with period 2/. The
corresponding fundamental frequency is the reciprocal of
110
CHAPTER 8. PARTIAL DIFFERENTIAL EQUATIONS
Again, we consider in turn the cases > 0, < 0 and = 0. For > 0, we
write = 2 and determine the general solution of (8.26) to be
() = cos + sin ,
so that taking the derivative
() = sin + cos .
Applying the bou
120
CHAPTER 8. PARTIAL DIFFERENTIAL EQUATIONS
Therefore,
(, ) =
2 2
2
0
2
()
,
2 2 cos ( ) + 2
an integral result for (, ) known as Poissons formula. As a trivial example,
consider the solution for (, ) if () = , a constant. Clearly, (, ) =
satisfies bo
8.7. THE LAPLACE EQUATION
119
which is an Euler equation. With the ansatz = , (8.59) reduces to the
algebraic equation ( 1) + 2 = 0, or 2 = 2 . Therefore, = , and
there are two real solutions when > 0 and degenerate solutions when = 0.
When > 0, the solut