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 La
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, a
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 bound
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 equa
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
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 th
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 (, )
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,
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
co
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 t
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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 (
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