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CS205 – Class 18
Covered in Class
: 1, 2, 3, 4
Readings
: Heath 11.111.2
Partial Differential Equations
1.
There are three types of PDE’s
a.
Elliptic, Hyperbolic, Parabolic
2.
The
Laplace Equation
is
2
2
2
1
n
i
i
p
p
pf
x
=
∂
∇=
Δ
=
=
∂
∑
is the model elliptic equation
a.
For PDE’s we often use subscript notation for derivatives.
E.g. in 2D it is
xx
yy
p
+=
, in 1D it is
xx
p
f
=
.
b.
Let’s look at a simpler case where
f
=0.
c.
In 1D we have
0
xx
p
=
.
The solution analytically is a straight line, i.e.,
p
ax
b
=
+
.
d.
To determine the exact formula for
p
we need certain constraints. Those usually
come in the form of boundary conditions, i.e. constraints on the value of
p
or its
derivatives for values of
x
belonging to the boundary of the domain in which we want
to solve.
e.
We can also have boundary conditions which involve the derivatives of the function
p
, called
Neumann
boundary conditions. We note that
(
)(
)
01
p
pb
′′
==
, thus if we
supply Neumann conditions for all boundary points then the function
p
is only
determined up to an additive constant. In order to determine the exact function
p
we
must supply
Dirichlet
conditions for at least one of the boundary points
i.
Take this plot
0.2
0.4
0.6
0.8
1
1
2
3
4
5
which represents the solution to the 1D
Laplace equation.
1.
We can get it by specifying two Dirichlet conditions
(0)
2
p
=
and
(1)
3
p
=
.
2.
We can also get it by specifying one Dirichlet p(0)=2 and one Neumann
p’(0)=1 (or indeed anywhere in 1D i.e.
() 1
x
pt
=
for any t).
3.
Specifying both Neumann end points i.e.
(0)
x
p
and
x
p
is troublesome.
a.
They need to be the same, because our solution is a line.
Similar
restrictions occur in multiD.
This is called the
compatibility
condition
.
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Additionally only can determine
p
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 Fall '07
 Fedkiw

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