Lecture 39
Finite Difference Method for Elliptic
PDEs
Examples of Elliptic PDEs
Elliptic
PDE’s are equations with second derivatives in space and no time derivative. The most
important examples are Laplace’s equation
Δ
u
=
u
xx
+
u
yy
+
u
zz
= 0
and the Poisson equation
Δ
u
=
f
(
x,y,z
)
.
These equations are used in a large variety of physical situations such as: steady state heat problems,
steady state chemical distributions, electrostatic potentials, elastic deformation and steady state
fluid flows.
For the sake of clarity we will only consider the two dimensional problem. A good model problem
in this dimension is the elastic deflection of a membrane. Suppose that a membrane such as a sheet
of rubber is stretched across a rectangular frame. If some of the edges of the frame are bent, or if
forces are applied to the sheet then it will deflect by an amount
u
(
x,y
) at each point (
x,y
). This
u
will satify the boundary value problem:
u
xx
+
u
yy
=
f
(
x,y
)
for
(
x,y
) in
R,
u
(
x,y
) =
g
(
x,y
)
for
(
x,y
) on
∂R,
(39.1)
where
R
is the rectangle,
∂R
is the edge of the rectangle,
f
(
x,y
) is the force density (pressure)
applied at each point and
g
(
x,y
) is the deflection at the edge.
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 Fall '08
 Young,T
 Equations, Derivative, Partial Differential Equations, Boundary value problem, Sturm–Liouville theory, Green's function, Elliptic PDEs

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