91
CHAPTER 9
NUMERICAL
PARTIAL
DIFFERENTIAL
EQUATIONS
(PDE)
A classic example of partial differential equations (PDE) is
the heat equation that governs the temperature distribution as
time evolves. For a uniform onedimensional object, the
equation takes the form of:
2
2
x
Q
D
t
Q
∂
∂
=
∂
∂
,
(*)
where
D
is called the diffusion coefficient and
Q
(
x
,
t
) is the
unknown temperature at location
x
and time
t
.
x
0
L
/2

L
/2
t
=0
Insulator
Hot spot
t
=
Uniform
t
>0
Diffusion
8
EE 3108 Semster B 2007/2008
S C Chan
92
The differentiations are said to be
partial
because they treat
the two independent variables separately. The equation is
referred to as
homogeneous
because it has no dependence on
x
nor
t
(except for the derivatives).
Similar to ODEs, the initial conditions at
t
=0 (for all
x
) are
typically given for an PDE problem. In addition, the
boundary conditions at
x
=
±
L
/2(for all
t
) should also be
given. Therefore, the problem is an initial value problem
(IVP) and also a boundary problem (BVP).
Two approaches are considered in this chapter, which are
based on the approximation of the firstorder timederivative.
The
explicit
method
uses
the
forwarddifference
approximation whilst the
implicit method
uses the backward
difference approximation. These methods can be modified for
solving other types of PDE.
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93
I. Physical background
A. Heat equation
The heat equation belongs to the class of parabolic PDE that
are frequently used in electrical engineering such as
semiconductor carrier transport and paraxial electromagnetic
wave propagation. The heat equation for a onedimensional
rod can be obtained by considering the heat flow
J.
x
Hot
Cold
Heat flow,
J
J
Absorbs heat
On the left, we can deduce that the heat flow is proportional
to the heat gradient:
x
Q
J
∂
∂
−
∝
On the right, we can deduce that the change of heat flow has
to be absorbed:
x
J
t
Q
∂
∂
−
∝
∂
∂
Combine the two equations, we have:
2
2
x
Q
D
t
Q
∂
∂
=
∂
∂
,
where
D
represents the proportionality constant. For copper,
D
= 1.16 cm
2
/s.
94
B. Boundary conditions
Suppose we use a copper rod of length
L
= 10 cm and put it
in between two insulators. No heat is permitted to flow at the
boundaries:
J
= 0
at
x
=
±
L
/2. Equivalently, in terms of
Q
,
we have the boundary conditions:
0
2
/
=
∂
∂
±
=
L
x
x
Q
.
Boundary conditions are usually expressed in terms of
Q
or
its derivatives.
C. Initial conditions
The center portion of the copper rod is first heated before
t
=
0. The PDE can be used to find the temperature distribution
for
t
> 0. We shall be interested in the time window from
t
=0
to
T
, where
T
= 10 s. The initial temperature distribution is
given as:
<
<
−
=
otherwise
25
2
/
2
/
for
100
)
0
,
(
a
x
a
x
Q
,
where
a
= 1 cm is the length of the heated region. The unit
for the temperature is chosen to be Celsius here.
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 Spring '07
 NelsonSzeChunChan
 Boundary value problem, Partial differential equation, Qn, Qn Qn

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