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Chapter 13
Lecture 17  Solving the heat
equation using fnite
diFerence methods
13.1
Approximating the Derivatives o± a ²unction
by ²inite DiFerences
Recall that the derivative of a function was deFned by taking the limit of a
di±erence quotient:
f
0
(
x
) = lim
Δ
x
→
0
f
(
x
+Δ
x
)
−
f
(
x
)
Δ
x
.
(13.1)
Now to use the computer to solve di±erential equations we go in the opposite
direction  we replace derivatives by appropriate di±erence quotients. If we
assume that the function can be di±erentiated many times then Taylor’s
Theorem is a very useful device in determining the appropriate di±erence
quotient to use. ²or example consider
f
(
x
x
)=
f
(
x
)+Δ
xf
0
(
x
)+
Δ
x
2
2!
f
0
(
x
Δ
x
3
3!
f
(3)
(
x
Δ
x
4
4!
f
(4)
(
x
...
(13.2)
Rearranging terms in (2) and dividing by Δ
x
we obtain
f
(
x
x
)
−
f
(
x
)
Δ
x
=
f
0
(
x
Δ
x
2
f
0
(
x
Δ
x
2
3!
f
(3)
(
x
....
85
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View Full DocumentLecture 17  Solving the heat equation using fnite diFerence methods
If we take the limit Δ
x
→
0 then we recover (1). But for our purposes it is
more useful to retain the approximation
f
(
x
+Δ
x
)
−
f
(
x
)
Δ
x
=
f
0
(
x
)+
Δ
x
2
f
0
(
ξ
)
(13.3)
=
f
0
(
x
O
(Δ
x
)
.
We retain the term
Δ
x
2
f
0
(
ξ
) in (3) as a measure of the error involved when
we approximate
f
0
(
x
) by the diFerence quotient
(
f
(
x
x
)
−
f
(
x
)
)
/
Δ
x
.
Notice that this error depends on how large
f
0
is in the interval [
x, x
x
]
(i.e. on the smoothness of
f
) and on the size of Δ
x
. Since we like to focus
on that part of the error we can control we say that the error term is of the
order Δ
x
– denoted by
O
(Δ
x
). Technically a term or function
E
(Δ
x
)i
s
O
(Δ
x
f
E
(Δ
x
)
Δ
x
Δ
x
→
0
→
const
.
Now the diFerence quotient (3) is not the only one that can be used to
approximate
f
0
(
x
). Indeed if we consider the expansion of
f
(
x
−
Δ
x
):
f
(
x
−
Δ
x
)=
f
(
x
)
−
Δ
xf
0
(
x
Δ
x
2
2!
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 Spring '11
 lo
 Math, Derivative

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