2.7: NUMERICAL APPROXIMATIONS:
EULER’S METHOD
KIAM HEONG KWA
Unless otherwise stated, we shall assume that the firstorder initial
value problem
(1)
dy
dt
=
f
(
t, y
)
, y
(
t
0
) =
y
0
,
has a unique solution
φ
(
t
) in an interval containing
t
0
. We have indi
cated in section 2.4 that the continuity of
f
and
∂f/∂y
in a rectangular
region containing (
t
0
, y
0
) is sufficient for this to be true. We shall see
how a numerical approximation of
φ
(
t
) can be made, tabulating (the
approximation of) the values of
φ
(
t
) at a sequence of mesh points
t
k
’s.
Euler’s method.
Recall that the tangent line
y
=
y
0
+
φ
prime
(
t
0
)(
t

t
0
) =
y
0
+
f
(
t
0
, y
0
)(
t

t
0
)
passing through the point (
t
0
, y
0
) approximates (the graph of) the func
tion
φ
(
t
) near
t
=
t
0
. To be more precise, it means if
t
1
is close to
t
0
,
then
φ
(
t
1
)
≈
y
0
+
f
(
t
0
, y
0
)(
t
1

t
0
)
.
We shall set as
y
1
the righthand side of the last
equation
, an approxi
mate value of
φ
(
t
1
).
More rigorously, the approximation can be obtained as follows pro
vided the rate function
f
is continuous. By the fundamental theorem
of calculus,
φ
(
t
1
) =
φ
(
t
0
) +
integraldisplay
t
1
t
0
φ
prime
(
s
)
ds
=
φ
(
t
0
) +
integraldisplay
t
1
t
0
f
(
s, φ
(
s
))
ds
On the other hand, by the mean value theorem, there exists some
t
*
∈
[
t
0
, t
1
] such that
integraldisplay
t
1
t
0
f
(
s, φ
(
s
))
ds
=
f
(
t
*
, φ
(
t
*
))(
t
1

t
0
)
.
Date
: January 8, 2011.
1
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KIAM HEONG KWA
Hence
φ
(
t
1
) =
φ
(
t
0
) +
f
(
t
*
, φ
(
t
*
))(
t
1

t
0
)
for some
t
*
∈
[
t
0
, t
1
]. Note that when
t
1
approaches
t
0
, so does
t
*
, from
which it follows that
lim
t
1
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 Winter '11
 Kwa
 Differential Equations, Numerical Analysis, Equations, Approximation, yk, tk, KIAM HEONG KWA

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