Summary from last lecture

Starting from the Taylor’s series expansion, we derive:

Euler’s method

Improvement of Euler’s method

Heun’s method

Midpoint method

Higher order Taylor method
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Higher order Taylor method
Euler method is only a first order approximation (use only
two terms in Taylor’s series with truncation error O(h
2
)).
It is obvious that if we want a better solution, we should
use more terms in the Taylor series in order to obtain a
higher order truncation error.
Suppose the solution y(t) to the initialvalue problem:
)
,
(
y
t
f
dt
dy
=
for a
≤
t
≤
b
y(t
o
) = y
o
has (n+1) continuous derivatives. If we expand the solution, y(t) in
therms of its nth Taylor polynomial about t:
1
)
1
(
1
)
(
2
1
)
(
)!
1
(
)
(
!
)
(
!
2
)
(
)
(
)
(
+
+
+
+
≤
≤
+
+
+
′
′
+
⋅
′
+
=
i
i
i
n
n
i
n
n
i
i
i
i
t
t
y
n
h
t
y
n
h
t
y
h
h
t
y
t
y
t
y
ξ
ξ
K
Higher order Taylor method
1
)
(
1
)
1
(
2
1
)
,
(
)!
1
(
)
,
(
!
)
,
(
!
2
)
,
(
)
(
)
(
+
+
−
+
≤
≤
+
+
+
′
+
⋅
+
=
i
i
i
i
n
n
i
i
n
n
i
i
i
i
i
i
t
t
y
f
n
h
y
t
f
n
h
y
t
f
h
y
t
f
h
t
y
t
y
ξ
ξ
K
Successive differentiation of the solution y(t) gives:
))
(
,
(
)
(
))
(
,
(
)
(
))
(
,
(
)
(
)
1
(
)
(
t
y
t
f
t
y
t
y
t
f
t
y
t
y
t
f
t
y
n
n
−
=
′
=
′
′
=
′
M
Substitute:
(
)
(
)
(
)
⎥
⎦
⎤
⎢
⎣
⎡
+
′
+
⋅
+
=
=
−
−
+
i
i
n
n
i
i
i
i
i
i
o
o
w
t
f
n
h
w
t
f
h
w
t
f
h
w
w
y
w
,
!
,
2
,
)
1
(
1
1
K
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document