Use the method of steps to show that the solution to the initial value problem
x
(
t
) =

x
(
t

1)
,
x
(
t
) = 1
on
[

1
,
0]
,
is given by
x
(
t
) =
n
k
=0
(

1)
k
[
t

(
k

1)]
k
k
!
,
for
n

1
≤
t
≤
n ,
where
n
is a nonnegative integer.
(This problem can also be solved using the
Laplace transform method of Chapter 7.)
(c)
Use the method of steps to compute the solution to the initial value problem given
in (0.1) on the interval 0
≤
t
≤
15 for
t
0
= 3.
Extrapolation
When precise information about the
form
of the error in an approximation is known, a
technique called
extrapolation
can be used to improve the rate of convergence.
Suppose the approximation method converges with rate
O
(
h
p
) as
h
→
0 (cf. Section 3.6).
From theoretical considerations, assume we know, more precisely, that
y
(
x
;
h
) =
φ
(
x
) +
h
p
a
p
(
x
) +
O
(
h
p
+1
)
,
(0.3)
where
y
(
x
;
h
) is the approximation to
φ
(
x
) using step size
h
and
a
p
(
x
) is some function
that is independent of
h
(typically, we do not know a formula for
a
p
(
x
), only that it
exists). Our goal is to obtain approximations that converge at the faster rate
O
(
h
p
+1
).
We start by replacing
h
by
h/
2 in (0.3) to get
y
x
;
h
2
=
φ
(
x
) +
h
p
2
p
a
p
(
x
) +
O
(
h
p
+1
)
.
If we multiply both sides by 2
p
and subtract equation (0.3), we find
2
p
y
x
;
h
2

y
(
x
;
h
) = (2
p

1)
φ
(
x
) +
O
(
h
p
+1
)
.
Solving for
φ
(
x
) yields
φ
(
x
) =
2
p
y
(
x
;
h/
2)

y
(
x
;
h
)
2
p

1
+
O
(
h
p
+1
)
.
Hence,
y
*
x
;
h
2
:=
2
p
y
(
x
;
h/
2)

y
(
x
;
h
)
2
p

1
has a rate of convergence of
O
(
h
p
+1
).
10
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
(a)
Assuming
y
*
x
;
h
2
=
φ
(
x
) +
h
p
+1
a
p
+1
(
x
) +
O
(
h
p
+2
)
,
show that
y
**
x
;
h
4
:=
2
p
+1
y
*
(
x
;
h/
4)

y
*
(
x
;
h/
2)
2
p
+1

1
has a rate of convergence of
O
(
h
p
+2
).
(b)
Assuming
y
**
x
;
h
4
=
φ
(
x
) +
h
p
+2
a
p
+2
(
x
) +
O
(
h
p
+3
)
,
show that
y
***
x
;
h
8
:=
2
p
+2
y
**
(
x
;
h/
8)

y
**
(
x
;
h/
4)
2
p
+2

1
has a rate of convergence of
O
(
h
p
+3
).
(c)
The results of using Euler’s method (with
h
= 1, 1
/
2, 1
/
4, 1
/
8) to approximate the
solution to the initial value problem
y
=
y,
y
(0) = 1
at
x
= 1 are given in Table 1.2, page 27. For Euler’s method, the extrapolation
procedure applies with
p
= 1. Use the results in Table 1.2 to find an approximation
to
e
=
y
(1) by computing
y
***
(1; 1
/
8). [Hint: Compute
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 MAZMANI
 Equations, Approximation, Constant of integration, Nonlinear system, van der Pol

Click to edit the document details