Lab 3 - Seth Gibson - MAT 275 Lab
Introduction to Numerical Methods for Solving ODEs
Contents
■ Exercise 1
■ Exercise 2
■ Exercise 3
■ Exercise 4
■ Exercise 5
Exercise 1
Part (a)
% Define ODE function f for ODE dy/dt
f = @ (t,y) (3*y);
f
(t,y) 3y.
% Define vector t of time-values over the interval [0,0.5] to compute
% analytical solution vector.
t = linspace(0,0.5,100);
% Create vector of analytical solution values at corresponding t values.
y = 2*exp(3*t);
% Solve IVP numerically using forward Euler's method with 5 timesteps.
[t5,y5] = euler(f, [0,0.5],2,5);
y5 (end) ;
% Solve IVP numerically using forward Euler's method with 50 timesteps.
[t50,y50] euler(f, [0,0.5],2,50);
y50 (end);
% Solve IVP numerically using forward Euler's method with 500 timesteps.
[t500,y500] = euler(f, [0,0.5] ,2,500);
y500 (end);
% Solve IVP numerically using forward Euler's method with 5000 timesteps.
[t5000,y5000] = euler(f, [0,0.5],2,5000);
y5000 (end);
% In the following steps, we define error as analytical (solution value
% - numerical solution value).
% Compute numerical solution error at t=0.5 for forward Euler with 5
% timesteps.
es = y(end) -yS(end);
% Compute numerical solution error at t=0.5 for forward Euler with 50
% timesteps.
e50 = y(end) -yS0(end);
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dy
=
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this
is
the
ODE
quiver
(T,Y,daT,dY)
%
draw
arrows
(t,y)->(ttdt,
ttdy)
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45
Part
(b)
figure
(1)
t
=
0:.3:5;7
y
=
-20:2:25
;
%
define
grid
of
values
in
t
and
y
direction
[T,Y]=meshgrid(t,y);
%
creates
2d
matrices
of
points
in
the
ty-plane
dT =
ones(size(T));
%
dt=1
for
all
points
dy
=
-3*Y;
%
dy
=
-3*y;
this
is
the
ODE
quiver
(T,Y,daT,dY)
%
draw
arrows
(t,y)->(ttdt,
ttdy)
axis
tight
%
adjust
look
hold
on
%
Define
vector
t
of
time-values
over
the
interval
[0,10]
to
define
analytical
solution
vector.
t
=
linspace(0,
5,
100);
%
Create
vector
of
analytical solution
values
at
corresponding
t
values.
y=2*exp
(-3*t)
;
%$
Plot
analytical
solution
vector
with
slopefield
from
(a).
plot
(t,y,
'k'
,
'linewidth'
,2)
hold
off
pode
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45
5
Part
(c)
figure
(2)
type
'euler.m'
t
=
0:.3:5;
y
=
-20:2:25
;
%&
define
grid
of
values
in
t
and
y
direction
[T,Y]=meshgrid(t,y);
%
creates
2d
matrices
of
points
in
the
ty-plane
dT =
ones(size(T));
%
dt=1
for
all
points
dY
=
-3*Y;
%
dy
=
-3*y;
this
is
the
ODE
quiver
(T,Y,daT,dY)
%
draw
arrows
(t,y)->(ttdt,
ttdy)
axis
tight
%
adjust
look
hold
on
x=linspace(0,5,100);
t=x;
y=2*exp
(-3*t)
;
plot
(t,y,'k-',
'linewidth',2)
%
Define
ODE
function.
£=@(t,y)
(-3*y);
$
Compute
numerical
solution
to
IVP
with
6
timesteps
using
forward
Euler.
[t6,
y6]=euler(f,
[0,5],2,6);
Plot
numerical
solution
with analytical solution
from
(b)
and
slopefield
from
(a)
using
circles
to
distinguish
between
the
approximated
data
de
oP
ol?
(i.e.,
the
numerical
solution
values)
and
actual
(analytical)
solution.
plot
(t6,y6,'ro-',
'linewidth',2)
hold
off;
%
end
plotting
in
this
figure
window
function
[t,y]
=
euler(f,tspan,y0,N)
m
=
length(y0);
tO
=
tspan(1);
tf
=
tspan(2);
h =
(tf-t0)/N;
%
evaluate
the
time
step size
t
=
linspace(t0,tf,N+1);
%
create
the
vector
of
t
values
y
=
zeros(m,N+1);
%
allocate
memory
for
the
output
y
y(:,1)
=
yO';
%
set
initial
condition
for
n=1:N
y(:,ntl)
=
y(:,n)
+
h*f(t(n),y(:,n));
%
implement
Euler's
method
end
3
t=
t';
y=y';
%
change
t
and
y
from
row
to
column
vectors
end
pe
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25
a
a5
4
45
5
The
applied
Euler's
method
for
this
function
appears
to
be
very
inaccurate
relative
to
the
exact
solution
function.
The
approximations
follow
the
slope
direction
field
at
certain
points,
and
each
approximation
will
only
follow
the
direction
field.
Because
the
direction
field
has
a
different
slope
at
points
further
away
from
the
exact
solution,
the
applied
Euler's
approximation
will
appear
to
oscilate
and
become
more
drastic
as
t-values
become
greater.
Part
(d)
%
Define
new
grid
of
t
and
y
values
at
which
to
plot
vectors
for
slope
%
field.
figure
(3)
nfs
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45
o
4
Because
the
function
decreases
so
quickly
near
the
beginning
values,
the
step
size
is
too
small
to
geometrically
appear
accurate
to
the
exact
solution,
as
the
slope
near
t=0
is
very
different
than
the
slop
at
t=0.5,
causing
the
applied
Euler's
method
to
'dive
down’
much
more
quickly
than
the
function.
The
applied
Euler's
method
oscilates
back
up
and
more
closely follows
the
exact
solution
after
t=0.5,
as
the
slope
does
not
change
drastically
with
respect
to
t.
Exercise
3
2
$
Display
contents
of
impeuler
M-file.
type
‘imeuler.m'
%
Define
ODE
function
for
dy/dt
=
f(t,y)
=
3y.
£=@(t,y)
(3*y);
oe
Compute
numerical
solution
to
IVP
with
5
timesteps
using
"improved
6
Euler."
[t5,y5] =
imeuler(f,
[0,.5],2,5);
function
[t,y]
=
imeuler(f,tspan,y0,N)
m
=
length(y0);
tO
=
tspan(1);
tf
=
tspan(2);
h =
(tf-t0)/N;
%
evaluate
the
time
step size
t
=
linspace(t0,tf,N+1);
%
create
the
vector
of
t
values
y
=
zeros(m,N+1);
%
allocate
memory
for
the
output
y
y(:,l)
=
yO';
%
set
initial
condition
for
n=1:N
fl=f(t(n),y(:,n));
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(b)
Because
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step
size
decreases
by
almost
10
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the
error
decreases
by
almost
100
in
the
last
column,
the
Improved
Euler's
method
is
of
order
h*2;
i.e.
as
the
step
size
decreases
by
a
factor
of
k,
the
error
decreases
by
a
factor
of
kA2.
Exercise
5
Part
(a)
same
as
in
Exercise
2
(see Exercise
2a)
Part
(b)
same
as
in
Exercise
2
(see Exercise
2b)
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axis
Eight
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approximated
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ae
0
(i.e.,
the
numerical
solution
values)
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actual
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(t6,y6,'ro-',
'linewidth',2)
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off;
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end
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in
this
figure
window
function
[t,y]
=
imeuler(f,tspan,y0,N)
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time
step size
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values
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memory
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the
output
y
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yO';
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(nt+1),y(:,n)+th*f£1);
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=
y(:,n)
+
h/2*(£1+f£2);
%
implement
Euler's
method
end
t=
t';
y=y';
%
change
t
and
y
from
row
to
column
vectors
end
30
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5
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1.5 \ t t ( t t ( t t t t t t t t \
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-0.5 I I I I I I I I I I I I I I J I
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0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
Figure 5 seems to have an applied Improve Euler's method that does not follow the field direction vectors whatsoever. I
do not understand why this is except by understanding the nature of the lmpoved Euler's method. However, because of
the average taken in the Improved Euler's method, we are able to have less 'dramatic' slopes that help the applied Euler's
method appear more geometrically accurate to the exact solutions in Figure 5 and 6.
Congratulations, you're done!
