This is about using the Matlab functions and scripts discussed in class.
First script:
ShowEuler.m
while input('Another Step Size Choice? (1=yes, 0=no). ');
fname = input('Enter Function Name: ');
h = input('Input Step Size ');
t0 = input('Enter initial time ');
tmax = input('Enter final time: ');
y0 = input('Enter initial state: ');
freq = input('Input print frequency ');
s = ['Euler(' fname
sprintf(',%6.3f,%6.3f,%6.3f,%6.3f)',y0,t0,tmax,h)];
disp([' tvals
' s])
disp(' ')
[tvals,yvals] = FixedEuler(fname,y0,t0,tmax,h);
N = size(tvals,1);
disp(sprintf(' %5.2f
%20.16f',tvals(1),yvals(1)))
for m = freq+1:freq:N
disp(sprintf(' %5.2f
%20.16f',tvals(m),yvals(m)))
end
disp(sprintf(' %5.2f
%20.16f',tvals(N),yvals(N)))
end
and the script
PlotEuler.m
while input('Another Step Size Choice? (1=yes, 0=no). ');
fname
= input('Enter Function Name: ');
truename = input('Enter True Solution Name ');
h
= input('Input Step Size ');
t0
= input('Enter Initial Time ');
tmax
= input('Enter Final Time: ');
y0
= input('Enter Initial State: ');
s = ['Euler(' fname
sprintf(',%6.3f,%6.3f,%6.3f,%6.3f)',y0,t0,tmax,h)];
disp([' tvals
' s])
disp(' ')
[tvals,yvals] = FixedEuler(fname,y0,t0,tmax,h);
N = size(tvals,1);
truevals = zeros(N,1);
for n=1:N
truevals(n) = feval(truename,t0,y0,tvals(n));
end
plot(tvals,truevals,tvals,yvals,'*');
print -dpng eulerplot
end
Now let’s do an example problem.
u’ = 2u
u(0) = 4
We know the true solution is
u(t) = 4 e2t
Let’s approximate solution using Euler’s method:
uhat(1) = 4
uhat(new) = uhat(old) + 2 uhat(old) h
where h is our chosen stepsize.
We can apply this using the MatLab script ShowEuler.
We use the MatLab function func.m seen below:
function y = func(t,x)
%
% right hand side dynamics
% t = time
% x = ODE variable
%
y = 2*x;
and we use the MatLab code for the trae solution in treufunc.m seen below:
function y = truefunc(t0,y0,t)
%
% true solution to y' = 2y; y(t_0) = y0
%
% t = time
% t0 = initial time
% y0 = initial condition
%
y = y0*exp(2*(t-t0));
The we run ShowEuler.m as follows:
>> ShowEuler
Another Step Size Choice? (1=yes, 0=no). 1
Enter Function Name: 'func'
Input Step Size .05
Enter initial time 0
Enter final time: 2
Enter initial state: 4
Input print frequency 5
tvals
Euler(func, 4.000, 0.000, 2.000, 0.050)
0.00
4.0000000000000000
0.25
6.4420400000000004
0.50
10.3749698404000000
0.75
16.7089926776626070
1.00
26.9099997973024050
1.25
43.3388237735534930
1.50
69.7976090755456510
1.75
112.4097473922570200
2.00
181.0370222727038400
2.00
181.0370222727038400
Another Step Size Choice? (1=yes, 0=no).
To see how the Euler approximate solution compares to the true solution,
We use the MatLab script PlotEuler.m.
>> PlotEuler
Another Step Size Choice? (1=yes, 0=no). 1
Enter Function Name: 'func'
Enter True Solution Name 'truefunc'
Input Step Size .05
Enter Initial Time 0
Enter Final Time: 2
Enter Initial State: 4
tvals
Euler(func, 4.000, 0.000, 2.000, 0.050)
Another Step Size Choice? (1=yes, 0=no).
The resulting plot is seen below:
Now let’s do a logistics problem.
u’ = .05 u ( 200 – u)
u(0) = 280
The new func.m file is
function y = func(t,x)
%
% right had side dynamics
% t = time
% x = ODE variable
% alpha = .05
% L = 200
alpha = .05;
L = 200;
y = alpha*x.*(L-x);
and the new truefunc.m is
function y = truefunc(t0,y0,t)
%
% true solution to
% y' = alpha y(L-y);
% y(t0) = y0
%
% t = time
% t0 = initial time
% y0 = initial condition
%
L = 200;
alpha = .05;
B = L/y0;
y = L./(1 + (B-1)*exp(-alpha*L*(t-t0)));
%
1
2
2
2
3
321
We approximate using ShowEuler.m.
>> ShowEuler
Another Step Size Choice? (1=yes, 0=no). 1
Enter Function Name: 'func'
Input Step Size .05
Enter initial time 0
Enter final time: 2
Enter initial state: 280
Input print frequency 2
tvals
Euler(func,280.000, 0.000, 2.000, 0.050)
0.00
280.0000000000000000
0.10
210.5600000000000000
0.20
202.4380775963033700
0.30
200.5984640080211500
0.40
200.1489457912178400
0.50
200.0371948718575400
0.60
200.0092961243013000
0.70
200.0023238690467300
0.80 200.0005809571360700
0.90
200.0001452386511900
1.00
200.0000363096232500
1.10
200.0000090774033500
1.20
200.0000022693506900
1.30
200.0000005673376500
1.40
200.0000001418343900
1.50
200.0000000354585800
1.60
200.0000000088646500
1.70
200.0000000022161500
1.80
200.0000000005540200
1.90
200.0000000001385000
2.00
200.0000000000346200
2.00
200.0000000000346200
Another Step Size Choice? (1=yes, 0=no). 1
Enter Function Name: 'func'
Input Step Size .2
Enter initial time 0
Enter final time: 2
Enter initial state: 280
Input print frequency 1
tvals
Euler(func,280.000, 0.000, 2.000, 0.200)
0.00
280.0000000000000000
0.20
56.0000000000000000
0.40 136.6400000000000100
0.60
223.2151040000000000
0.80
171.3954854626918500
1.00
220.4223320181574200
1.20
175.4069515312440000
1.40
218.5448681388901900
1.60
178.0160105182215500
1.80
217.1510315464290100
2.00
179.9073896225049500
2.00
179.9073896225049500
Another Step Size Choice? (1=yes, 0=no). 0
>>
We now use PlotEuler.m to see results graphically.
>> PlotEuler
Another Step Size Choice? (1=yes, 0=no). 1
Enter Function Name: 'func'
Enter True Solution Name 'truefunc'
Input Step Size .2
Enter Initial Time 0
Enter Final Time: 2
Enter Initial State: 280
tvals
Euler(func,280.000, 0.000, 2.000, 0.200)
Another Step Size Choice? (1=yes, 0=no).
With graph generated below: