Question

in MATLAB Write a program to plot the proportional error, the ratio of the simulation value minus the exact solution to the exact solution. Produce the plot.

Answer 1

%Script that demonstrates Euler integration for a first order problem using

%MATLAB.

%The problem to be solved is:

%y'(t)+2*y(t)=2-exp(-4*t)

%Note: This problem has a known exact solution

%y(t)=1+0.58*exp(-4*t)-0.5*exp(-2*t)

function ystar = Eulermethod20(n)

a=0;

b=5;

h=(b-a)/n;

t=0:h:5;%Initialize time variable

clear ystar;%wipe out old variable

ystar(1)=1.0;%Initial condition (same for approximation)

for i=1:length(t)-1, %Set up "for" loop

k1=2-exp(-4*t(i))-2*ystar(i); %Calculate the derivative

ystar(i+1)=ystar(i)+h*k1;%Estimate new value of y

end

%Exact solution

y=1+0.5*exp(-4*t)-0.5*exp(-2*t);

%Error calculation

percent_error=100*abs(y-ystar)./y;

%Plot approximate and exact solutions

plot(t,ystar,'b--',t,y,'r-',t,percent_error,'g');

legend('Approximate','Exact','Error');

title('Euler Approximation n=5000');

xlabel('Time');

ylabel('y*(t), y(t)');