Given imeuler.m and fmass.m below.  Answer the following questions:
 

• Write down the differential equation and the initial value of the solution.

 
 
• If I want to use fmass.m to evaluate the function value at  t=2, y=8, how do we do in Matlab?

 
 
• Which file contains Matlab commands, and which one is a Matlab function? How do we use them differently?

 
 
• What are the initial time, the final time, and the step size? How many steps are used to get the solution at the final time.

 
 
• Which method is used to solve the ODE-IVP? Mark the lines whether the method is used.

 
 
• Copy the line where fmass is called to the space below.

 
 
• Write down the formula of the improved Euler's method.

 
 
 
• What are the order of accuracy of the Euler's method and improved Euler's method? If we double the total number of steps, how do the errors change?

 
 
• Which variables in the code  represent the time and the velocity?

 
 
• How do we print out the numerical values of time and the velocity?

 
 
 
 

---


 

fmass.m

function fmass= fmass(t,y)

     fmass = -32 + .1*abs(y) + .001*y*y;

---

imeuler.m

%your name, your student number, lesson number
clear;
y(1) = 0.; T = 100.; KK = 100 h = T/KK; t(1)= 0.;

for k = 1:KK
     y(k+1) = y(k) + h*fmass(t(k),y(k));
     t(k+1) = t(k) + h;
     y(k+1) = y(k) + .5*h*(fmass(t(k),y(k)) + fmass(t(k+1),y(k+1)));
end

plot(t,y)
title('your name, your student number, lesson number')
xlabel('time');    ylabel('speed')