Question

MATLAB

I need the input code and the output. Thanks.

7. Modify the Euler's method MATLAB code presented in the Learning activity video called Using Euler's Method on Matlab (located in the Blackboard Modue#10:: Nomerical Solution to ODE: part 1) to plot and compare the approximate solution using the modified Euler method, for a step size of 0.1 and 0.01

Answer 1

The MatLab code for the Euler's method -

function Euler(func, y0, del, tf)
t = 0: delt: tf;

// t= initial point
y(1) = y0;
for i = 1:length(t)-1
//Assuming the function is dependent only on the x-axis, and not on y axis or it's values
y(i+1) = y(i) + del/2*(feval(func, t(i)) + feval(func, t(i+1)));
end
plot(t, y);
xlabel('time')
ylabel('y')
disp(y(end))

Modified Euler Algorithm:
The modified Euler method starts with an Euler step, giving a provisional value for wi+1 at the next time ti+1:

The step actually taken looks like an Euler step, but with f replaced by the average of f at the starting point of the step and f at the provisional point:

function euler(func, y0, delt, tf)
t = 0:delt:tf;
y(1) = y0;
for i = 1:length(t)-1
% Assuming function is dependent only on the x-axis, and not on corresponding y-values
y(i+1) = y(i) + delt/2*(feval(func, t(i)) + feval(func, t(i+1)));
end
plot(t, y);
xlabel('time')
ylabel('y')
disp(y(end))

Modified Euler Algorithm:
The modified Euler method starts with an Euler step, giving a provisional value for wi+1 at the next time ti+1:
where -

W' i+1= Wi+hf(ti,Wi)

and Wi+1 = Wi + h/2(f(ti,Wi) + f( ti+1 , W' i+1)

where h varies from 0.1 to .01