midpoint rule for matlab

Question

Hello I was asked to create a matlab code for the midpoint rule. What I have is the code for eulers method, so I have to make some modifications, but I am struggling to do it I have the following

function H = heun(f,a,b,ya,M)
h = (b-a)/M;
T = zeros(1,M+1);
Y = zeros(1,M+1);
T = a:h:b;
Y(1) = ya;
for j = 1 : M
k1=feval(f,T(j),Y(j));
k2=feval(f,T(j+1),Y(j)+h*k1);
Y(j+1)=Y(j)+(h/2)*(k1+k2);
end
H = [T' Y'];
function f = dif1(t,y)
f=(t-y)/2;

So yea my function is y'=(t-y)/2 and the interval and other stuff i define in the command window..

If it is possible to make the for loop into a mid point rule I think that is the way to go, any help would be appreciated.

Answer 1

Below is an implementation in MATLAB I have done of the Euler's Method for solving a pair of coupled 1st order DE's. It solves a harmonic oscillator of represented by the following:

y1(t+h) = y1(t) + h*y2(t)

y2(t+h) = y2(t) + h*(-A/M y1(t) -B/M y1(t)/|y1(t)|)

% Do the integration using the Euler Method
    while(T<=T1)
        % Update the position of the pt mass using current velocity
        Y1(i+1) = Y1(i) + H*Y2(i);

        % Update the velocity of the pt mass checking if we are turning 
        % within the C/A band
        if ( (abs(Y2(i) < th) && abs(Y1(i)) < C/A) )
            Y2(i+1) = Y2(i);
        else
            Y2(i+1) = Y2(i) + H * ( ((-A/M)*Y1(i)) - (B/M)*sign(Y2(i)) );
        end

        % Incriment the time by H 
        T = T + H;
        % Increase the looping index variable
        i = i + 1;
    end

Without explicitly solving YOUR homework question, I hope this example helps. A few things to note: the if statement is accounting for static friction in this particular example -- so disregard that and only look at what happens after the 'else'.

Also note that I have initial conditions y1(0) and y2(0) defined as Y1(i=1) and Y2(i=1), so starting with Yj(i+1) gives Yj(2). Note that T1 is the ending time of the simulation.

Below is the same problem using the Improved Euler Method. If you derive the update equations for this system you should be able to walk through the code.

% Do the integration using the Improved Euler Method
    % Ki_j = K^i_j
    while(T<=T1)
        % Calculate K^i_j's

        K1_1 = Y2(i);

        % Must check if we are turning within C/A
        if ( (abs(Y2(i) < th) && abs(Y1(i)) < C/A) )
            K1_2 = 0;
        else
            K1_2 = (-A/M)*Y1(i) - (B/M)*sign(Y2(i));
        end

        K2_1 = Y2(i)+H*K1_2;

        % Checking if we are turning within C/A
        if ( (abs(Y2(i) < th) && abs(Y1(i)) < C/A) )
            K2_2 = 0;
        else
            K2_2 = (-A/M)*(Y1(i) + H*K1_1) - (B/M)*sign(Y2(i)+ H*K1_2);
        end


        % Update the position and velocity
        Y1(i+1) = Y1(i)+ (H/2)*(K1_1+K2_1);
        Y2(i+1) = Y2(i) + (H/2)*(K1_2+K2_2);

        % Incriment the time by H 
        T = T + H;
        % Increase the looping index variable
        i = i + 1;
    end