How i implement Adams Predictor-Corrector Method from general code ?

Below is the Adams predictor-corrector formula and general code. How can I adapt this code to the above question? Can you please help?

%------------------------------------------------------
% 2-step Predictor-Corrector
% [T,Y]=dd2(f,definition,y,h); definition=[t1,tfinal]
%------------------------------------------------------
function [T,Y]=dd2(f,definition,Y1,h)
t1=definition(1);tfinal=definition(2);T=t1;Y=Y1;
t2=t1+h;
definition=[t1,t2];
[T,Y]=rk2(f,definition,Y1,h) ;
Y2=Y(2);
while t2 <tfinal
t3=t2+h;
P=Y2+h*(3/2*f(t2,Y2)-1/2*f(t1,Y1));
Y3=Y2+h/12*(5* f(t3,P)+8*f(t2,Y2)-f(t1,Y1));
Y1=Y2; Y2=Y3;t1=t2;t2=t3;
T=[T;t3];Y=[Y;Y3];
end
%
%----------------------------------------------

[Hazel Can]

Would this kind of code be correct for the BDF method? @Lateef Adewale Kareem

[Torsten]

You know the correct result of your differential equation.

If you plot Y against T in the calling program and compare the plot with the analytical solution, both should be approximately the same.

If yes, your code is (most probably) correct, if not, it's not.

Answer 1

[Lateef Adewale Kareem]

clc; clear all;
h = 0.01;
mu = 20;
f_m = @(t,y) mu*(y-cos(t))-sin(t);
exact = @(t) exp(mu*t)+cos(t);
[t,y_m] = dd2(f_m,[0, 1],exact(0), exact(h), h);
plot(t, exact(t)); hold

Current plot held

plot(t,y_m);
%plot(t,y,'-o');
legend('Exact Solution','Adams predictor-corrector formula')
xlabel('t')
ylabel('y')
title('When h = 0.01 and µ=20')

%------------------------------------------------------
% 2-step Predictor-Corrector
% [T,Y]=dd2(f,definition,y,h); definition=[t1,tfinal]
%------------------------------------------------------
function [T,Y] = dd2(f, definition, Y1, Y2, h)
    t1 = definition(1); tfinal = definition(2); t = t1:h:tfinal;
    T = t(1:2)'; Y = [Y1;Y2];
    for i = 2:numel(t)-1
        P = Y(i) + h/2*(3*f(t(i),Y(i))-f(t(i-1),Y(i-1)));
        Y(i+1) = Y(i) + h/12*(5*f(t(i+1), P) + 8*f(t(i),Y(i)) - f(t(i-1),Y(i-1)));
        T=[T;t(i+1)];
    end
end

%

[Torsten]

@Lateef Adewale Kareem

You implemented rk4, not rk2.

[Hazel Can]

I replaced RK4 with RK2. The solution has moved a little away from the exact solution. Was there any particular reason to use RK4 or would RK2 be correct as well?

@Torsten @Lateef Adewale Kareem

[Torsten]

As far as I read in your assignment, you should use the exact solution for y1. So neither rk2 nor rk4 is needed.

[Lateef Adewale Kareem]

yeah. he should have sent it in. I have modified the solution to use the exact solution at h