Two Step Adam Bashford Method

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B
B on 21 Sep 2021
Commented: B on 22 Sep 2021
I am trying to make a function that implements the two step Adam Bashford Method to solve an ODE
function [t, w, h] = abs2(f, a, b, alpha, n)
%AB2 Two-step Adams Bashforth method
% [t, w, h] = ab2(f, a, b, alpha, n) performs the two-step Adams Bashforth
% method for solving the IVP y' = f(t,y) with initial condition y(a) = alpha
% taking n steps from t = a to t = b. The first step from t = a to t = a + h
% is performed using the modified Euler method.
h=(b-a)/n;
t=a:h:b;
w=zeros(1,length(t));
w(1)=alpha;
%modified euler method
for i=2
k1=h*f(t(i),w(i));
k2=h*f(t(i)+h,w(i)+k1);
w(i+1)=w(i)+1/2*(k1+k2);
end
for i=3:length(t)
w(i+1)=w(i)+(3/2)*h*f(t(i),w(i))-.5*h*f(t(i-1),w(i-1));
end
end
Code to call the function
f=@(t,y) 3*t+y/t;
alpha=5;
a=1;
b=2;
n=3;
[t, w, h] = ab2(f, a, b, alpha, n)
%% Output %%
t = 1×4
1.0000 1.3333 1.6667 2.0000
w = 1×5
5.0000 0 1.6333 3.9567 6.9492
h = 0.3333
As you can the output doesn't look right at all. Any help would be appreciated

Accepted Answer

Alan Stevens
Alan Stevens on 21 Sep 2021
As follows
f=@(t,y) 3*t+y/t;
alpha=5;
a=1;
b=2;
n=3;
[t, w, h] = abs2(f, a, b, alpha, n);
plot(t,w,'-o'),grid
xlabel('t'),ylabel('w')
disp(['h = ' num2str(h)])
h = 0.33333
function [t, w, h] = abs2(f, a, b, alpha, n)
%AB2 Two-step Adams Bashforth method
% [t, w, h] = ab2(f, a, b, alpha, n) performs the two-step Adams Bashforth
% method for solving the IVP y' = f(t,y) with initial condition y(a) = alpha
% taking n steps from t = a to t = b. The first step from t = a to t = a + h
% is performed using the modified Euler method.
h=(b-a)/n;
t=a:h:b;
w=zeros(1,length(t));
w(1)=alpha;
%modified euler method
k1=h*f(t(1),w(1));
k2=h*f(t(1)+h,w(1)+k1);
w(2)=w(1)+1/2*(k1+k2);
for i=2:length(t)-1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
w(i+1)=w(i)+(3/2)*h*f(t(i),w(i))-.5*h*f(t(i-1),w(i-1));
end
end

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