Adams-Bashforth 4th order

Hi everyone,

I'm trying to implement the AB4 to solve ODE's, and my code seems to be wrong as it's not compiling: I get the error message "Index exceeds the number of array elements (4)."

I tried to check the range of implementation for the code in the AB part (the second loop in the code). I used R-K4 to find the initial condition, as they are four.

Here is my code:

function adams_bashforth_method(a,b,s,f,y0)
% s is the number of subinterval for the interpolation

k=3;                    % steps-1

x=linspace(a,b, s+1);   %network set creation

h=1/(s+1);              %discretization of the network set

u=zeros(1,s+1);         %network function creation

u(1)=y0;                %initial condition of Cauchy problem

for i=1:k
    k_1=f(x(i),u(i));                            %k_1 
    k_2=f(x(i)+h./2,u(i)+((h./2).*k_1));         %k_2
    k_3=f(x(i)+h./2,u(i)+((h./2).*k_2));         %k_3
    k_4=f(x(i)+h,u(i)+(h.*k_3));                 %k_4
    u(i+1)=u(i)+(h/6).*(k_1+2.*(k_2+k_3)+k_4);   %Runge-Kutta
end
%Now that I know the initial four condition, as I set the step constant, I
%can evaluate the others u's by AB4 formula
for i=4:s 
    u(i+1)=u(i)+(h./24).*(55.*(f(x(i+3),u(i+3)))-59.*(f(x(i+2),u(i+2)))+37.*(f(x(i+1),u(i+1)))-9.*(f(x(i),u(i))));
end
plot(x,u,'r');
title('Adams Bashforth 4th order method')
hold on
end

As a supply, I'll leave here the Cauchy problem I'm facing:

[Star Strider]

It would help to have the arguments: ‘a’, ‘b’, ‘s’, ‘f’, ‘y0’ .

.

[LUCA SCARPELLI]

yh, sorry, I'll leave it here

% f=@(x,y) x*exp(-x*y);
f=@(t,y) (y-t^2+1); %differential equation we interest to solve 
a=-10;
b=10;
s=100;
y0=0.5;
adams_bashforth_method(a,b,s,f,y0)
% opt2=odeset('Stats','on');
% ode113(f,[a,b],y0)

[Torsten]

To dimension x,h and u, you must work with s instead of k.

Further, you must include k_1 = f(x(i),u(i)) in the for-loop in the RK section.

Further, your update in the AB loop for u is wrong:

u(i+1) = u(i) + h/24*(55*f(x(i),u(i))-59*f(x(i-1),u(i-1))+37*f(x(i-2),u(i-2))-9*f(x(i-3),u(i-3)))

Further, you should return u to the calling program from the adams bashforth function.

[LUCA SCARPELLI]

ok, that was very useful. Thank you very much!!

I didn't really wrote the return value, as I simply write

function [u]=adams_bashforth_method(a,b,s,f,y0)

It should be the same in this way, isn't it!?