(a) Modify the following function so that it performs N steps of the Adams-Bashf

Question

(a) Modify the following function so that it performs N steps of the Adams-Bashforth-Moulton combination in PECE mode.

function [x,y] = ab4 (x,y,h,N)

% The first 3 steps are taken using a four-stage explicit

% Runge-Kutta method

for i=1:3

k1 = fn(x,y);

k2 = fn(x+0.5*h,y+0.5*h*k1);

k3 = fn(x+0.5*h,y+0.5*h*k2);

k4 = fn(x+h,y+h*k3);

y = y + h/6*(k1+2*k2+2*k3+k4);

x = x + h; f(:,i) = k1;

end

f(:,4) = fn(x,y);

% Now do the rest of the steps using the four-step

% Adams Bashforth

for i=4:N

y = y + h/24*(55*f(:,4)-59*f(:,3)+37*f(:,2)-9*f(:,1));

x= x + h;

f(:,1) = f(:,2);

f(:,2) = f(:,3);

f(:,3) = f(:,4);

f(:,4) = fn(x,y);

end

This question is about using the four-step Adams-Bashforth and Adams-Moulton methods in PECE mode to estimate y(10) for IVP1. Therefore estimate y(10) for IVP1.

Explanation / Answer

% The first 3 steps are taken using a four-stage explicit

% Runge-Kutta method

for i=1:3

k1 = fn(x,y);

k2 = fn(x+0.5*h,y+0.5*h*k1);

k3 = fn(x+0.5*h,y+0.5*h*k2);

k4 = fn(x+h,y+h*k3);

y = y + h/6*(k1+2*k2+2*k3+k4);

x = x + h; f(:,i) = k1;

end

f(:,4) = fn(x,y);

% Now do the rest of the steps using the four-step

% Adams Bashforth

for i=4:N

y = y + h/24*(55*f(:,4)-59*f(:,3)+37*f(:,2)-9*f(:,1));

x= x + h;

f(:,1) = f(:,2);

f(:,2) = f(:,3);

f(:,3) = f(:,4);

f(:,4) = fn(x,y);

end

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