Please answer with Matlab, Thank you

ab4.m:

function [wi, ti] = ab4 ( RHS, t0, x0, tf, N )

%AB4 approximate the solution of the initial value problem

%

% x'(t) = RHS( t, x ), x(t0) = x0

%

% using the fourth-order Adams-Bashforth method

% - this routine will work for a system of first-order

% equations as well as for a single equation

%

% the classical fourth-order Runge-Kutta method is used to

% initialize the multistep method

%

%

% calling sequences:

% [wi, ti] = ab4 ( RHS, t0, x0, tf, N )

% ab4 ( RHS, t0, x0, tf, N )

%

% inputs:

% RHS string containing name of m-file defining the

% right-hand side of the differential equation; the

% m-file must take two inputs - first, the value of

% the independent variable; second, the value of the

% dependent variable

% t0 initial value of the independent variable

% x0 initial value of the dependent variable(s)

% if solving a system of equations, this should be a

% row vector containing all initial values

% tf final value of the independent variable

% N number of uniformly sized time steps to be taken to

% advance the solution from t = t0 to t = tf

%

% output:

% wi vector / matrix containing values of the approximate

% solution to the differential equation

% ti vector containing the values of the independent

% variable at which an approximate solution has been

% obtained

%

neqn = length ( x0 );

ti = linspace ( t0, tf, N+1 );

wi = [ zeros( neqn, N+1 ) ];

wi(1:neqn, 1) = x0';

h = ( tf - t0 ) / N;

oldf = zeros(3,neqn);

%

% generate starting values using classical 4th order RK method

% remember to save function values

%

for i = 1:3

oldf(i,1:neqn) = feval ( RHS, t0, x0 );

k1 = h * oldf(i,:);

k2 = h * feval ( RHS, t0 + h/2, x0 + k1/2 );

k3 = h * feval ( RHS, t0 + h/2, x0 + k2/2 );

k4 = h * feval ( RHS, t0 + h, x0 + k3 );

x0 = x0 + ( k1 + 2*k2 + 2*k3 + k4 ) / 6;

t0 = t0 + h;

wi(1:neqn,i+1) = x0';

end;

%

% continue time stepping with 4th order Adams Predictor / Corrector

%

for i = 4:N

fnew = feval ( RHS, t0, x0 );

x0 = x0 + (h/24) * ( 55*fnew - 59*oldf(3,:) + 37*oldf(2,:) - 9*oldf(1,:) );

oldf(1,1:neqn) = oldf(2,1:neqn);

oldf(2,1:neqn) = oldf(3,1:neqn);

oldf(3,1:neqn) = fnew;

t0 = t0 + h;

wi(1:neqn,i+1) = x0';

end;

Problem 2. The MATLAB code ab4.m from the class web-site implements the 4-step 4th-order Adams-Bashforth method (AB4) given by wi+1 = tri + -| 55 f(ti, ui)-59 f(4-1, tvi.) + 37 f(4-2, wi-2)-9 f(4-3, wi-3) 24 Modify this code to create the code pred_corr_4.m (or write your own program from scracth) that implements the predictor-corrector method based on the AB4 method above and the 3-step 4th-order Adams-Moulton method (AM3) given by 24 (a) Attach a printout of your MATLAB code. (b) Test your code on the initial-value problem y(t) =-y + sin t , t E [T,2n] , whose exact solution is yexact(t- (e-t + sint-cost). Compute the error ly(2) yexact (2T)| for N - 10, 100, 1000; here N is the number of intervals in which the interval r.2] is divided, i.e., the last argument of the function ab4 (note that the arrays wi and ti created by ab4.m are of size N 1) Attach a printout of your MATLAB session Problem 2. The MATLAB code ab4.m from the class web-site implements the 4-step 4th-order Adams-Bashforth method (AB4) given by wi+1 = tri + -| 55 f(ti, ui)-59 f(4-1, tvi.) + 37 f(4-2, wi-2)-9 f(4-3, wi-3) 24 Modify this code to create the code pred_corr_4.m (or write your own program from scracth) that implements the predictor-corrector method based on the AB4 method above and the 3-step 4th-order Adams-Moulton method (AM3) given by 24 (a) Attach a printout of your MATLAB code. (b) Test your code on the initial-value problem y(t) =-y + sin t , t E [T,2n] , whose exact solution is yexact(t- (e-t + sint-cost). Compute the error ly(2) yexact (2T)| for N - 10, 100, 1000; here N is the number of intervals in which the interval r.2] is divided, i.e., the last argument of the function ab4 (note that the arrays wi and ti created by ab4.m are of size N 1) Attach a printout of your MATLAB session