Question

ME 32200 Programming course (MATLAB)

4. Please finish the following Matlab code for solving the ODE: dy = y(1+1) dt I.C. y(0) = 0 with the multi-step 4th order Milne's Method, and apply 4th order Runge Kutta method to the first 4 points (1 boundary point and the next 3 points). (Hint: 4th order Milne's Method Predictor: 7i+ = Y-3 +h(2f;- fi- +25,-2) Corrector: y = y + + +0. +45j + fi-) Where f; = f(t;,y,) and Fit = f(t) +h, Vi.)) clear all; close all; h=0.01; t0=0; tn-1; t=t0:httn; n=(tn-to) /h+1; y=zeros (n,1); y (1)=0; % 4th Runge Kutta Method for i=1:3 kle k2 k3= k4 y(i+1) = end % 4th Milne's Method for i=4:n-1 X bar= y(i+1) = end plot (tebena')

Answer 1

I have completed the code. Please find a screenshot of the code and output and the text of the code. Please note that this is a script file to be saved as MRK.m in the MATLAB root directory.

CODE

+ 1 2 3 4 5 7 8 9 10 11 12 13 basiliskplotter.m x MRK.m X rk4calc.m S008.m X Q function MRK) clear all; close all; fun=@(t,y) (y* (1+t)); h = 0.01; to = 0; tn = 1; t = to:h:tn; n = ((tn - to)/h) +1 ; y=zeros (n,1); v(1)=1; $ 4th Runga Kutta Method o for i=1:3 kl=fun (t(i),y(i)); k2=fun ((t(i) +0.5*h), (y(i) +0.5*kl)); k3=fun ((t(i) +0.5*h), (y(i) +0.5*k2)); k4=fun((t(i)+h), (y(i) +k3)); y(i+1)=y (i)+(h* (kl+2*k2+2*k3+k4)/6); Fend $4th Milne Method V for i=4:n-1 fi = fun(t(i),y(i)); fil = fun(t(i-1),y(i-1)); fi2 = fun(t(1-2),y(i-2)); 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 III y_bar = y(i-3) + ( 4*h* (2*fi - fil + 2*fi2) 13); fbaril = fun(t(i) + h, v_bar); y(i+1) = y(i-1) + (h* (fbaril + 4*fi + fil)/3); - end plot(t,v,'b-'); -end 30

function MRK()
clear all; close all;
fun=@(t,y)(y*(1+t));
h = 0.01;
t0 = 0;
tn = 1;
t = t0:h:tn;
n = ((tn - t0)/h) +1 ;
y=zeros(n,1);
y(1)=1;
% 4th Runga Kutta Method
for i=1:3
   k1=fun(t(i),y(i));
   k2=fun((t(i)+0.5*h),(y(i)+0.5*k1));
    k3=fun((t(i)+0.5*h),(y(i)+0.5*k2));  
   k4=fun((t(i)+h),(y(i)+k3));
   y(i+1)=y(i)+(h*(k1+2*k2+2*k3+k4)/6);
end
%4th Milne Method
for i=4:n-1
fi = fun(t(i),y(i));
fi1 = fun(t(i-1),y(i-1));
fi2 = fun(t(i-2),y(i-2));
  
y_bar = y(i-3) + ( 4*h*(2*fi - fi1 + 2*fi2) /3);
fbari1 = fun(t(i) + h,y_bar);
y(i+1) = y(i-1) + (h*(fbari1 + 4*fi + fi1)/3);
end
plot(t,y,'b-');
end

OUTPUT

5 4.5 4. - 3.5 2.5 2 1.5 1 - 1 1 0 0.1 0.2 0.3 0.4 0.5 0.6. 0.7 0.8 10.9 1

If my answer helped you, please consider leaving an upvote. Thank you! :)