MATLAB Program for Midpoint method

% Midpoint method

 % Example 1:  Approximate the solution to the initial-value problem

 % dy/dt=e^t ; 0<=t<=2 ; y(0)=1;

 % Example 2:  Approximate the solution to the initial-value problem

 % dy/dt=y-t^2+1 ; 0<=t<=2 ; y(0)=0.5;

 %f = @(t,y) (0*y+exp(t));  %Example 1

 f = @(t,y) (y-t^2+1);       %Example 2

 a = input('Enter left end ponit, a:  ');

 b = input('Enter right end point, b:  ');

 n = input('Enter no. of subintervals, n: ');               

 alpha = input('Enter the initial condition, alpha:  ');

 h = (b-a)/n;                                               

 t=[a zeros(1,n)];

 w=[alpha zeros(1,n)];

 for i = 1:n+1

  t(i+1)=t(i)+h;

  wprime=w(i)+(h/2)*f(t(i),w(i));

  %w(i+1)=w(i)+(h/2)*f( t(i), w(i)+ f(t(i+1),wprime) );

  w(i+1)=w(i)+h*f( t(i)+(h/2), wprime );

  fprintf('%5.4f  %11.8f\n', t(i), w(i));

  plot(t(i),w(i),'r*'); grid on;

  xlabel('t values'); ylabel('w values');

  hold on;

 end

OUTPUT:

>> midpoint_method

Enter left end ponit, a:  0

Enter right end point, b:  2

Enter no. of subintervals, n: 10

Enter the initial condition, alpha:  .5

0.0000   0.50000000

0.2000   0.82800000

0.4000   1.21136000

0.6000   1.64465920

0.8000   2.12128422

1.0000   2.63316675

1.2000   3.17046344

1.4000   3.72116540

1.6000   4.27062178

1.8000   4.80095857

2.0000   5.29036946

>>