MATLAB Program for Backward Euler's method

MATLAB Program:

% Backward Euler's 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*f(t(i),w(i));

 w(i+1)=w(i)+h*f(t(i+1),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:

>> backwardmodifiedEulermethod

Enter left end ponit, a:  0

Enter right end point, b:  2

Enter no. of subintervals, n: 10

Enter the initial condition, alpha:  1

0.0000   1.00000000

0.2000   1.47200000

0.4000   2.03168000

0.6000   2.68088320

0.8000   3.42189517

1.0000   4.25755001

1.2000   5.19136201

1.4000   6.22768889

1.6000   7.37193423

1.8000   8.63079844

2.0000  10.01259007

>>