diff --git a/modified_euler.m b/modified_euler.m new file mode 100644 index 0000000000000000000000000000000000000000..8b2f124da0c0f0680ebaf541f9b07f577df7ed1d --- /dev/null +++ b/modified_euler.m @@ -0,0 +1,73 @@ +% Improved Euler Method to find an approximate solution of y'=f(x,y) +% with y(x(0))=y(0) +clear all +%input +f_in = input('Enter the function f(x,y) = ','s'); +f = inline(f_in); +x(1) = input('Enter the initial value of x = '); +y(1) = input('Enter the initial value of y = '); +h = input('Enter the step length h = '); +xn = input('enter the x value for which y is to be evaluated = '); +% main program +x = x(1):h:xn; % generating x values +N=length(x); % number of iterations to be performed +fprintf('itr \t x(i) \t y(i)\n') +for i=1:N + fprintf('%d\t %f\t %f\n',i-1, x(i), y(i)); % print (x, y) in each iteration + k1=h*(f(x(i),y(i))); + k2= h*f(x(i)+h,y(i)+k1); % avoids repeated calculations + y(i+1)= y(i)+(k1+k2)/2; % y(i+1) by Improved EM +end +% simply you can print by using [x', y'] at the end by taking proper care +% in the index !! + +% OUTPUT 1 -------------------------------------------------------------- +% Enter the function f(x,y) = x+y +% Enter the initial value of x = 0 +% Enter the initial value of y = 1 +% Enter the step length h = .2 +% enter the x value for which y is to be evaluated = 1 +% itr x(i) y(i) +% 0 0.000000 1.000000 +% 1 0.200000 1.240000 +% 2 0.400000 1.576800 +% 3 0.600000 2.031696 +% 4 0.800000 2.630669 +% 5 1.000000 3.405416 +% OUTPUT 2 -------------------------------------------------------------- +% Enter the function f(x,y) = (y-x)/(y+x) +% Enter the initial value of x = 0 +% Enter the initial value of y = 1 +% Enter the step length h = 0.02 +% enter the x value for which y is to be evaluated = 0.1 +% itr x(i) y(i) +% 0 0.000000 1.000000 +% 1 0.020000 1.019615 +% 2 0.040000 1.038489 +% 3 0.060000 1.056673 +% 4 0.080000 1.074213 +% 5 0.100000 1.091148 + +% OUTPUT 3 -------------------------------------------------------------- +% Enter the function f(x,y) = x*(x+y^2) +% Enter the initial value of x = 1 +% Enter the initial value of y = 1 +% Enter the step length h = .1 +% enter the x value for which y is to be evaluated = 1.3 +% itr x(i) y(i) +% 0 1.000000 1.000000 +% 1 1.100000 1.239700 +% 2 1.200000 1.597136 +% 3 1.300000 2.179113 + +% OUTPUT 4 -------------------------------------------------------------- +% Enter the function f(x,y) = x^2*(x+y) +% Enter the initial value of x = 1 +% Enter the initial value of y = 1 +% Enter the step length h = .1 +% enter the x value for which y is to be evaluated = 1.3 +% itr x(i) y(i) +% 0 1.000000 1.000000 +% 1 1.100000 1.239150 +% 2 1.200000 1.576666 +% 3 1.300000 2.053451