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modified_euler.m
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modified_euler.m
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% 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