Question

y' = 2 + t^2 + y^2 0<t<1 y(0)=0
use the euler method to determine step size (h) to keep global truncation error below .0001

Answer 1

clear all
close all

%function for which Euler method have to calculate
f=@(y,t) 2+t.^2+y.^2;
err=1;
N=10; count=0;
c1(1)=inf;
tol=0.0001;
while err>tol
  
    count=count+1;
    [y1_result,t_result] = euler_method(f,0,0,1,N);
    err=abs(c1(count)-y1_result(end));
    c1(count+1)=y1_result(end);
    N=N+1;
  
end
plot(t_result,y1_result)
xlabel('time')
ylabel('y(t)')
title('y(t) vs. t plot')
box on
fprintf('Step size required for error less than 0.0001 is %f\n',(1/N))

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%Matlab code for Euler's forward
function [y1_result,t_result] = euler_method(f,y0,t0,tend,N)
%function for Euler equation solution
  
    %all step size
    h=(tend-t0)/N;
    %Initial values

    %t end values
    tn=t0:h:tend;
    % Euler steps
    y1_result(1)=y0;
    t_result(1)=t0;
  
    for i=1:length(tn)-1
        t_result(i+1)= t_result(i)+h;
        y1_result(i+1)=y1_result(i)+h*double(f(y1_result(i),t_result(i)));
    end
end

%%%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%