Question

Plot the Euler’s Method approximate solution on [0,1] for the differential equation
y* = 1 + y^2 and initial condition (a) y0 = 0 (b) y0 = 1, along with the exact solution (see
Exercise 7). Use step sizes h = 0.1 and 0.05. The exact solution is y = tan(t + c)

Answer 1

%%Matlab code for solving ode using Euler Improved Euler and RK4
clear all
close all

%function for which Solution have to do
fun=@(t,y) 1+y.^2;
%initial guess
tinit=0; yinit=0;

%Answering Question for initial condition y(0)=0.
syms y(t)
eqn = diff(y,t) == 1+y.^2;
cond = y(0) == yinit;
ySol(t) = dsolve(eqn,cond);

%displaying result
fprintf('Exact solution y(t)=')
disp(ySol)

%Answering Question
%All step size
hh=[0.1 0.05];

fprintf('\nSolution Using Euler Method\n')
%loop for all step size and tend
    for jj=1:length(hh)
        h=hh(jj);
        tend=1;
      
        [t_euler,y_euler]=Euler(fun,tinit,yinit,tend,h);
        y_ext=double(ySol(tend));
      
        error = abs(y_ext-y_euler(end));
        fprintf('\tFor h=%2.4f at t=%2.2f value of y=%f.\n ',h,t_euler(end),y_euler(end))
        fprintf('\t Error E = %f.\n\n',error)
        figure(1)
        hold on
        plot(t_euler,y_euler)
    end
plot(t_euler,ySol(t_euler))
legend('Euler h=0.1','Euler h=0.05','Actual Solution')
xlabel('t')
ylabel('y(t)')
title('Solution plot using Euler for y(0)=0')
box on

clear all
%function for which Solution have to do
fun=@(t,y) 1+y.^2;
%initial guess
tinit=0; yinit=1;
%Answering Question for initial condition y(0)=0.
syms y(t)
eqn = diff(y,t) == 1+y.^2;
cond = y(0) == yinit;
ySol(t) = dsolve(eqn,cond);
%displaying result
fprintf('Exact solution y(t)=')
disp(ySol)
%Answering Question
%All step size
hh=[0.1 0.05];
fprintf('\nSolution Using Euler Method\n')
%loop for all step size and tend
    for jj=1:length(hh)
        h=hh(jj);
        tend=1;
        [t_euler,y_euler]=Euler(fun,tinit,yinit,tend,h);
        y_ext=double(ySol(tend));    
        error = abs(y_ext-y_euler(end));
        fprintf('\tFor h=%2.4f at t=%2.2f value of y=%f.\n ',h,t_euler(end),y_euler(end))
        fprintf('\t Error E = %f.\n\n',error)
        figure(2)
        hold on
        plot(t_euler,y_euler)
    end
plot(t_euler,ySol(t_euler))
legend('Euler h=0.1','Euler h=0.05','Actual Solution')
xlabel('t')
ylabel('y(t)')
title('Solution plot using Euler for y(0)=1')
box on

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%Matlab function for Euler Method
function [t_euler,y_euler]=Euler(f,tinit,yinit,tend,h)
    %Euler method
    % h amount of intervals
    t=tinit;         % initial t
    y=yinit;         % initial y
    t_eval=tend;     % at what point we have to evaluate
    n=(t_eval-t)/h; % Number of steps
    t_euler(1)=t;
    y_euler(1)=y;
    for i=1:n
        %Eular Steps
        m=double(f(t,y));
        t=t+h;
        y=y+h*m;
        t_euler(i+1)=t;
        y_euler(i+1)=y;
    end  
end
%%%%%%%%%%%%%%%%%%%%%% End of Code %%%%%%%%%%%%%%%%%%%%%%%