Question

Problem 1: Consider the following Initial Value Problem (IVP) where ? is the dependent variable and ? is the independent variable: ?′=sin(?)∗(1−?) with ?(0)=?0 and ?≥0
Note: the analytic solution for this IVP is: ?(?)=1+(?_0−1)?^cos(?)−1

Part 1A: Approximate the solution to the IVP using Euler’s method with the following conditions: Initial condition ?_0=−1/2; time step ℎ=1/16; and time interval ?∈[0,20]
+ Derive the recursive formula for Euler’s method applied to this IVP
+ Plot the Euler’s method approximation
+ Plot the absolute error between the approximation and the exact solution using a semilog plot

Coding errors, please type out.

Answer 1

%%Matlab code for Euler's forward
clear all
close all
%function for Euler equation solution
f=@(t,y) sin(t)*(1-y);
%displaying the function
fprintf('Function for which differential equation have to solve\n')
disp(f)
%Initial values
t0=0;
y0=-1/2;
%t_end values
t_end=0.4;

    %step size
    h=1/16;
    %x end values

    xn=t0:h:t_end;
    % Euler steps
    y_r(1)=y0;
    t_r(1)=t0;
    fprintf(' \nSolving Differential equation using Euler method for step size h=%f\n',h)
    fprintf(' Initial condition t=%f ; y_1= %f.\n',t_r(1),y_r(1))
        for i=1:length(xn)-1
          
            t_r(i+1)= t_r(i)+h;
            y_r(i+1)= y_r(i)+h*double(f(t_r(i),y_r(i)));         
          
        end     

%exact solution
syms y(t)
eqn = diff(y,t) == sin(t)*(1-y);
cond = y(0) == y0;
ySol(t) = dsolve(eqn,cond);

fprintf('\n\nExact solution is y(t)= ')
disp(ySol(t))

figure(1)
plot(t_r,y_r)
hold on
plot(t_r,ySol(t_r))
xlabel('t')
ylabel('y(t)')
title('y(t) vs. t plot')
legend('Euler method','Exact solution')

figure(2)
err=abs(ySol(t_r)-y_r);
semilogy(t_r,err)
title('semilog plot of Euler error')
xlabel('t')
ylabel('error')

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