Question

i really just need help with part c and d. thank you!

(a)Use Euler method to find the difference equation for the following IVP (initial value problem). Please Type your work. (Due on March 5th) dt(, yo 0.01 (b) Calculate the numerical solution for 0 s t S T using k and M T where k = and T = 9 for M 32,64, 128. Using programming languages such as Ct+, MATLAB, eto. (c) Graph those numerical solutions versus exact solution. (Due on March 19th) (d) Find the maximum error for each M 32, 64, 128 where maximum error eo naxj 1, M+1|y(5)-m y(t,) exact solution of the IVP at t - tj; y, solution of difference equation at t t, calculated by the computer. (Due on April 5th)

Answer 1

(c) and (d)

%% Matlab code %%

clc;
close all;
clear all;
format long;
f=@(t,y)y*(1-y);
y(1)=0.01;

%%%% Exact solution
[t1 y1]=ode45(f,[0 9],y(1));
figure;
plot(t1,y1,'*');
hold on
% Eular therom
M=[32 64 128];
T=9;
fprintf(' M Max error \n' );
for n=1:length(M)
k=T/M(n);
t=0:k:T;
for h=1:length(t)-1
y(h+1)=y(h)+k*f(t(h),y(h));
end
plot(t,y);
hold on
%%% Exact solution
[t y2]=ode45(f,t,y(1));
Mer=max(abs(y2-y'));
fprintf(' %d %f \n',M(n),Mer);
end

legend('Exact solution','M=30','M=64','M=128');
xlabel('t');
ylabel('y(t)');

OUTPUT:

M Max error
32 0.109855   
64 0.056016   
128 0.028231