Numerical Methods for Time Integration of Scalar ODEs

1 - Matlab Implementation code for the comparison between the 4th-order Runge–Kutta method and the exact solution.
2 - Matlab Implementation code for the comparison between the ode45 Matlab method and the exact solution.

Also write the numerical result as detailed explanation.

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%final time
tf=2;
disp(['final time = ' num2str(tf)]);

disp('table ODE45')

Abstols=5*10.^[-3:-3:-12];
nsteps=[];
MaxError=[];
for k=1:length(Abstols)
    Abstol=Abstols(k);
    [nsteps(k),MaxError45(k)]=P(tf,Abstol);
end

disp('ODE45: Tol MaxErr Nsteps ')
for k=1:length(Abstols)
    fprintf('%g %g %d \n',Abstols(k),MaxError45(k),nsteps(k));
end

%this is just to generate a plot of timestep used by ode45
to_plot=1;
P(tf,Abstol(end),to_plot);

disp('table ERK')
MaxError=[];
AbsoluteError=[];
Ns=10*2.^[3:7];
for k=1:length(Ns)
    n=Ns(k);
    [h(k),MaxError(k),AbsoluteError(k)]=erk(n,tf);
end

disp('ERK: MaxErr Nsteps h ')
for k=1:length(Ns)
    n=Ns(k);
    fprintf('%g %d %g \n',MaxError(k), n,h(k));
end


figure;
loglog(nsteps,MaxError45,Ns,MaxError,'Linewidth',2);
xlabel('N Steps');
ylabel('Max Error');
legend('ode45','erk','Location','Best');
title_str=['Comparison ODE45 and ERK, Final Time = ' num2str(tf)];
title(title_str);


%final time 8
tf=8;
disp(['final time = ' num2str(tf)]);

disp('table ODE45')...