clear t %Clear old time steps
clear y %Clear y values from previous runs
a=0; % Initial tau
b=2; % Final tau
f_an = @(tau) 1./(1+tau);
h=[0.1 0.2 0.5 1.0]; %Incremental time step
y0=1; % Initial value y(a)
figure('Name','Backward Euler:Numerical-Analytical solution comparison plot','NumberTitle','off');
y_an = f_an(a:h(1):b);
plot(a:h(1):b,y_an,'Color',rand(1,3),'LineWidth',2);
legendInfo{1}='Analytical solution';
hold on;
for i=1:length(h)
N=(b-a)/h(i); % Number of time steps
tau(1) = a; % assign initial time
y_n(1) = y0; % assign initial condition
for n=1:N % For loop, sets next t,y values
   tau(n+1) = tau(n)+h(i);
   % here newton-Raphson will be used to iteratively solve y(n+1)
   % two new variables are introduced for N-R.
   % ynr1 will represent the kth iteration and ynr2 the (k+1) iteration
   ynr1 = y_n(n);%initialize NR
   loopcount = 0; diff = 1;
   while abs(diff)>0.0005
         loopcount = loopcount+1;
         ynr2=ynr1 - (h(i)*(ynr1^2)+ynr1-y_n(n))/(1+2*h(i)*ynr1); %N-R iteration
         diff = (ynr2-ynr1)/ynr2; %evaluates convergence
         ynr1=ynr2

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