% build a vector of ones 
n = 20; 
e1=ones(n,1); 

% construct spatial discretization matrix A 
A=spdiags([e1 -2*e1 e1],[-1 0 1],n,n); 

% periodic boundaries: 
A(1,n)=1; A(n,1)=1; 

% define the domain and time span 
dx = (1-(-1))/(n); 
x = (-1:dx:1-dx); 
tspan = [0 10]; 

% initial conditions (gaussian) 
u0 = exp(-x.^2);

% solve ODE in time 
k = 1.2; 
[t,y]=ode45('rhs',tspan,u0,[],k,dx,A); 

% plot initial condition, solution after one time 
% step and the solution at end time 
figure(1)
set(gca, 'FontSize', 20); 
plot(x,u0,'r*-',x,y(1,:),'g*-',x,y(end,:),'b*-'); 
legend('Initial condition', 'Solution at t = dt\_1', 'Solution at t = 10')
xlabel('x'); ylabel('u');

% plot solutions as a function of time at x = 0 
figure(2)
set(gca, 'FontSize', 20); 
I = find(x == 0); 
plot(t,y(:,I),'b*-');
xlabel('t'); ylabel('u');