Problem with my forward euler scheme

I am trying to make a code to run the explicit forward euler scheme to solve the 1d wave equation(Ut=c*Ux) with no boundary conditions and I cant get the indexing right and it is leading to an error. The last term where it is j-1 is the problem. It wont run unless I switch it to j= 2:Nt but that is leaving me with an incorrect answer. If i left it as j the answer still doesnt seem right but it at least looks like a plot that would be close. It might not be the indexing and my code could just be completely wrong, but I have no idea. Any help would be greatly appreciated.

clear all
close all

%parameters
Nx = 50;          % Number of spatial steps
xl = 0;          % Left boundary
xr = 4;           % Right boundary
dx = 0.04;        % Spatial step
x = xl:dx:xr;     % x
dt=0.02;           % time step
tf=2;              % final time
Nt = round(tf/dt); % Number of time steps
t = 0:dt:2;        % t 
c=0.5;             %CFL condition
u=zeros(length(x)); %initialize u

%initial condition
for i = 1:Nx+1
    if x<0
        u(i,1) = 0; % u(x,0) 
    else 
        u(i,1) = 1; % u(x,0)
    end
end

%explicit forward euler
for j = 1:Nt                    % Time Loop
    for i = 2:Nx                
        u(i+1,j)=u(i,j)+c*(dt/(2*dx))*(u(i,j+1)-u(i,j-1));
    end
end
plot(u(:,1))
xlabel('Time')
ylabel('Solution')
title('Forward Euler Scheme, part a, dx=0.04 dt=0.02')

Answer 1

[Torsten]

%parameters
c = 0.05;     %Velocity
Nx = 50;         % Number of spatial steps
xl = 0;          % Left boundary
xr = 4;           % Right boundary
x = linspace(xl,xr,Nx) ;    % x
dx = (xr-xl)/(Nx-1);  % Spatial Step
dt = 0.9*dx/c;         % CFL condition
tf=20;              % final time
t = 0:dt:tf;        % t
Nt = numel(t);     % Number of time steps
u=zeros(numel(x),numel(t)); %initialize u
%initial condition
u(2:nx,1) = 1.0;
%explicit forward euler
for j = 1:Nt-1               % Time Loop
  u(2:Nx,j+1) = u(2:Nx,j)-c*dt/dx*(u(2:Nx,j)-u(1:Nx-1,j));
end
plot(x,[u(:,10),u(:,end)])
xlabel('Time')
ylabel('Solution')
title('Forward Euler Scheme, part a, dx=0.04 dt=0.02')

[Vezzaz]

Thank you!