%{
# Derivation of the equation
 Here we use Chebychev differentiation matrix chebdif.m and compare with the finite differences
%}
%{
# First we use the finite differences to find the first and second derivation of cos(x)
%}

% finite differences
% the grid
% parameters 
L=4*pi;             % domain length 
N=50;           % number of grid points

x=linspace(0,L,N)';
h=x(2)-x(1); % the grid size

% first derivative
D=zeros(N,N);
D(1,1:3)=[-3/2, 2, -1/2]/h;
for ind=2:N-1
    D(ind,ind-1:ind+1)=[-1/2, 0, 1/2]/h;
end
D(end,end-2:end)=[1/2, -2, 3/2]/h;

% second derivative
DD=zeros(N,N);
DD(1,1:3)=[1, -2, 1]/h^2;
for ind=2:N-1
    DD(ind,ind-1:ind+1)=[1, -2, 1]/h^2;
end
DD(end,end-2:end)=[1, -2, 1]/h^2;
f=cos(x);
fxx=DD*f;
fx=D*f;

plot(x,f,'b',x,fx,'r',x,fxx,'g' );

 %{
Here is the figure of cos(x)(blue) , first derivation of cos(x)(red) and second derivation of cos(x)(green) by using finite differences:
<center>
![validation](/sandbox/easystab/finite fifferences.png)

</center>
%}

%{
# Second we use the chebychev to find the first and second derivation of cos(x)
%}

% test de la matrice de chebychev
% parameters 
L=4*pi;             % domain length 
N=50;           % number of grid points


% differentiation and integration 
scale=-2/L; [x,DM] = chebdif(N,2);
D=DM(:,:,1)*scale; DD=DM(:,:,2)*scale^2; x=(x-1)/scale;

% build a function 
f=cos(x);
fx=D*f;
fxx=DD*f;
plot(x,f,'b',x,fx,'k',x,fxx,'g') 

%{
Here is the figure of cos(x)(blue) , first derivation of cos(x)(black) and second derivation of cos(x) (green)by using chebychev:
<center>
![validation](/sandbox/easystab/chebychev.png)</center>
%}