Wanting an explanation to the solution method of the system of differential equations.

Please can anybody help me with explaining, in detail, why and how the author solves the system of differential equations by such a step (see below)? How the equation 6.3 is formed step by step? Why such a column basis is valid? It would be extremely helpful if someone knows the name of the method or could give an exemplification.

The terms multiplied by $\mu$ & $\nu$ are modeling diffusion (this is a standard center difference scheme). What we know about the continuous diffusion operator (with appropriate boundary conditions) is that its eigenvectors are sines &/or cosines. If you prefer, they're exponentials with imaginary powers (the two are equivalent by Euler's formula). This is just the discrete analog: note that the vectors employed for the change of coordinates are exactly such complex exponentials evaluated at the grid points. You can check yourself that they are eigenvectors of the discretized diffusion matrix.
• Sure, the discretized diffusion matrix is a tridiagonal matrix with all diagonal entries equal to $-2$ and all superdiagonal and subdiagonal entries equal to $1$. (See also your ODE system (6.2) above.) Caveat: boundary conditions affect the top and bottom rows (this is reflected in how $x_{-1}$ and $y_{-1}$ are defined). Note, additionally, that (6.3) will only diagonalize the part of your linear ODE system pertaining to diffusion; it will not do so for the part relating to $a$, $b$, $c$ and $d$. – automaton 3 Sep 30 '13 at 12:06
• You use periodic BCs, so I expect the eigenfunctions of the diffusion operator to be translation invariant. No wonder you didn't get the exponentials. I don't have access to MATLAB right now so I can't do it myself, but I'd suggest that you calculate the action of the diffusion matrix on said exponential and see whether you get a multiple of it. As for the constant vector (=spatially uniform pattern in the continuous case), of course that's an eigenvector (corresponding eigenvalue: zero) for your matrix $[-2,1,0,0,1 ; 1,-2,1,0,0 ; 0,1-2,1,0 ; 0,0,1,-2,1 ; 1,0,0-2,1]$! Check your numerics! – automaton 3 Sep 30 '13 at 22:08