Similarly, if we calculate the discrete problem of the left hand side equation in (3.2) using the same central difference operator
by taking [[xi].sub.i] = [u.sup.2.sub.i], where u is the central difference computed solution for equation (2.1), we observe that LHS is negative wherever the solution is smooth enough and positive where we have boundary layers (or oscillation in the computed solution of equation (2.1)).
Recently, they  proposed a novel numerical method for the space Riesz fractional advection-dispersion equation based on fractional central difference operator. Ozdemir et al.
In , Tuan and Gorenflo introduced the following left fractional central difference operator:
As u(x, t) with respect to x belongs to [L.sub.1] (R), then the Fourier transform of the fractional average central difference operator (33) exists and has the following form:
uniformly holds for x [member of] R, where 8X denotes second-order central difference operator with respect to x and is defined by [[delta].sup.2.sub.x] u([x.sub.j], t) = u([x.sub.j + 1],t) - 2u([x.sub.j], t) + u([x.sub.j - 1], t).