The mathematical modeling of phase separation of a binary alloy is formulated by a nonlinear evolution equation, known as Cahn-Hilliard equation with various free energy functions. The Cahn-Hilliard equation generally is a fourth-order nonlinear equation and its analytical solution does not exist except trivial solutions. In this paper, a second--order splitting finite difference scheme, based on 2D Crank--Nicolson method, is investigated to approximate the solution of the 2D Cahn-Hilliard equation with a logarithmic free energy function under Neumann boundary conditions. We have proved that the scheme has a unique solution and we have studied the error estimation for the numerical scheme. Also, we have shown that the proposed method preserves the conservation of mass and the decreasing of the total energy. Finally, three numerical experiments are carried out to test the performance of proposed scheme.

The paper investigates a class of exactly solvable third order nonlinear evolution equation [16]. A list of unknown function F(u) is reported for which considered equation contains the nontrivial Lie point symmetries. Moreover, nonlinear self-adjointness is discussed and it is examined that it is not strictly self-adjoint equation for physical parameter A 6= 0 but quasi self-adjoint or more generally nonlinear self-adjoint. Additionally, it is observed that Calogero-Degasperis-Fokas (CDF) equation admits a minimal set of Lie algebra under invariance criteria of Lie groups. These classes are utilized one by one to construct the similarity variables to reduce the dimension of the discussed equation. Additionally, Lie symmetries are used to exhibit the associated conservation laws. Henceforth, Lie symmetry reductions of CDF equation are reported with the help of an optimal system. Meantime, this Lie symmetry method reduces the considered equation into ordinary differential equations. Moreover, well known (G’/G)-expansion method is used to get the exact solutions. The obtained new periodic and solitary wave solutions can be widely used to provide many attractive complex physical phenomena in the different fields of sciences.