Numerical Approximation of the Space Fractional Cahn-Hilliard Equation

In this paper, a second-order accurate (in time) energy stable Fourier spectral scheme for the fractional-in-space Cahn-Hilliard (CH) equation is considered. The time is discretized by the implicit backward differentiation formula (BDF), along with a linear stabilized term which represents a second-order Douglas-Dupont-type regularization. The semidiscrete schemes are shown to be energy stable and to be mass conservative. Then we further use Fourier-spectral methods to discretize the space. Some numerical examples are included to testify the effectiveness of our proposed method. In addition, it shows that the fractional order controls the thickness and the lifetime of the interface, which is typically diffusive in integer order case.

An efficient numerical method for solving nonlinear Thomas-Fermi equation

In this paper, the nonlinear Thomas-Fermi equation for neutral atoms by using the fractional order of rational Chebyshev functions of the second kind (FRC2), ${\rm{FU}}_{\rm{n}}^\alpha \left( {{\rm{t}},{\rm{L}}} \right)$ (t, L), on an unbounded domain is solved, where L is an arbitrary parameter. Boyd (Chebyshev and Fourier Spectral Methods, 2ed, 2000) has presented a method for calculating the optimal approximate amount of L and we have used the same method for calculating the amount of L. With the aid of quasilinearization and FRC2 collocation methods, the equation is converted to a sequence of linear algebraic equations. An excellent approximation solution of y(t), y′ (t), and y ′ (0) is obtained.