Second-order Stable Finite Difference Schemes for the Time-fractional Diffusion-wave Equation

We propose two stable and one conditionally stable finite difference schemes of second-order in both time and space for the time-fractional diffusion-wave equation. In the first scheme, we apply the fractional trapezoidal rule in time and the central difference in space. We use the generalized Newton-Gregory formula in time for the second scheme and its modification for the third scheme. While the second scheme is conditionally stable, the first and the third schemes are stable. We apply the methodology to the considered equation with also linear advection-reaction terms and also obtain second-order schemes both in time and space. Numerical examples with comparisons among the proposed schemes and the existing ones verify the theoretical analysis and show that the present schemes exhibit better performances than the known ones

Inferring the time-dependent complex Ginzburg-Landau equation from modulus data

We present a formalism for inferring the equation of evolution of a complex wave field that is known to obey an otherwise unspecified (2+1)-dimensional time-dependent complex Ginzburg-Landau equation, given field moduli over three closely-spaced planes. The phase of the complex wave field is retrieved via a non-interferometric method, and all terms in the equation of evolution are determined using only the magnitude of the complex wave field. The formalism is tested using simulated data for a generalized nonlinear system with a single-component complex wave field. The method can be generalized to multi-component complex fields.Comment: 9 pages, 9 figure