Numerical simulation of fractal wave propagation of a multi-dimensional nonlinear fractional-in-space Schrödinger equation

Abstract

This paper studies a quantum particle traveling in a fractal space-time, which can be modelled by a fractional modification of the Schrödinger equation with variable coefficients. The Fourier spectral method is used to reveal the solution properties numerically, and the fractal properties are illustrated graphically by choosing different coefficients and different fractional orders. Some novel isosurface plots of the dynamics of pattern formation in fractional Schrödinger equation with variable coefficients are shown. Keywords: Fractal wave propagation; Multi-dimensional; Schrödinger equation; Fractal symmetry; Fourier spectral method.

An efficient numerical simulation of chaos dynamical behaviors for fractional-order Rössler chaotic systems with Caputo fractional derivative

“…Some fractional derivative [9][10][11] and corresponding numerical method have been widely employed in numerical solution of fractal-fractional systems, [12][13][14][15] such as Adams-Bashforth-Moulton algorithm, 1 predictor-correctors scheme, 2 highprecision difference scheme, 16,17 local discontinuous Galerkin method, 18 finite difference method, 19,20 reproducing kernel method, 21,22,23 spectral method, [24][25][26][27] homotopy perturbation method, [28][29][30] Li-He's modified homotopy perturbation method, 31,32 and variational iteration method. The variational iteration method was proposed by Ji-huan He and was applied to a kind of nonlinear oscillators, 33,34 autonomous ordinary differential systems, 35 the Kaup-Newell system, 36 the nanobeam-based N/MEMS system, 37 and fractal pull-in motion of electrostatic MEMS resonators.…”