Well-posedness of space fractional Ginzburg–Landau equations involving the fractional Laplacian arising in a Bose–Einstein condensation and its kernel based approximation

Abstract

This study aims to investigate some theoretical results, numerical study and a real-word application of the SFGLE, involving the fractional Laplacian. First, we describe the application of SFGLE in the Bose–Einstein condensate, where the energy of bosons is transported over long distances. Then, the existence, uniqueness and regularity of the local weak solutions are briefly analyzed. These theoretical results are presented for the first time. Next, the new Green reproducing kernels are introduced for utilization in the numerical solution of SFGLE. The finite difference scheme, introduced by Meerschaer in 2004, is employed to estimate the fractional derivative. For the temporal discretization, an implicit midpoint difference method is applied. The stability, well-posedness of the numerical solution, and the doubled L2–convergence order are attended in details. Our numerical method has the ability to handle high dimensions for numerically solve the equations defined in the hypercube domain. The accuracy and efficiency of our method are evaluated by presenting various 2D and 3D examples and the results will be validated with the previous related works. Ultimately, the utilization of an artificial neural network and empirical outcomes is employed to anticipate the BEC dynamics of Strontium by SFGLE at a specific instance. Subsequently, a comparison between the obtained consequences and experimental data is carried out to demonstrate the authentic applicability of FGLE in precisely describing several physical phenomena.