A Convergent Three-Step Numerical Method to Solve a Double-Fractional Two-Component Bose–Einstein Condensate

Adán J Serna-Reyes,Nuria Reguera,Jorge E Macías-Díaz

doi:10.3390/math9121412

Abstract

This manuscript introduces a discrete technique to estimate the solution of a double-fractional two-component Bose–Einstein condensate. The system consists of two coupled nonlinear parabolic partial differential equations whose solutions are two complex functions, and the spatial fractional derivatives are interpreted in the Riesz sense. Initial and homogeneous Dirichlet boundary data are imposed on a multidimensional spatial domain. To approximate the solutions, we employ a finite difference methodology. We rigorously establish the existence of numerical solutions along with the main numerical properties. Concretely, we show that the scheme is consistent in both space and time as well as stable and convergent. Numerical simulations in the one-dimensional scenario are presented in order to show the performance of the scheme. For the sake of convenience, A MATLAB code of the numerical model is provided in the appendix at the end of this work.

Highlights

P ∈ N represents the number of spatial dimensions, and T ∈ R+ defines a period of time
Suppose that f is a real function defined in all of R and assume that n ∈ N ∪ {0} and α ∈ R are such that n − 1 < α < n
The code is provided in MATLAB for the sake of convenience, and the nomenclature used to describe it follows that of the theoretical description

Summary

Introduction

P ∈ N represents the number of spatial dimensions, and T ∈ R+ defines a period of time. Suppose that α and n are a real and a natural number, respectively, satisfying the properties in the previous definition Whenever it exists, the space-fractional partial derivative in the sense of Riesz of order α of ψ with respect to the variable xi at (x, t) ∈ ΩT is defined as. Riesz fractional derivatives have been discretized consistently in various fashions using fractional-order centered differences [11,28] and weighted-shifted Grünwald differences [29,30] Those discrete approaches have been studied to determine their analytical properties, and they have been used extensively to provide discrete models to solve Riesz space-fractional conservative/dissipative space-fractional wave equations [31], a Hamiltonian fractional nonlinear elastic string equation [32], an energy-preserving double fractional Klein–Gordon–Zakharov system [33], and even a Riesz space-fractional generalization with generalized time-dependent diffusion coefficient and potential of the Higgs boson equation in the de Sitter space-time [32], among other complex systems. For the sake of convenience to the reader, an appendix is provided at the end of this paper, in which we present the numerical scheme used to produce the simulations

Numerical Method

Numerical Properties

Illustrative Simulations

Conclusions