Abstract
This manuscript studies a double fractional extended p-dimensional coupled Gross–Pitaevskii-type system. This system consists of two parabolic partial differential equations with equal interaction constants, coupling terms, and spatial derivatives of the Riesz type. Associated with the mathematical model, there are energy and non-negative mass functions which are conserved throughout time. Motivated by this fact, we propose a finite-difference discretization of the double fractional Gross–Pitaevskii system which inherits the energy and mass conservation properties. As the continuous model, the mass is a non-negative constant and the solutions are bounded under suitable numerical parameter assumptions. We prove rigorously the existence of solutions for any set of initial conditions. As in the continuous system, the discretization has a discrete Hamiltonian associated. The method is implicit, multi-consistent, stable and quadratically convergent. Finally, we implemented the scheme computationally to confirm the validity of the mass and energy conservation properties, obtaining satisfactory results.
Highlights
Fractional calculus emerged as a generalization of conventional calculus
The classical differential and integral operators proposed by Leibniz more than three centuries ago have been extended to the fractional scenario in various different forms
Various applications have been proposed to within mathematics, such as Cauchy problems with Caputo Hadamard fractional derivatives [1], the synchronization for a class of fractionalorder hyperchaotic systems [2], the inequality estimates for the boundedness of multilinear singular and fractional integral operators [3], the analysis of unstructured-mesh Galerkin finite element method for the two-dimensional multi-term time–space fractional Bloch– Torrey equations on irregular convex domains [4], and the design of numerically efficient and conservative model for a Riesz space-fractional Klein–Gordon–Zakharov system [5], among various other interesting problems [6]
Summary
Fractional calculus emerged as a generalization of conventional calculus. The classical differential and integral operators proposed by Leibniz more than three centuries ago have been extended to the fractional scenario in various different forms. It is worth pointing out that each fractional operator is fully characterized by its own special kernel, and they can be used in a range of particular problems In this respect, the analysis on the uniqueness of solutions for fractional-order differential equations may be carried out through the use of integral inequalities of fractional order, which obviously depend on the type of operator. We introduce the function of mass density of the continuous fractional model (4) as R(ψ1, ψ2) = R(x, t) = R1(x, t) + R2(x, t), for each (x, t) ∈ ΩT In these expressions, R1(x, t) = |ψ2(x, t)|2 and R2(x, t) = |ψ2(x, t)|2, for each (x, t) ∈ ΩT. We design a numerical method which is able of preserving the discrete energy and mass of the system, and such that the solutions satisfy similar boundedness properties. The proposed numerical model has an easy implementation, as shown by the code in Appendix A
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