Abstract
In this work, we introduce and theoretically analyze a relatively simple numerical algorithm to solve a double-fractional condensate model. The mathematical system is a generalization of the famous Gross–Pitaevskii equation, which is a model consisting of two nonlinear complex-valued diffusive differential equations. The continuous model studied in this manuscript is a multidimensional system that includes Riesz-type spatial fractional derivatives. We prove here the relevant features of the numerical algorithm, and illustrative simulations will be shown to verify the quadratic order of convergence in both the space and time variables.
Highlights
Accepted: 19 October 2021There have been dramatic developments in the area of fractional calculus in recent decades [1], and many areas in applied and theoretical mathematics have benefited from these developments [2,3]
From a theoretical point of view, theoretical analyses of conservative finitedifference schemes to solve the Riesz space-fractional Gross–Pitaevskii system have been proposed in the literature [4], along with convergent three-step numerical methods to solve double-fractional condensates, explicit dissipation-preserving methods for Riesz space-fractional nonlinear wave equations in multiple dimensions [5], energy conservative difference schemes for nonlinear fractional Schrödinger equations [6], conservative difference schemes for the Riesz space-fractional sine-Gordon equation [7], high-order central difference schemes for Caputo fractional derivatives [8], among other examples
It is important to point out that most of the methods mentioned above refer to discretizations for partial differential equations with fractional derivatives in space
Summary
There have been dramatic developments in the area of fractional calculus in recent decades [1], and many areas in applied and theoretical mathematics have benefited from these developments [2,3]. There are reports on the numerical solutions of conservative nonlinear Klein-Gordon [14] and sine-Gordon [15,16] equations, symplectic methods for the Schrödinger equation [17], fast and structure-preserving schemes for partial differential equations based on the discrete variational derivative method [18], structure-preserving numerical methods for partial differential equations [19], dissipative or conservative Galerkin methods using discrete partial derivatives for nonlinear evolution equations [20], among other reports [21] These approaches have been extended to the fractional-case scenario, and have been helpful in designing numerical models that are able to preserve the mass and the energy of nonlinear systems [22,23,24]. Other discretizations [4,25] are more difficult to implement computationally, in view that a fixed point technique should be coded along with the computational algorithm Those schemes provide more complicated conditions in order to guarantee the convergence. For almost all x ∈ R (see [29])
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