Abstract
To improve the computing efficiency, a fourth-order difference scheme is proposed and a fast algorithm is designed to simulate the nonlinear fractional Schrödinger (FNLS) equation oriented from the fractional quantum mechanics. The numerical analysis and experiments conducted in this article show that the proposed difference scheme has the optimal second-order and fourth-order convergence rates in time and space respectively, reduces its computation cost to mathcal{O}(Mlog M), and recognizes accurately its physical feature of FNLS such as the mass balance.
Highlights
1 Introduction As is well known, numerous experiments have recognized that the fractional calculus can provide more flexible descriptions than the counterpart of integer-order for the real-world phenomena arising in various fields of science and engineering such as transmission of malaria disease [1], the constrained systems [2], the exothermic reactions model [3], and the spring pendulum [4], which has attracted a mounting number of valuable research work both mathematically and numerically during the last few years, see [5,6,7,8,9,10,11,12,13,14,15,16] and the references therein
As one of the most significant applications of the fractional calculus in quantum mechanics, the fractional Schrödinger equation (FNLS) was derived from the Lévy path integrals instead of the Brownian path integrals as done in the classical Schrödinger equation given by Feynman and Hibbs [17]
The goals of this article are as follows: (1) to adopt a fourth-order difference scheme by applying the Crank–Nicolson scheme to discrete the temporal derivative and truncating the weighted and shifted difference formula, [25] to discrete the fractional Laplacian operator; (2) to prove the solvability of the scheme and conduct numerical analysis to verify the convergence rates, as well as to show that the proposed numerical scheme can inherit the physical feature of FNLS such as the mass balance; (3) to design an efficient numerical algorithm through combining the Toeplitz structure of the coefficient matrix and the fast Fourier transform, which will reduce the computation cost from O(M2) to O(M log M); and (4) to conduct numerical experiments to verify the theoretical results and the physical properties
Summary
Numerous experiments have recognized that the fractional calculus can provide more flexible descriptions than the counterpart of integer-order for the real-world phenomena arising in various fields of science and engineering such as transmission of malaria disease [1], the constrained systems [2], the exothermic reactions model [3], and the spring pendulum [4], which has attracted a mounting number of valuable research work both mathematically and numerically during the last few years, see [5,6,7,8,9,10,11,12,13,14,15,16] and the references therein.As one of the most significant applications of the fractional calculus in quantum mechanics, the fractional Schrödinger equation (FNLS) was derived from the Lévy path integrals instead of the Brownian path integrals as done in the classical Schrödinger equation given by Feynman and Hibbs [17]. Numerous experiments have recognized that the fractional calculus can provide more flexible descriptions than the counterpart of integer-order for the real-world phenomena arising in various fields of science and engineering such as transmission of malaria disease [1], the constrained systems [2], the exothermic reactions model [3], and the spring pendulum [4], which has attracted a mounting number of valuable research work both mathematically and numerically during the last few years, see [5,6,7,8,9,10,11,12,13,14,15,16] and the references therein. Checking carefully the existing numerical methods, we find that the difference methods are implemented, they possess low computing accuracy, which motivates. Chang and Chen Advances in Difference Equations (2020) 2020:4 us to design a high-order scheme to numerically solve the fractional Schrödinger. We find that the nonlocality of the fractional Laplacian operator in FNLS often generates a non-sparse matrix of the discrete system, which makes the computation cost to be O(M2) if CG-like algorithms are used
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