The construction of higher-order numerical approximation formula for Riesz derivative and its application to nonlinear fractional differential equations (I)

The main goal of this paper is to construct high-order numerical differential formulas approximating the Riesz derivative and apply them to the numerical solution of the nonlinear space fractional Ginzburg–Landau equations. Firstly, we introduce a novel second-order fractional central difference ope...

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Bibliographic Details
Published in:Communications in nonlinear science & numerical simulation Vol. 110; p. 106394
Main Authors: Ding, Hengfei, Yi, Qian
Format: Journal Article
Language:English
Published: Amsterdam Elsevier B.V 01.07.2022
Elsevier Science Ltd
Subjects:
Approximation
Convergence
Differential equations
Energy methods
Finite difference method
Finite differences
Fractional calculus
Fractional-compact numerical algorithm
Fractions
Landau-Ginzburg equations
Mathematical analysis
Nonlinear space fractional Ginzburg–Landau equations
Norms
Operators (mathematics)
Riesz derivative
Stability
Riesz derivative
Fractional-compact numerical algorithm
Stability
Nonlinear space fractional Ginzburg–Landau equations
Convergence
ISSN:1007-5704, 1878-7274
Online Access:Get full text
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Summary:The main goal of this paper is to construct high-order numerical differential formulas approximating the Riesz derivative and apply them to the numerical solution of the nonlinear space fractional Ginzburg–Landau equations. Firstly, we introduce a novel second-order fractional central difference operator for the approximation of the Riesz derivative with order α∈(1,2]. Moreover, based on the difference operator and the compact technique, a novel fourth-order fractional compact difference operator is also derived. Secondly, using the fourth-order difference operator in space and the Crank–Nicolson method in time, a high-order difference scheme is proposed for the nonlinear space fractional Ginzburg–Landau equations. Thirdly, besides the standard energy method, some new techniques and important lemmas are developed to prove the unique solvability, stability and convergence in the sense of different norms. It is proved that the difference scheme is unconditionally stable and convergent with order Oτ2+h4 for α∈(1,1.5), where τ and h are the temporal step size and spatial step size, respectively. Finally, some numerical examples are given to show the efficiency and accuracy of the numerical differential formulas and finite difference scheme. •A fourth-order fractional compact numerical differential formula is constructed.•Some important inequalities are established.•The important properties of the coefficients are studied.•An implicit difference scheme is established.•The new techniques for analyzing stability and convergenceare proposed.
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ISSN:1007-5704
1878-7274
DOI:10.1016/j.cnsns.2022.106394