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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| Published in: | Communications in nonlinear science & numerical simulation Vol. 110; p. 106394 |
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| Main Authors: | , |
| Format: | Journal Article |
| Language: | English |
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Amsterdam
Elsevier B.V
01.07.2022
Elsevier Science Ltd |
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| ISSN: | 1007-5704, 1878-7274 |
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| Abstract | 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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| AbstractList | 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 ... 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.(ProQuest: ... denotes formulae omitted.) 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. |
| ArticleNumber | 106394 |
| Author | Yi, Qian Ding, Hengfei |
| Author_xml | – sequence: 1 givenname: Hengfei surname: Ding fullname: Ding, Hengfei email: dinghf05@163.com organization: School of Mathematics and Statistics, Tianshui Normal University, Tianshui 741001, China – sequence: 2 givenname: Qian orcidid: 0000-0003-3664-6519 surname: Yi fullname: Yi, Qian email: yiqian@i.shu.edu.cn organization: School of Mathematics and Statistics, Guangxi Normal University, Guilin 541006, China |
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| Cites_doi | 10.1016/j.jcp.2016.02.018 10.1007/s11075-017-0466-y 10.1016/j.apm.2009.04.006 10.1002/num.21763 10.1016/j.camwa.2017.12.005 10.1103/RevModPhys.74.99 10.1016/j.jcp.2013.02.037 10.1016/j.jde.2015.06.028 10.1016/j.cnsns.2016.03.008 10.1007/s00220-012-1621-x 10.1002/num.22076 10.1137/S003614299528554X 10.1080/00036811.2018.1469008 10.1063/1.2197167 10.1016/j.ijleo.2018.02.022 10.1016/j.jcp.2014.10.053 10.1137/110833294 10.1016/j.jcp.2014.06.007 10.1137/17M1116222 10.1137/1034003 10.2140/pjm.1977.68.241 10.1007/s10444-021-09862-x 10.4208/cicp.250112.061212a 10.1002/num.20535 10.2478/s13540-013-0014-y 10.1016/j.jcp.2010.10.007 10.1088/0305-4470/39/26/008 10.1140/epjp/i2018-11846-x 10.1007/s10444-005-9004-x 10.1016/j.jcp.2013.03.007 10.1137/140961560 10.1002/mma.5852 10.1016/j.physa.2005.02.047 10.1103/PhysRevLett.79.4047 10.1080/00036811.2018.1466281 10.1016/j.apnum.2017.03.003 10.1016/j.cam.2020.113355 10.1103/PhysRevE.62.3135 10.1080/01630563.2010.510974 10.1016/j.aml.2020.106710 10.1090/S0002-9939-1995-1273525-5 10.1093/imanum/13.1.115 10.1007/s00205-014-0776-3 10.1080/00036811.2011.614601 10.1007/s10543-018-0698-9 |
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| Keywords | Riesz derivative Fractional-compact numerical algorithm Stability Nonlinear space fractional Ginzburg–Landau equations Convergence |
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| References | Lü, Lu (b36) 2007; 27 Zhang (b23) 2010; 31 Fei, Huang, Wang, Zhang (b32) 2021; 15 Zhang, Sun, Wang (b24) 2013; 29 Aranson, Kramer (b7) 2002; 74 Yang, Liu, Turner (b5) 2010; 34 Gu, Shi, Liu (b17) 2019; 98 Xu, Chang (b22) 2011; 27 Tarasov, Zaslavsky (b11) 2006; 16 Zhang, Li, Wang (b34) 2019; 98 Gradshteyn, Ryzhik (b41) 2007 Li, Cai (b45) 2019 Kincaid, Cheney (b49) 1991 Sjölin (b44) 1995; 123 Laskin (b3) 2000; 62 Zhang, Hesthaven, Sun, Ren (b39) 2021; 47 Kuang (b42) 2012 Wang, Xiao, Yang (b4) 2013; 242 Du, Gunzburger, Lehoucq (b9) 2012; 54 He, Pan (b2) 2018; 79 Ginzburg, Landau (b10) 1950; 20 Hao, Sun, Cao (b27) 2015; 281 Wang (b37) 2021; 112 Gao, Sun (b48) 2011; 230 Lu, Bates, Lü, Zhang (b18) 2015; 259 Tarasov (b13) 2006; 39 Wang, Huang (b26) 2018; 58 Millot, Sire (b16) 2015; 215 Mvogo, Tambue, Ben-Bolie, Kofane (b25) 2016; 39 Kirkpatrick, Lenzmann, Staffilani (b40) 2013; 317 Arshed (b19) 2018; 160 Wang, Huang (b1) 2016; 312 Deng, Li, Tian (b47) 2018; 16 Akhmediev, Ankiewicz, Soto-Crespo (b6) 1997; 79 Zhang, Zhang, Sun (b38) 2021; 389 Ding, Li, Chen (b29) 2015; 293 Du, Gunzburger, Peterson (b46) 1992; 34 Guo, Huo (b15) 2013; 16 Akrivis (b50) 1993; 13 Bao, Tang (b8) 2013; 14 Hao, Sun (b28) 2017; 33 Russo (b43) 1977; 68 Wang, Guo, Xu (b21) 2013; 243 Wang, Huang (b33) 2018; 75 Tarasov, Zaslavsky (b12) 2005; 354 Pu, Guo (b14) 2013; 92 Lord (b20) 1997; 34 Li, Huang, Wang (b31) 2017; 118 Mohebbi (b35) 2018; 133 Zhao, Sun, Hao (b30) 2014; 36 Wang (10.1016/j.cnsns.2022.106394_b26) 2018; 58 Du (10.1016/j.cnsns.2022.106394_b9) 2012; 54 Laskin (10.1016/j.cnsns.2022.106394_b3) 2000; 62 Kirkpatrick (10.1016/j.cnsns.2022.106394_b40) 2013; 317 Zhang (10.1016/j.cnsns.2022.106394_b24) 2013; 29 Akhmediev (10.1016/j.cnsns.2022.106394_b6) 1997; 79 Kincaid (10.1016/j.cnsns.2022.106394_b49) 1991 Li (10.1016/j.cnsns.2022.106394_b31) 2017; 118 Gao (10.1016/j.cnsns.2022.106394_b48) 2011; 230 Mohebbi (10.1016/j.cnsns.2022.106394_b35) 2018; 133 Akrivis (10.1016/j.cnsns.2022.106394_b50) 1993; 13 Tarasov (10.1016/j.cnsns.2022.106394_b13) 2006; 39 Du (10.1016/j.cnsns.2022.106394_b46) 1992; 34 Bao (10.1016/j.cnsns.2022.106394_b8) 2013; 14 Tarasov (10.1016/j.cnsns.2022.106394_b11) 2006; 16 Li (10.1016/j.cnsns.2022.106394_b45) 2019 Wang (10.1016/j.cnsns.2022.106394_b4) 2013; 242 Deng (10.1016/j.cnsns.2022.106394_b47) 2018; 16 Tarasov (10.1016/j.cnsns.2022.106394_b12) 2005; 354 Ding (10.1016/j.cnsns.2022.106394_b29) 2015; 293 Russo (10.1016/j.cnsns.2022.106394_b43) 1977; 68 Zhao (10.1016/j.cnsns.2022.106394_b30) 2014; 36 Xu (10.1016/j.cnsns.2022.106394_b22) 2011; 27 Millot (10.1016/j.cnsns.2022.106394_b16) 2015; 215 Zhang (10.1016/j.cnsns.2022.106394_b34) 2019; 98 Aranson (10.1016/j.cnsns.2022.106394_b7) 2002; 74 Zhang (10.1016/j.cnsns.2022.106394_b39) 2021; 47 Lu (10.1016/j.cnsns.2022.106394_b18) 2015; 259 He (10.1016/j.cnsns.2022.106394_b2) 2018; 79 Wang (10.1016/j.cnsns.2022.106394_b37) 2021; 112 Ginzburg (10.1016/j.cnsns.2022.106394_b10) 1950; 20 Wang (10.1016/j.cnsns.2022.106394_b21) 2013; 243 Hao (10.1016/j.cnsns.2022.106394_b27) 2015; 281 Sjölin (10.1016/j.cnsns.2022.106394_b44) 1995; 123 Wang (10.1016/j.cnsns.2022.106394_b33) 2018; 75 Mvogo (10.1016/j.cnsns.2022.106394_b25) 2016; 39 Pu (10.1016/j.cnsns.2022.106394_b14) 2013; 92 Lü (10.1016/j.cnsns.2022.106394_b36) 2007; 27 Hao (10.1016/j.cnsns.2022.106394_b28) 2017; 33 Fei (10.1016/j.cnsns.2022.106394_b32) 2021; 15 Kuang (10.1016/j.cnsns.2022.106394_b42) 2012 Yang (10.1016/j.cnsns.2022.106394_b5) 2010; 34 Zhang (10.1016/j.cnsns.2022.106394_b38) 2021; 389 Guo (10.1016/j.cnsns.2022.106394_b15) 2013; 16 Arshed (10.1016/j.cnsns.2022.106394_b19) 2018; 160 Lord (10.1016/j.cnsns.2022.106394_b20) 1997; 34 Zhang (10.1016/j.cnsns.2022.106394_b23) 2010; 31 Gu (10.1016/j.cnsns.2022.106394_b17) 2019; 98 Gradshteyn (10.1016/j.cnsns.2022.106394_b41) 2007 Wang (10.1016/j.cnsns.2022.106394_b1) 2016; 312 |
| References_xml | – volume: 79 start-page: 899 year: 2018 end-page: 925 ident: b2 article-title: An unconditionally stable linearized difference scheme for the fractional Ginzburg–Landau equation publication-title: Numer. Algorithms – volume: 79 start-page: 4047 year: 1997 end-page: 4051 ident: b6 article-title: Multisoliton solutions of the complex Ginzburg–Landau equation publication-title: Phys Rev Lett – volume: 259 start-page: 5276 year: 2015 end-page: 5301 ident: b18 article-title: Dynamics of the 3-D fractional complex Ginzburg–Landau equation publication-title: J. Differ. Equ. – year: 2007 ident: b41 article-title: Table of integrals, series, and products – volume: 47 start-page: 1 year: 2021 end-page: 36 ident: b39 article-title: Pointwise error estimate in difference setting for the two-dimensional nonlinear fractional complex Ginzburg–Landau equation publication-title: Adv Comput Math – volume: 39 start-page: 396 year: 2016 end-page: 410 ident: b25 article-title: Localized numerical impulse solutions in diffuse neural networks modeled by the complex fractional Ginzburg–Landau equation publication-title: Commun. Nonlinear Sci. – year: 2012 ident: b42 article-title: Applied inequalities – volume: 389 year: 2021 ident: b38 article-title: A three-level finite difference method with preconditioning technique for two-dimensional nonlinear fractional complex Ginzburg–Landau equations publication-title: J Comput Appl Math – volume: 31 start-page: 1190 year: 2010 end-page: 1211 ident: b23 article-title: Long time behavior of difference approximations for the two-dimensional complex Ginzburg–Landau equation publication-title: Numer Funct Anal Optim – volume: 34 start-page: 1483 year: 1997 end-page: 1512 ident: b20 article-title: Attractors and inertial manifolds for finite-difference approximations of the complex Ginzburg–Landau equation publication-title: SIAM J Numer Anal – volume: 243 start-page: 382 year: 2013 end-page: 399 ident: b21 article-title: Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions publication-title: J Comput Phys – volume: 75 start-page: 2223 year: 2018 end-page: 2242 ident: b33 article-title: An efficient split-step quasi-compact finite difference method for the nonlinear fractional Ginzburg–Landau equations publication-title: Comput Math Appl – volume: 20 start-page: 1064 year: 1950 end-page: 1082 ident: b10 article-title: On the theory of superconductivity publication-title: Zh Eksp Teor Fiz – volume: 98 start-page: 2648 year: 2019 end-page: 2667 ident: b34 article-title: A linearized Crank–Nicolson Galerkin FEMs for the nonlinear fractional Ginzburg–Landau equation publication-title: Appl Anal – volume: 16 year: 2006 ident: b11 article-title: Fractional dynamics of coupled oscillators with long-range interaction publication-title: Chaos – volume: 281 start-page: 787 year: 2015 end-page: 805 ident: b27 article-title: A fourth-order approximation of fractional derivatives with its applications publication-title: J Comput Phys – volume: 27 start-page: 293 year: 2007 end-page: 318 ident: b36 article-title: Fourier spectral approximation to long-time behavior of three dimensional Ginzburg–Landau type equation publication-title: Adv Comput Math – volume: 33 start-page: 105 year: 2017 end-page: 124 ident: b28 article-title: A linearized high-order difference scheme for the fractional Ginzburg–Landau equation publication-title: Numer Methods Partial Differential Equations – volume: 230 start-page: 586 year: 2011 end-page: 595 ident: b48 article-title: A compact finite difference scheme for the fractional sub-diffusion equations publication-title: J Comput Phys – volume: 36 start-page: 2865 year: 2014 end-page: 2886 ident: b30 article-title: A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrödinger equation publication-title: SIAM J Sci Comput – volume: 160 start-page: 322 year: 2018 end-page: 332 ident: b19 article-title: Soliton solutions of fractional complex Ginzburg–Landau equation with Kerr law and non-Kerr law media publication-title: Optik – volume: 293 start-page: 218 year: 2015 end-page: 237 ident: b29 article-title: High-order algorithms for Riesz derivative and their applications (II) publication-title: J Comput Phys – volume: 54 start-page: 667 year: 2012 end-page: 696 ident: b9 article-title: Analysis and approximation of nonlocal diffusion problems with volume constraints publication-title: SIAM Rev – volume: 16 start-page: 125 year: 2018 end-page: 149 ident: b47 article-title: Boundary problems for the fractional and tempered fractional operators publication-title: Multiscale Model Simul – volume: 133 start-page: 67 year: 2018 ident: b35 article-title: Fast and high-order numerical algorithms for the solution of multidimensional nonlinear fractional Ginzburg–Landau equation publication-title: Eur Phys J Plus – volume: 312 start-page: 31 year: 2016 end-page: 49 ident: b1 article-title: An implicit midpoint difference scheme for the fractional Ginzburg–Landau equation publication-title: J Comput Phys – year: 2019 ident: b45 article-title: Theory and numerical approximations of fractional integrals and derivatives – year: 1991 ident: b49 article-title: Numerical analysis – volume: 27 start-page: 507 year: 2011 end-page: 528 ident: b22 article-title: Difference methods for computing the Ginzburg–Landau equation in two dimensions publication-title: Numer Methods Partial Differential Equations – volume: 112 year: 2021 ident: b37 article-title: Fast exponential time differencing/spectral-Galerkin method for the nonlinear fractional Ginzburg publication-title: Appl Math Lett – volume: 354 start-page: 249 year: 2005 end-page: 261 ident: b12 article-title: Fractional Ginzburg–Landau equation for fractal media publication-title: Phys. A – volume: 68 start-page: 241 year: 1977 end-page: 253 ident: b43 article-title: On the Hausdorff–Young theorem for integral operators publication-title: Pacific J Math – volume: 317 start-page: 563 year: 2013 end-page: 591 ident: b40 article-title: On the continuum limit for discrete NLS with longrange lattice interactions publication-title: Comm Math Phys – volume: 92 start-page: 318 year: 2013 end-page: 334 ident: b14 article-title: Well-posedness and dynamics for the fractional Ginzburg–Landau equation publication-title: Appl Anal – volume: 242 start-page: 670 year: 2013 end-page: 681 ident: b4 article-title: Crank-nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative publication-title: J Comput Phys – volume: 14 start-page: 819 year: 2013 end-page: 850 ident: b8 article-title: Numerical study of quantized vortex interaction in the Ginzburg–Landau equation on bounded domains publication-title: Commun Comput Phys – volume: 34 start-page: 54 year: 1992 end-page: 81 ident: b46 article-title: Analysis and approximation of the Ginzburg–Landau model of superconductivity publication-title: SIAM Rev – volume: 74 start-page: 99 year: 2002 end-page: 143 ident: b7 article-title: The world of the complex Ginzburg–Landau equation publication-title: Rev. Modern Phys. – volume: 34 start-page: 200 year: 2010 end-page: 218 ident: b5 article-title: Numerical methods for fractional partial differential equations with Riesz space fractional derivatives publication-title: Appl Math Model – volume: 39 start-page: 8395 year: 2006 end-page: 8407 ident: b13 article-title: Psi-series solution of fractional Ginzburg–Landau equation publication-title: J Phys A: Math Gen – volume: 123 start-page: 3085 year: 1995 end-page: 3088 ident: b44 article-title: A remark on the Hausdorff–Young inequality publication-title: Proc Amer Math Soc – volume: 58 start-page: 783 year: 2018 end-page: 805 ident: b26 article-title: An efficient fourth-order in space difference scheme for the nonlinear fractional Ginzburg–Landau equation publication-title: BIT – volume: 215 start-page: 125 year: 2015 end-page: 210 ident: b16 article-title: On a fractional Ginzburg–Landau equation and 1/2-harmonic maps into spheres publication-title: Arch. Ration. Mech. Anal. – volume: 15 start-page: 2711 year: 2021 end-page: 2730 ident: b32 article-title: Galerkin-Legendre spectral method for the nonlinear Ginzburg–Landau equation with the Riesz fractional derivative publication-title: Math Methods Appl Sci – volume: 98 start-page: 2545 year: 2019 end-page: 2558 ident: b17 article-title: Well-posedness of the fractional Ginzburg–Landau equation publication-title: Appl Anal – volume: 29 start-page: 1487 year: 2013 end-page: 1503 ident: b24 article-title: Convergence analysis of a linearized Crank–Nicolson scheme for the two-dimensional complex Ginzburg–Landau equation publication-title: Numer Methods Partial Differential Equations – volume: 62 start-page: 3135 year: 2000 ident: b3 article-title: Fractional quantum mechanics publication-title: Phys. Rev. E. – volume: 13 start-page: 115 year: 1993 end-page: 124 ident: b50 article-title: Finite difference discretization of the cubic Schrödinger equation publication-title: IMA J Numer Anal – volume: 16 start-page: 226 year: 2013 end-page: 242 ident: b15 article-title: Well-posedness for the nonlinear fractional Schrödinger equation and inviscid limit behavior of solution for the fractional Ginzburg–Landau equation publication-title: Fract Calc Appl Anal – volume: 118 start-page: 131 year: 2017 end-page: 149 ident: b31 article-title: Galerkin element method for the nonlinear fractional Ginzburg–Landau equation publication-title: Appl Numer Math – volume: 312 start-page: 31 year: 2016 ident: 10.1016/j.cnsns.2022.106394_b1 article-title: An implicit midpoint difference scheme for the fractional Ginzburg–Landau equation publication-title: J Comput Phys doi: 10.1016/j.jcp.2016.02.018 – volume: 79 start-page: 899 issue: 3 year: 2018 ident: 10.1016/j.cnsns.2022.106394_b2 article-title: An unconditionally stable linearized difference scheme for the fractional Ginzburg–Landau equation publication-title: Numer. Algorithms doi: 10.1007/s11075-017-0466-y – year: 2019 ident: 10.1016/j.cnsns.2022.106394_b45 – volume: 34 start-page: 200 year: 2010 ident: 10.1016/j.cnsns.2022.106394_b5 article-title: Numerical methods for fractional partial differential equations with Riesz space fractional derivatives publication-title: Appl Math Model doi: 10.1016/j.apm.2009.04.006 – volume: 29 start-page: 1487 year: 2013 ident: 10.1016/j.cnsns.2022.106394_b24 article-title: Convergence analysis of a linearized Crank–Nicolson scheme for the two-dimensional complex Ginzburg–Landau equation publication-title: Numer Methods Partial Differential Equations doi: 10.1002/num.21763 – volume: 75 start-page: 2223 year: 2018 ident: 10.1016/j.cnsns.2022.106394_b33 article-title: An efficient split-step quasi-compact finite difference method for the nonlinear fractional Ginzburg–Landau equations publication-title: Comput Math Appl doi: 10.1016/j.camwa.2017.12.005 – volume: 74 start-page: 99 year: 2002 ident: 10.1016/j.cnsns.2022.106394_b7 article-title: The world of the complex Ginzburg–Landau equation publication-title: Rev. Modern Phys. doi: 10.1103/RevModPhys.74.99 – volume: 242 start-page: 670 year: 2013 ident: 10.1016/j.cnsns.2022.106394_b4 article-title: Crank-nicolson difference scheme for the coupled nonlinear Schrödinger equations with the Riesz space fractional derivative publication-title: J Comput Phys doi: 10.1016/j.jcp.2013.02.037 – year: 1991 ident: 10.1016/j.cnsns.2022.106394_b49 – volume: 259 start-page: 5276 year: 2015 ident: 10.1016/j.cnsns.2022.106394_b18 article-title: Dynamics of the 3-D fractional complex Ginzburg–Landau equation publication-title: J. Differ. Equ. doi: 10.1016/j.jde.2015.06.028 – volume: 39 start-page: 396 year: 2016 ident: 10.1016/j.cnsns.2022.106394_b25 article-title: Localized numerical impulse solutions in diffuse neural networks modeled by the complex fractional Ginzburg–Landau equation publication-title: Commun. Nonlinear Sci. doi: 10.1016/j.cnsns.2016.03.008 – volume: 317 start-page: 563 year: 2013 ident: 10.1016/j.cnsns.2022.106394_b40 article-title: On the continuum limit for discrete NLS with longrange lattice interactions publication-title: Comm Math Phys doi: 10.1007/s00220-012-1621-x – volume: 33 start-page: 105 year: 2017 ident: 10.1016/j.cnsns.2022.106394_b28 article-title: A linearized high-order difference scheme for the fractional Ginzburg–Landau equation publication-title: Numer Methods Partial Differential Equations doi: 10.1002/num.22076 – volume: 34 start-page: 1483 year: 1997 ident: 10.1016/j.cnsns.2022.106394_b20 article-title: Attractors and inertial manifolds for finite-difference approximations of the complex Ginzburg–Landau equation publication-title: SIAM J Numer Anal doi: 10.1137/S003614299528554X – volume: 98 start-page: 2648 issue: 15 year: 2019 ident: 10.1016/j.cnsns.2022.106394_b34 article-title: A linearized Crank–Nicolson Galerkin FEMs for the nonlinear fractional Ginzburg–Landau equation publication-title: Appl Anal doi: 10.1080/00036811.2018.1469008 – volume: 20 start-page: 1064 year: 1950 ident: 10.1016/j.cnsns.2022.106394_b10 article-title: On the theory of superconductivity publication-title: Zh Eksp Teor Fiz – volume: 16 year: 2006 ident: 10.1016/j.cnsns.2022.106394_b11 article-title: Fractional dynamics of coupled oscillators with long-range interaction publication-title: Chaos doi: 10.1063/1.2197167 – volume: 160 start-page: 322 year: 2018 ident: 10.1016/j.cnsns.2022.106394_b19 article-title: Soliton solutions of fractional complex Ginzburg–Landau equation with Kerr law and non-Kerr law media publication-title: Optik doi: 10.1016/j.ijleo.2018.02.022 – volume: 281 start-page: 787 year: 2015 ident: 10.1016/j.cnsns.2022.106394_b27 article-title: A fourth-order approximation of fractional derivatives with its applications publication-title: J Comput Phys doi: 10.1016/j.jcp.2014.10.053 – volume: 54 start-page: 667 year: 2012 ident: 10.1016/j.cnsns.2022.106394_b9 article-title: Analysis and approximation of nonlocal diffusion problems with volume constraints publication-title: SIAM Rev doi: 10.1137/110833294 – volume: 293 start-page: 218 year: 2015 ident: 10.1016/j.cnsns.2022.106394_b29 article-title: High-order algorithms for Riesz derivative and their applications (II) publication-title: J Comput Phys doi: 10.1016/j.jcp.2014.06.007 – volume: 16 start-page: 125 year: 2018 ident: 10.1016/j.cnsns.2022.106394_b47 article-title: Boundary problems for the fractional and tempered fractional operators publication-title: Multiscale Model Simul doi: 10.1137/17M1116222 – volume: 34 start-page: 54 issue: 1 year: 1992 ident: 10.1016/j.cnsns.2022.106394_b46 article-title: Analysis and approximation of the Ginzburg–Landau model of superconductivity publication-title: SIAM Rev doi: 10.1137/1034003 – volume: 68 start-page: 241 year: 1977 ident: 10.1016/j.cnsns.2022.106394_b43 article-title: On the Hausdorff–Young theorem for integral operators publication-title: Pacific J Math doi: 10.2140/pjm.1977.68.241 – volume: 47 start-page: 1 year: 2021 ident: 10.1016/j.cnsns.2022.106394_b39 article-title: Pointwise error estimate in difference setting for the two-dimensional nonlinear fractional complex Ginzburg–Landau equation publication-title: Adv Comput Math doi: 10.1007/s10444-021-09862-x – volume: 14 start-page: 819 issue: 3 year: 2013 ident: 10.1016/j.cnsns.2022.106394_b8 article-title: Numerical study of quantized vortex interaction in the Ginzburg–Landau equation on bounded domains publication-title: Commun Comput Phys doi: 10.4208/cicp.250112.061212a – volume: 27 start-page: 507 year: 2011 ident: 10.1016/j.cnsns.2022.106394_b22 article-title: Difference methods for computing the Ginzburg–Landau equation in two dimensions publication-title: Numer Methods Partial Differential Equations doi: 10.1002/num.20535 – volume: 16 start-page: 226 year: 2013 ident: 10.1016/j.cnsns.2022.106394_b15 article-title: Well-posedness for the nonlinear fractional Schrödinger equation and inviscid limit behavior of solution for the fractional Ginzburg–Landau equation publication-title: Fract Calc Appl Anal doi: 10.2478/s13540-013-0014-y – volume: 230 start-page: 586 year: 2011 ident: 10.1016/j.cnsns.2022.106394_b48 article-title: A compact finite difference scheme for the fractional sub-diffusion equations publication-title: J Comput Phys doi: 10.1016/j.jcp.2010.10.007 – volume: 39 start-page: 8395 year: 2006 ident: 10.1016/j.cnsns.2022.106394_b13 article-title: Psi-series solution of fractional Ginzburg–Landau equation publication-title: J Phys A: Math Gen doi: 10.1088/0305-4470/39/26/008 – volume: 133 start-page: 67 issue: 2 year: 2018 ident: 10.1016/j.cnsns.2022.106394_b35 article-title: Fast and high-order numerical algorithms for the solution of multidimensional nonlinear fractional Ginzburg–Landau equation publication-title: Eur Phys J Plus doi: 10.1140/epjp/i2018-11846-x – volume: 27 start-page: 293 year: 2007 ident: 10.1016/j.cnsns.2022.106394_b36 article-title: Fourier spectral approximation to long-time behavior of three dimensional Ginzburg–Landau type equation publication-title: Adv Comput Math doi: 10.1007/s10444-005-9004-x – volume: 243 start-page: 382 year: 2013 ident: 10.1016/j.cnsns.2022.106394_b21 article-title: Fourth-order compact and energy conservative difference schemes for the nonlinear Schrödinger equation in two dimensions publication-title: J Comput Phys doi: 10.1016/j.jcp.2013.03.007 – volume: 36 start-page: 2865 issue: 6 year: 2014 ident: 10.1016/j.cnsns.2022.106394_b30 article-title: A fourth-order compact ADI scheme for two-dimensional nonlinear space fractional Schrödinger equation publication-title: SIAM J Sci Comput doi: 10.1137/140961560 – volume: 15 start-page: 2711 year: 2021 ident: 10.1016/j.cnsns.2022.106394_b32 article-title: Galerkin-Legendre spectral method for the nonlinear Ginzburg–Landau equation with the Riesz fractional derivative publication-title: Math Methods Appl Sci doi: 10.1002/mma.5852 – year: 2012 ident: 10.1016/j.cnsns.2022.106394_b42 – volume: 354 start-page: 249 year: 2005 ident: 10.1016/j.cnsns.2022.106394_b12 article-title: Fractional Ginzburg–Landau equation for fractal media publication-title: Phys. A doi: 10.1016/j.physa.2005.02.047 – volume: 79 start-page: 4047 issue: 21 year: 1997 ident: 10.1016/j.cnsns.2022.106394_b6 article-title: Multisoliton solutions of the complex Ginzburg–Landau equation publication-title: Phys Rev Lett doi: 10.1103/PhysRevLett.79.4047 – volume: 98 start-page: 2545 issue: 14 year: 2019 ident: 10.1016/j.cnsns.2022.106394_b17 article-title: Well-posedness of the fractional Ginzburg–Landau equation publication-title: Appl Anal doi: 10.1080/00036811.2018.1466281 – volume: 118 start-page: 131 year: 2017 ident: 10.1016/j.cnsns.2022.106394_b31 article-title: Galerkin element method for the nonlinear fractional Ginzburg–Landau equation publication-title: Appl Numer Math doi: 10.1016/j.apnum.2017.03.003 – volume: 389 year: 2021 ident: 10.1016/j.cnsns.2022.106394_b38 article-title: A three-level finite difference method with preconditioning technique for two-dimensional nonlinear fractional complex Ginzburg–Landau equations publication-title: J Comput Appl Math doi: 10.1016/j.cam.2020.113355 – volume: 62 start-page: 3135 issue: 3 year: 2000 ident: 10.1016/j.cnsns.2022.106394_b3 article-title: Fractional quantum mechanics publication-title: Phys. Rev. E. doi: 10.1103/PhysRevE.62.3135 – volume: 31 start-page: 1190 year: 2010 ident: 10.1016/j.cnsns.2022.106394_b23 article-title: Long time behavior of difference approximations for the two-dimensional complex Ginzburg–Landau equation publication-title: Numer Funct Anal Optim doi: 10.1080/01630563.2010.510974 – volume: 112 year: 2021 ident: 10.1016/j.cnsns.2022.106394_b37 article-title: Fast exponential time differencing/spectral-Galerkin method for the nonlinear fractional Ginzburg publication-title: Appl Math Lett doi: 10.1016/j.aml.2020.106710 – volume: 123 start-page: 3085 year: 1995 ident: 10.1016/j.cnsns.2022.106394_b44 article-title: A remark on the Hausdorff–Young inequality publication-title: Proc Amer Math Soc doi: 10.1090/S0002-9939-1995-1273525-5 – year: 2007 ident: 10.1016/j.cnsns.2022.106394_b41 – volume: 13 start-page: 115 year: 1993 ident: 10.1016/j.cnsns.2022.106394_b50 article-title: Finite difference discretization of the cubic Schrödinger equation publication-title: IMA J Numer Anal doi: 10.1093/imanum/13.1.115 – volume: 215 start-page: 125 year: 2015 ident: 10.1016/j.cnsns.2022.106394_b16 article-title: On a fractional Ginzburg–Landau equation and 1/2-harmonic maps into spheres publication-title: Arch. Ration. Mech. Anal. doi: 10.1007/s00205-014-0776-3 – volume: 92 start-page: 318 year: 2013 ident: 10.1016/j.cnsns.2022.106394_b14 article-title: Well-posedness and dynamics for the fractional Ginzburg–Landau equation publication-title: Appl Anal doi: 10.1080/00036811.2011.614601 – volume: 58 start-page: 783 issue: 3 year: 2018 ident: 10.1016/j.cnsns.2022.106394_b26 article-title: An efficient fourth-order in space difference scheme for the nonlinear fractional Ginzburg–Landau equation publication-title: BIT doi: 10.1007/s10543-018-0698-9 |
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| SubjectTerms | 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 |
| Title | The construction of higher-order numerical approximation formula for Riesz derivative and its application to nonlinear fractional differential equations (I) |
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