FracSym

FracSym: automated symbolic computation of Lie symmetries of fractional differential equations. In this paper, we present an algorithm for the systematic calculation of Lie point symmetries for fractional order differential equations (FDEs) using the method as described by E. Buckwar and Y. Luchko [J. Math. Anal. Appl. 227, No. 1, 81–97, Art. No. AY986078 (1998; Zbl 0932.58038)] and R. K. Gazizov et al. [“Continuous transformation groups of fractional differential equations”, Vestn. USATU 9, No. 21, 125–135 (2007); “Symmetry properties of fractional diffusion equations”, Phys. Scr. 2009, T136, Article ID 014016, 6 p. (2009; doi:10.1088/0031-8949/2009/T136/014016); in: Nonlinear science and complexity. Based on the 2nd conference on nonlinear science and complexity, NSC ’08, Porto, Portugal, 2008. Berlin: Springer. 51–59 (2011; Zbl 1217.37066)]. The method has been generalised here to allow for the determination of symmetries for FDEs with n independent variables and for systems of partial FDEs. The algorithm has been implemented in the new MAPLE package FracSym [the authors, Comput. Phys. Commun. 184, No. 3, 1045–1063 (2013; Zbl 1306.65267)] which uses routines from the MAPLE symmetry packages DESOLVII [K. T. Vu et al., Comput. Phys. Commun. 183, No. 4, 1044–1054 (2012; Zbl 1308.35002)] and ASP [the authors, Comput. Phys. Commun. 184, No. 3, 1045–1063 (2013; Zbl 1306.65267)]. We introduce FracSym by investigating the symmetries of a number of FDEs; specific forms of any arbitrary functions, which may extend the symmetry algebras, are also determined. For each of the FDEs discussed, selected invariant solutions are then presented.


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  1. Hejazi, S. Reza; Saberi, Elaheh; Mohammadizadeh, Fatemeh: Anisotropic non-linear time-fractional diffusion equation with a source term: classification via Lie point symmetries, analytic solutions and numerical simulation (2021)
  2. Kadkhoda, Nematollah; Hadjian, Armin: Group classification and constructing conservation laws for time-fractional Sawada-Kotera-Parker-Dye equation (2021)
  3. Zhang, Zhi-Yong; Li, Guo-Fang: Invariant analysis and conservation laws of the time-fractional (b)-family peakon equations (2021)
  4. Zhang, Zhi-Yong; Zheng, Jia: Symmetry structure of multi-dimensional time-fractional partial differential equations (2021)
  5. Kaur, Bikramjeet; Gupta, R. K.: Time fractional (2+1)-dimensional Wu-Zhang system: dispersion analysis, similarity reductions, conservation laws, and exact solutions (2020)
  6. Ma, Wen-Xiu; Ali, Mohamed R.; Sadat, R.: Analytical solutions for nonlinear dispersive physical model (2020)
  7. Wang, Zhenli; Zhang, Lihua; Liu, Hanze: Lie symmetry analysis to Fisher’s equation with time fractional order (2020)
  8. Kasatkin, Alexey A.; Gainetdinova, Aliya A.: Symbolic and numerical methods for searching symmetries of ordinary differential equations with a small parameter and reducing its order (2019)
  9. Li, Changzhao; Zhang, Juan: Lie symmetry analysis and exact solutions of generalized fractional Zakharov-Kuznetsov equations (2019)
  10. Nass, Aminu M.: Lie symmetry analysis and exact solutions of fractional ordinary differential equations with neutral delay (2019)
  11. Sadat, R.; Kassem, M. M.: Lie analysis and novel analytical solutions for the time-fractional coupled Whitham-Broer-Kaup equations (2019)
  12. Wang, Zhenli; Zhang, Lihua; Li, Chuanzhong: Lie symmetry analysis to the weakly coupled Kaup-Kupershmidt equation with time fractional order (2019)
  13. Baleanu, Dumitru; Inc, Mustafa; Yusuf, Abdullahi; Aliyu, Aliyu Isa: Lie symmetry analysis, exact solutions and conservation laws for the time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation (2018)
  14. Dorjgotov, Khongorzul; Ochiai, Hiroyuki; Zunderiya, Uuganbayar: Lie symmetry analysis of a class of time fractional nonlinear evolution systems (2018)
  15. Dorjgotov, Khongorzul; Ochiai, Hiroyuki; Zunderiya, Uuganbayar: Exact solutions to a class of time fractional evolution systems with variable coefficients (2018)
  16. Kaur, Bikramjeet; Gupta, R. K.: Invariance properties, conservation laws, and soliton solutions of the time-fractional ((2+1))-dimensional new coupled ZK system in magnetized dusty plasmas (2018)
  17. Zhao, Zhonglong; Han, Bo: Symmetry analysis and conservation laws of the time fractional Kaup-Kupershmidt equation from capillary gravity waves (2018)
  18. Akbulut, Arzu; Taşcan, Filiz: Lie symmetries, symmetry reductions and conservation laws of time fractional modified Korteweg-de Vries (mKdV) equation (2017)
  19. Naeem, Imran; Khan, M. D.: Symmetry classification of time-fractional diffusion equation (2017)
  20. Pan, Mingyang; Zheng, Liancun; Liu, Chunyan; Liu, Fawang: Symmetry analysis and conservation laws to the space-fractional Prandtl equation (2017)

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