LIE SYMMETRY ANALYSIS TO FISHER'S EQUATION WITH TIME FRACTIONAL ORDER

Zhenli Wang; Lihua Zhang; Hanze Liu

doi:10.11948/20190323

2020 Volume 10 Issue 5

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Zhenli Wang, Lihua Zhang, Hanze Liu. LIE SYMMETRY ANALYSIS TO FISHER'S EQUATION WITH TIME FRACTIONAL ORDER[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 2058-2067. doi: 10.11948/20190323

Citation:

Zhenli Wang, Lihua Zhang, Hanze Liu. LIE SYMMETRY ANALYSIS TO FISHER'S EQUATION WITH TIME FRACTIONAL ORDER[J]. Journal of Applied Analysis & Computation, 2020, 10(5): 2058-2067. doi: 10.11948/20190323

LIE SYMMETRY ANALYSIS TO FISHER'S EQUATION WITH TIME FRACTIONAL ORDER

1.
School of Science, Nanjing University of Science and Technology, Nanjing, 210094, Jiangsu, China
2.
Hebei University of Economics and Business, Shijiazhuang, 050061, Hebei, China
3.
School of Mathematical Sciences, Dezhou University, Dezhou 253023, Shandong, China
4.
School of Mathematical Sciences, Liaocheng University, Liaocheng 252059, China

Corresponding author: Email address:zzlh100@163.com(L. Zhang)
Fund Project: The authors were supported by National Natural Science Foundation of China(11471174), Natural Science Foundation of Ningbo Grant(2014A610018) and National Science Foundation of Shandong Province(ZR2017LA012), Science and Technology Program of Colleges and Universities in Shandong(J17KA156)

Abstract

Abstract

The aim of this letter is to apply the Lie group analysis method to the Fisher's equation with time fractional order. We considered the symmetry analysis, explicit solutions to the time fractional Fisher's(TFF) equations with Riemann-Liouville (R-L) derivative. The time fractional Fisher's is reduced to respective nonlinear ordinary differential equation(ODE) of fractional order. We solve the reduced fractional ODE using an explicit power series method.
- Lie symmetry analysis /
- time fractional Fisher's equation /
- Riemann-Liouville derivative /
- Erdelyi-Kober operators /
- explicit solutions

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References

[1]	Y. Chen, H. An, Numerical solutions of a new type of fractional coupled nonlinear equations, Commun. Theor. Phys., 2008, 49, 839-844. doi: 10.1088/0253-6102/49/4/07 CrossRef Google Scholar
[2]	Y. Chen, H. An, Numerical solutions of coupled Burgers equations with time-and space-fractional derivatives, Appl. Math. Comput., 2008, 200, 87-95. Google Scholar
[3]	V. D. Djordjevic, T. M. Atanackovic, Similarity solutions to nonlinear heat conduction and Burgers/Korteweg-deVries fractional equations, Comput. Appl. Math., 2008, 212, 701-714. Google Scholar
[4]	A. M. A. El-Sayed, M. Gaber, The Adomian decomposition method for solving partial differential equations of fractal order in finite domains, Phys. Lett.A., 2006, 359, 175-182. doi: 10.1016/j.physleta.2006.06.024 CrossRef Google Scholar
[5]	S. Guo, L. Mei, Y. Li, Y. Sun, The improved fractional sub-equation method and its applications to the space-time fractional differential equations in fluid mechanics, Phys. Lett. A., 2012, 376, 407-411. doi: 10.1016/j.physleta.2011.10.056 CrossRef Google Scholar
[6]	R. K. Gazizov, A. A. Kasatkin, S. Yu. Lukashchuk, Symmetry properties of fractional diffusion equations, Phys. Scr. T., 2009, 136, 014-016. Google Scholar
[7]	V. A. Galaktionov, S. R. Svirshchevskii, Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, Chapman and Hall/CRC, Boca Raton, Florida, 2006. Google Scholar
[8]	S. Hu, W. Chen, X. Gou, Modal analysis of fractional derivative damping model of frequency dependent viscoelastic soft matter, Advancesin Vibration Engineering, 2011, 10, 187-196. Google Scholar
[9]	J. He, A coupling method of a homotopy technique and a perturbation technique for nonlinear problems, J. Non-LinearMech., 2000, 35, 37-43. doi: 10.1016/S0020-7462(98)00085-7 CrossRef Google Scholar
[10]	G. Jumarie, Modied Riemann-Liouville derivative and fractional Taylor series of non-differentiable functions further results, Comput. Math. Appl., 2006, 51, 1367-1376. doi: 10.1016/j.camwa.2006.02.001 CrossRef Google Scholar
[11]	G. F. Jefferson, J. Carminati, FracSym: automated symbolic computation of Lie symmetries of fractional differential equations, Comput. Phys. Commun., 2014, 185, 430-441. doi: 10.1016/j.cpc.2013.09.019 CrossRef Google Scholar
[12]	J. Klafter and R. Metzler, The fractional Fokker-Planck equation: dispersive transport in an external force field, Journal of Molecular Liquids, 2000, 86(1), 219-228. Google Scholar
[13]	V. Kiryakova, Generalized fractional calculus and applications, Pitman Res. Notesin Math., 1994, 301. Google Scholar
[14]	Y. Liang, W. Chen, A survey on numerical evaluation of Lvy stable distributions and a new MATLAB tool box, Signal Processing, 2013, 93, 242-251. doi: 10.1016/j.sigpro.2012.07.035 CrossRef Google Scholar
[15]	B. Lu, Baklund transformation of fractional Riccati equation and its applications to nonlinear fractional partial differential equations, Phys. Lett. A., 2012, 376, 2045-2048. doi: 10.1016/j.physleta.2012.05.013 CrossRef Google Scholar
[16]	X. Li, W. Chen, Analytical study on the fractional anomalous diffusion in a half-plane, J. Phys. A: Math. Theor., 2010, 43, 495206. doi: 10.1088/1751-8113/43/49/495206 CrossRef Google Scholar
[17]	S. Momani, Z. Odibat, Comparison between homotopy perturbation method and the variational iteration method for linear fractional partial differential equations, Comput. Math. Appl., 2007, 54, 910-919. doi: 10.1016/j.camwa.2006.12.037 CrossRef Google Scholar
[18]	S. Momani, Z. Odibat, Homotopy perturbation method for nonlinear partial differential equations of fractional order, Phys. Lett. A., 2007, 365, 345-350. doi: 10.1016/j.physleta.2007.01.046 CrossRef Google Scholar
[19]	K. S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley, New York, 1993. Google Scholar
[20]	Z. Odibat, S. Momani, Modified homotopy perturbation method: Application to quadratic Riccati differential equation of fractional order, Chaos Solitons Fractals, 2008, 36, 167-174. doi: 10.1016/j.chaos.2006.06.041 CrossRef Google Scholar
[21]	Z. Odibat, S. Momani, A generalized differential transform method for linear partial differential equations of fractional order, Appl. Math. Lett., 2008, 21, 194-199. doi: 10.1016/j.aml.2007.02.022 CrossRef Google Scholar
[22]	W. Peng, S. Tian, et al, RiemannšCHilbert method and multi-soliton solutions for three-component coupled nonlinear Schrödinger equations, J. Geom. Phys., 2019, 146, 103508. doi: 10.1016/j.geomphys.2019.103508 CrossRef Google Scholar
[23]	W. Peng, S. Tian, X. Wang, T. Zhang, Characteristics of rogue waves on a periodic background for the Hirota equation, Wave Motion, 2020, 93, 102454. doi: 10.1016/j.wavemoti.2019.102454 CrossRef Google Scholar
[24]	W. Peng, S. Tian, et al, Dynamics of breather waves and higher-order rogue waves in a coupled nonlinear Schrödinger equation, EPL (Europhysics Letters), 2018, 123(5), 50005. doi: 10.1209/0295-5075/123/50005 CrossRef Google Scholar
[25]	W. Peng, S. Tian, T. Zhang, Breather waves, high-order rogue waves and their dynamics in the coupled nonlinear Schrödinger equations with alternate signs of nonlinearities, EPL (Europhysics Letters), 2019, 127(5), 50005. doi: 10.1209/0295-5075/127/50005 CrossRef Google Scholar
[26]	I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, CA, 1999. Google Scholar
[27]	C. Qin, S. Tian, X. Wang, T. Zhang, Lie symmetry analysis, conservation laws and analytical solutions for a generalized time-fractional modified KdV equation, Waves Rand Compl. Med., 2019, 29(3), 456-476. doi: 10.1080/17455030.2018.1450538 CrossRef Google Scholar
[28]	C. Qin, S. Tian, et al, Lie symmetry analysis, conservation laws and exact solutions of fourth-order time fractional Burgers equation, J. Appl. Anal. Comput., 2018, 8(6), 1727-1746. Google Scholar
[29]	R. Sahadevan, T. Bakkyaraj, Invariant analysis of time fractional generalized Burgers and Korteweg-de Vries equations, J. Math. Anal. Appl., 2012, 393, 341-347. doi: 10.1016/j.jmaa.2012.04.006 CrossRef Google Scholar
[30]	S. Tian, Lie symmetry analysis, conservation laws and solitary wave solutions to a fourth-order nonlinear generalized Boussinesq water wave equation, Appl. Math. Lett., 2020, 100, 106056. doi: 10.1016/j.aml.2019.106056 CrossRef Google Scholar
[31]	S. Tian, H. Zhang, On the integrability of a generalized variable-coefficient Kadomtsev-Petviashvili equation, J. Phys. A: Math. Gen., 2012, 192(1), 35-43. Google Scholar
[32]	G. Wu, E. W. M. Lee, Fractional variational iteration method and its application, Phys. Lett. A., 2010, 374, 2506-2509. doi: 10.1016/j.physleta.2010.04.034 CrossRef Google Scholar
[33]	G. Wang, X. Liu, Y. Zhang, Lie symmetry analysis to the time fractional generalized fifth-order KdV equation, Commun. Nonlinear Sci. Numer. Simul., 2013, 18, 2321-2326. doi: 10.1016/j.cnsns.2012.11.032 CrossRef Google Scholar
[34]	G. Wang, T. Xu, Invariant analysis and exact solutions of nonlinear time fractional Sharma-Tasso-Olver equation by Lie group analysis, Nonlinear Dyn., 2014, 76, 571-580. doi: 10.1007/s11071-013-1150-y CrossRef Google Scholar
[35]	E. Yasar, Y. Yildirim, C.M. Khalique, Lie symmetry analysis, conservation laws and exact solutions of the seventh-order time fractional sawada-koteraito equation, Results Phys., 2016, 6, 322-328. doi: 10.1016/j.rinp.2016.06.003 CrossRef Google Scholar
[36]	X. Yan, S. Tian, et al, Rogue Waves and Their Dynamics on Bright-Dark Soliton Background of the Coupled Higher Order Nonlinear Schrödinger Equation, J. Phys. Soc. Japan., 2019, 88(7), 074004. doi: 10.7566/JPSJ.88.074004 CrossRef Google Scholar
[37]	G. M. Zaslavsky, Chaos, fractional kinetics, and anomalous transport, Phys., 2002, 371, 461-580. Google Scholar
[38]	S. Zhang, H. Zhang, Fractional sub-equation method and its applications to nonlinear fractional PDEs, Phys. Lett. A., 2011, 375, 1069-1073. doi: 10.1016/j.physleta.2011.01.029 CrossRef Google Scholar
[39]	L. Zou, Z. Yu, S. Tian, et al, Lie point symmetries, conservation laws, and analytical solutions of a generalized time-fractional Sawada-Kotera equation, Waves Rand Compl. Med., 2018, 3, 1-14. Google Scholar
[40]	T. Zhang, On Lie symmetry analysis, conservation laws and solitary waves to a longitudinal wave motion equation, Appl. Math. Lett., 2019, 98, 199-205. doi: 10.1016/j.aml.2019.06.016 CrossRef Google Scholar