Lie symmetry group, exact solutions and conservation laws for multi-term time fractional differential equations

Miao Yang; Lizhen Wang; Miao Yang; Lizhen Wang

doi:10.3934/math.20231536

AIMS Mathematics

2023, Volume 8, Issue 12: 30038-30058. doi: 10.3934/math.20231536

Previous Article Next Article

Research article

Lie symmetry group, exact solutions and conservation laws for multi-term time fractional differential equations

Miao Yang ,
Lizhen Wang ^,

Center for Nonlinear Studies, School of Mathematics, Northwest University, Xi'an, 710127, China

Received: 28 August 2023 Revised: 20 October 2023 Accepted: 24 October 2023 Published: 06 November 2023
MSC : 26A33, 76M60

In this paper, the time fractional Benjamin-Bona-Mahony-Peregrine (BBMP) equation and time-fractional Novikov equation with the Riemann-Liouville derivative are investigated through the use of Lie symmetry analysis and the new Noether's theorem. Then, we construct their group-invariant solutions by means of Lie symmetry reduction. In addition, the power-series solutions are also obtained with the help of the Erdélyi-Kober (E-K) fractional differential operator. Furthermore, the conservation laws for the time-fractional BBMP equation are established by utilizing the new Noether's theorem.
- Lie symmetry analysis,
- time-fractional BBMP equation,
- time-fractional Novikov equation,
- group-invariant solutions,
- conservation laws
Citation: Miao Yang, Lizhen Wang. Lie symmetry group, exact solutions and conservation laws for multi-term time fractional differential equations[J]. AIMS Mathematics, 2023, 8(12): 30038-30058. doi: 10.3934/math.20231536

Related Papers:

Abstract

In this paper, the time fractional Benjamin-Bona-Mahony-Peregrine (BBMP) equation and time-fractional Novikov equation with the Riemann-Liouville derivative are investigated through the use of Lie symmetry analysis and the new Noether's theorem. Then, we construct their group-invariant solutions by means of Lie symmetry reduction. In addition, the power-series solutions are also obtained with the help of the Erdélyi-Kober (E-K) fractional differential operator. Furthermore, the conservation laws for the time-fractional BBMP equation are established by utilizing the new Noether's theorem.

References

[1]	R. K. Gazizov, N. H. Ibragimov, Lie symmetry analysis of differential equations in finance, Nonlinear Dynam., 17 (1998), 387–407. https://doi.org/10.1023/A:1008304132308 doi: 10.1023/A:1008304132308
[2]	H. Liu, J. B. Li, Q. X. Zhang, Lie symmetry analysis and exact explicit solutions for general Burgers' equation, J. Comput. Appl. Math., 228 (2009), 1–9. https://doi.org/10.1016/j.cam.2008.06.009 doi: 10.1016/j.cam.2008.06.009
[3]	A. Jannelli, M. Ruggieri, M. P. Speciale, Exact and numerical solutions of time-fractional advection-diffusion equation with a nonlinear source term by means of the Lie symmetries, Nonlinear Dynam., 92 (2018), 543–555. https://doi.org/10.1007/s11071-018-4074-8 doi: 10.1007/s11071-018-4074-8
[4]	A. Jannelli, M. P. Speciale, Exact and numerical solutions of two-dimensional time-fractional diffusion-reaction equations through the Lie symmetries, Nonlinear Dynam., 105 (2021), 2375–2385. https://doi.org/10.1007/s11071-021-06697-5 doi: 10.1007/s11071-021-06697-5
[5]	A. Jannelli, M. Ruggieri, M. P. Speciale, Numerical solutions of space-fractional advection-diffusion equations with nonlinear source term, Appl. Numer. Math., 155 (2020), 93–102. https://doi.org/10.1016/j.apnum.2020.01.016 doi: 10.1016/j.apnum.2020.01.016
[6]	S. Kazem, Exact solution of some linear fractional differential equations by Laplace transform, Int. J. Nonlinear Sci., 16 (2013), 3–11.
[7]	A. M. Wazwaz, A. Gorguis, An analytic study of Fisher's equation by using Adomian decomposition method, Appl. Math. Comput., 154 (2004), 609–620. https://doi.org/10.1016/S0096-3003(03)00738-0 doi: 10.1016/S0096-3003(03)00738-0
[8]	J. H. He, Variational iteration method-a kind of non-linear analytical technique: some examples, Internat. J. Non-Linear Mech., 34 (1999), 699–708. https://doi.org/10.1016/s0020-7462(98)00048-1 doi: 10.1016/s0020-7462(98)00048-1
[9]	S. J. Liao, On the homotopy analysis method for nonlinear problems, Appl. Math. Comput., 147 (2004), 499–513. https://doi.org/10.1016/s0096-3003(02)00790-7 doi: 10.1016/s0096-3003(02)00790-7
[10]	W. X. Zhong, X. X. Zhong, Method of separation of variables and Hamiltonian system, Numer. Methods Partial Differential Equations, 9 (1993), 63–75. https://doi.org/10.1002/num.1690090107 doi: 10.1002/num.1690090107
[11]	W. X. Ma, A refined invariant subspace method and applications to evolution equations, Sci. China Math., 55 (2012), 1769–1778. https://doi.org/10.1007/s11425-012-4408-9 doi: 10.1007/s11425-012-4408-9
[12]	F. Tasnim, M. A. Akbar, M. S. Osman, The extended direct algebraic method for extracting analytical solitons solutions to the cubic nonlinear Schrodinger equation involving Beta derivatives in space and time, Fractal Fract., 7 (2023), 426. https://doi.org/10.3390/fractalfract7060426 doi: 10.3390/fractalfract7060426
[13]	K. K. Ali, M. A. Abd El Salam, E. M. H. Mohamed, B. Samet, S. Kumar, M. S. Osman, Numerical solution for generalized nonlinear fractional integro-differential equations with linear functional arguments using Chebyshev series, Adv. Differ. Equ., 2020 (2020), 494. https://doi.org/10.1186/s13662-020-02951-z doi: 10.1186/s13662-020-02951-z
[14]	H. Z. Liu, J. B. Li, Lie symmetry analysis and exact solutions for the short pulse equation, Nonlinear Anal., 71 (2009), 2126–2133. https://doi.org/10.1016/j.na.2009.01.075 doi: 10.1016/j.na.2009.01.075
[15]	M. Craddock, K. A. Lennox, Lie symmetry methods for multi-dimensional parabolic PDEs and diffusions, J. Differ. Equ., 252 (2012), 56–90. https://doi.org/10.1016/j.jde.2011.09.024 doi: 10.1016/j.jde.2011.09.024
[16]	J. Hu, Y. J. Ye, S. F. Shen, J. Zhang, Lie symmetry analysis of the time fractional KdV-type equation, Appl. Math. Comput., 233 (2014), 439–444. https://doi.org/10.1016/j.amc.2014.02.010 doi: 10.1016/j.amc.2014.02.010
[17]	R. A. Leo, G. Sicuro, P. Tempesta, A foundational approach to the Lie theory for fractional order partial differential equations, Fract. Calc. Appl. Anal., 20 (2017), 212–231. https://doi.org/10.1515/fca-2017-0011 doi: 10.1515/fca-2017-0011
[18]	L. Z. Wang, D. J. Wang, S. F. Shen, Q. Huang, Lie point symmetry analysis of the Harry-Dym type equation with Riemann-Liouville fractional derivative, Acta Math. Appl. Sin. Engl. Ser., 34 (2018), 469–477. https://doi.org/10.1007/s10255-018-0760-z doi: 10.1007/s10255-018-0760-z
[19]	E. Lashkarian, S. R. Hejazi, N. Habibi, A. Motamednezhad, Symmetry properties, conservation laws, reduction and numerical approximations of time-fractional cylindrical-Burgers equation, Commun. Nonlinear Sci. Numer. Simul., 67 (2019), 176–191. https://doi.org/10.1016/j.cnsns.2018.06.025 doi: 10.1016/j.cnsns.2018.06.025
[20]	X. Y. Cheng, L. Z Wang, Invariant analysis, exact solutions and conservation laws of (2+1)-dimensional time fractional Navier-Stokes equations, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 477 (2021), 20210220. https://doi.org/10.1098/rspa.2021.0220 doi: 10.1098/rspa.2021.0220
[21]	Z. Y. Zhang, G. F. Li, Invariant analysis and conservation laws of the time-fractional b-family peakon equations, Commun. Nonlinear Sci. Numer. Simul., 103 (2021), 106010. https://doi.org/10.1016/j.cnsns.2021.106010 doi: 10.1016/j.cnsns.2021.106010
[22]	E. Nöether, Invariant variation problems, Transport Theory Statist. Phys., 1 (1971), 186–207. https://doi.org/10.1080/00411457108231446 doi: 10.1080/00411457108231446
[23]	N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., 333 (2007), 311–328. https://doi.org/10.1016/j.jmaa.2006.10.078 doi: 10.1016/j.jmaa.2006.10.078
[24]	S. F. Tian, Lie symmetry analysis, conservation laws and solitary wave solutions to a fourth-order nonlinear generalized Boussinesq water wave equation, Appl. Math. Lett., 100 (2020), 106056. https://doi.org/10.1016/j.aml.2019.106056 doi: 10.1016/j.aml.2019.106056
[25]	O. O. Vaneeva, A. G. Johnpillai, R. O. Popovych, C. Sophocleous, Enhanced group analysis and conservation laws of variable coefficient reaction-diffusion equations with power nonlinearities, J. Math. Anal. Appl., 330 (2007), 1363–1386. https://doi.org/10.1016/j.jmaa.2006.08.056 doi: 10.1016/j.jmaa.2006.08.056
[26]	P. Razborova, A. H. Kara, A. Biswas, Additional conservation laws for Rosenau-KdV-RLW equation with power law nonlinearity by Lie symmetry, Nonlinear Dynam., 79 (2015), 743–748. https://doi.org/10.1007/s11071-014-1700-y doi: 10.1007/s11071-014-1700-y
[27]	M. Inc, A. Yusuf, A. I. Aliyu, D. Baleanu, Time-fractional Cahn-Allen and time-fractional Klein-Gordon equations: Lie symmetry analysis, explicit solutions and convergence analysis, Phys. A, 493 (2018), 94–106. https://doi.org/10.1016/j.physa.2017.10.010 doi: 10.1016/j.physa.2017.10.010
[28]	D. Baleanu, M. Inc, A. Yusuf, A. I. Aliyu, Lie symmetry analysis, exact solutions and conservation laws for the time fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation, Commun. Nonlinear Sci. Numer. Simulat., 59 (2018), 222–234. https://doi.org/10.1016/j.cnsns.2017.11.015 doi: 10.1016/j.cnsns.2017.11.015
[29]	V. A Galaktionov, S. R. Svirshchevskii, Exact solutions and invariant subspaces of nonlinear partial differential equations in mechanics and physics, New York: Chapman and Hall/CRC, 2006.
[30]	D. H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech., 25 (1966), 321–330. https://doi.org/10.1017/S0022112066001678 doi: 10.1017/S0022112066001678
[31]	J. D. Meiss, W. Horton, Solitary drift waves in the presence of magnetic shear, Phys. Fluids, 26 (1983), 990–997. https://doi.org/10.1063/1.864251 doi: 10.1063/1.864251
[32]	C. T. Yan, Regularized long wave equation and inverse scattering transform, J. Math. Phys., 34 (1993), 2618–2630. https://doi.org/10.1063/1.530087 doi: 10.1063/1.530087
[33]	C. M. Khalique, Solutions and conservation laws of Benjamin-Bona-Mahony-Peregrine equation with power-law and dual power-law nonlinearities, Pramana, 80 (2013), 413–427. https://doi.org/10.1007/s12043-012-0489-9 doi: 10.1007/s12043-012-0489-9
[34]	V. Novikov, Generalizations of the Camassa-Holm equation, J. Phys. A: Math. Theor., 42 (2009), 342002. https://doi.org/10.1088/1751-8113/42/34/342002 doi: 10.1088/1751-8113/42/34/342002
[35]	I. Podlubny, Fractional differential equations, San Diego: Academic Press, 1999.
[36]	A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier Science, 2006.
[37]	Q. Huang, R. Zhdanov, Symmetries and exact solutions of the time fractional Harry-Dym equation with Riemann-Liouville derivative, Physica. A, 409 (2014), 110–118. https://doi.org/10.1016/j.physa.2014.04.043 doi: 10.1016/j.physa.2014.04.043
[38]	B. Gao, Y. Zhang, Symmetries and conservation laws of the Yao-Zeng two-component short-pulse equation, Bound. Value Probl., 2019 (2019), 45. https://doi.org/10.1186/s13661-019-1156-6 doi: 10.1186/s13661-019-1156-6
[39]	W. Rudin, Principles of mathematia analysis, McGraw Hill, 1953.
[40]	N. H. Ibragimov, E. D. Avdonina, Nonlinear self-adjointness, conservation laws, and the construction of solutions of partial differential equations using conservation laws, Russ. Math. Surv., 68 (2013), 889. https://doi.org/10.1070/RM2013v068n05ABEH004860 doi: 10.1070/RM2013v068n05ABEH004860

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)