In this work, we studied Lie symmetry and the conservation law of the conformable fractional Burgers' equation. We discussed the physical significance of the conformable Burgers' equation, as well as the behavior of the fluid that satisfies this equation. As an application, we calculated the group invariant solutions corresponding to an optimal system of one-dimensional subalgebras of the Lie algebra of infinitesimal point symmetries of this equation under the condition that the three spatial derivative orders of the equation are consistent.
Citation: Zhihan Liu, Qi Wang. Lie symmetry and conservation law of the conformable fractional Burgers' equation[J]. AIMS Mathematics, 2026, 11(3): 8332-8354. doi: 10.3934/math.2026342
In this work, we studied Lie symmetry and the conservation law of the conformable fractional Burgers' equation. We discussed the physical significance of the conformable Burgers' equation, as well as the behavior of the fluid that satisfies this equation. As an application, we calculated the group invariant solutions corresponding to an optimal system of one-dimensional subalgebras of the Lie algebra of infinitesimal point symmetries of this equation under the condition that the three spatial derivative orders of the equation are consistent.
| [1] |
R. Khalil, M. Al Horani, A. Yousef, M. Sababheh, A new definition of fractional derivative, J. Comput. Appl. Math., 264 (2014), 65–70. https://doi.org/10.1016/j.cam.2014.01.002 doi: 10.1016/j.cam.2014.01.002
|
| [2] |
T. Abdeljawad, On conformable fractional calculus, J. Comput. Appl. Math., 279 (2015), 57–66. https://doi.org/10.1016/j.cam.2014.10.016 doi: 10.1016/j.cam.2014.10.016
|
| [3] |
A. Atangana, D. Baleanu, A. Alsaedi, New properties of conformable derivative, Open Math., 13 (2015), 889–898. https://doi.org/10.1515/math-2015-0081 doi: 10.1515/math-2015-0081
|
| [4] |
D. Zhao, M. Luo, General conformable fractional derivative and its physical interpretation, Calcolo, 54 (2017), 903–917. https://doi.org/10.1007/s10092-017-0213-8 doi: 10.1007/s10092-017-0213-8
|
| [5] |
M. Chaudhary, R. Kumar, M. K. Singh, Fractional convection-dispersion equation with conformable derivative approach, Chaos Solitons Fract., 141 (2020), 110426. https://doi.org/10.1016/j.chaos.2020.110426 doi: 10.1016/j.chaos.2020.110426
|
| [6] |
G. Tian, X. Meng, Exact solutions to fractional Schrödinger-Hirota equation using auxiliary equation method, Axioms, 13 (2024), 663. https://doi.org/10.3390/axioms13100663 doi: 10.3390/axioms13100663
|
| [7] |
N. Wangyu, Z. Wang, Y. Jiang, X. Meng, Lie symmetries and conservation laws of the conformable one-dimensional heat equation, Discrete Contin. Dyn. Syst. Ser. B, 34 (2026), 216–242. https://doi.org/10.3934/dcdsb.2025170 doi: 10.3934/dcdsb.2025170
|
| [8] | L. V. Ovsiannikov, Group analysis of differential equations, Academic Press, 1982. https://doi.org/10.1016/C2013-0-07470-1 |
| [9] | P. J. Olver, Applications of Lie groups to differential equations, New York: Springer, 1986. https://doi.org/10.1007/978-1-4612-4350-2 |
| [10] |
Y. Chatibi, E. H. El Kinani, A. Ouhadan, Lie symmetry analysis of conformable differential equations, AIMS Mathematics, 4 (2019), 1133–1144. http://dx.doi.org/10.3934/math.2019.4.1133 doi: 10.3934/math.2019.4.1133
|
| [11] |
B. A. Tayyan, A. H. Sakka, Lie symmetry analysis of some conformable fractional partial differential equations, Arab. J. Math., 9 (2020), 201–212. https://doi.org/10.1007/s40065-018-0230-8 doi: 10.1007/s40065-018-0230-8
|
| [12] |
Y. Çenesiz, D. Baleanu, A. Kurt, O. Tasbozan, New exact solutions of Burgers' type equations with conformable derivative, Waves Random Complex Media, 27 (2017), 103–116. https://doi.org/10.1080/17455030.2016.1205237 doi: 10.1080/17455030.2016.1205237
|
| [13] |
M. Kaplan, A. Akbulut, The analysis of the soliton-type solutions of conformable equations by using generalized Kudryashov method, Opt. Quant. Electron., 53 (2021), 498. https://doi.org/10.1007/s11082-021-03144-y doi: 10.1007/s11082-021-03144-y
|
| [14] |
A. M. Ayebire, A. Pasrija, M. S. Manshahia, S. Arora, A novel hybrid computational technique to study conformable Burgers' equation, Math. Comput. Appl., 29 (2024), 114. https://doi.org/10.3390/mca29060114 doi: 10.3390/mca29060114
|
| [15] | A. Ouhadan, T. Mekkaoui, E. Elkinani, Invariant and some new exact solutions of Burgers equation, Int. J. Nonlinear Sci., 11 (2011), 51–58. |
Zhihan Liu, Qi Wang. Lie symmetry and conservation law of the conformable fractional Burgers' equation[J]. AIMS Mathematics, 2026, 11(3): 8332-8354. doi: 10.3934/math.2026342
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