| Citation: |
Bikramjeet Kaur, R.K. Gupta. DISPERSION AND FRACTIONAL LIE GROUP ANALYSIS OF TIME FRACTIONAL EQUATION FROM BURGERS HIERARCHY[J]. Journal of Applied Analysis & Computation, 2021, 11(1): 1-22. doi: 10.11948/20180152
|
DISPERSION AND FRACTIONAL LIE GROUP ANALYSIS OF TIME FRACTIONAL EQUATION FROM BURGERS HIERARCHY
-
Abstract
The paper presents the analysis of time fractional $ 5^{th} $ order equation from Burgers hierarchy. We discuss the dispersion relation and provide the complete analysis of the phase velocity and group velocity along with the nature of wave dispersion. Similarity reductions are carried out using infinitesimal symmetries to obtain nonlinear fractional ordinary differential equations having Erd$ \acute{e} $lyi-Kober fractional differential operator. The explicit power series solution is obtained for reduced fractional ordinary differential equation and its convergence is discussed. The solution is appeared in the form of singular kink wave and further analysed graphically for various values of fractional order $ \alpha $. The new conservation theorem is applied to derive the conservation laws. -
-
References
[1] E. A. B. Abdel Salam and G. F. Hassan, Multiwave solutions of fractional 4th and 5th order Burgers equations., Tur. J. Phys., 2015, 39(3), 227-241. [2] A. Abourabia, K. Hassan and E. Selima, Painlevé analysis and new analytical solutions for compound KdV-Burgers equation with variable coefficients, Can. J. Phys., 2010, 88(3), 211-221. doi: 10.1139/P10-003 [3] A. Abourabia and A. Morad, Exact traveling wave solutions of the van der Waals normal form for fluidized granular matter, Physica A, 2015, 437, 333-350. doi: 10.1016/j.physa.2015.06.005 [4] S. C. Anco and G. Bluman, Direct construction method for conservation laws of partial differential equations Part Ⅰ: Examples of conservation law classifications, Eur. J. Appl. Math., 2002, 13(5), 545-566. doi: 10.1017/S095679250100465X [5] T. M. Atanacković, S. Konjik, S. Pilipović and S. Simić, Variational problems with fractional derivatives: Invariance conditions and Noether's theorem, Nonlinear Anal., 2009, 71(5), 1504-1517. [6] A. Bekir, Ö. Güner and A. C. Cevikel, Fractional complex transform and expfunction methods for fractional differential equations, Abstr. Appl. Anal., 2013, 2013. [7] A. Bekir, Ö. Güner and Ö. Unsal, The first integral method for exact solutions of nonlinear fractional differential equations, J. Comput. Nonlinear Dyn., 2015, 10(2), 021020. doi: 10.1115/1.4028065 [8] Z. Bin, G'/G-expansion method for solving fractional partial differential equations in the theory of mathematical physics, Commun. Theor. Phys., 2012, 58(5), 623. doi: 10.1088/0253-6102/58/5/02 [9] A. Biswas, Solitary wave solution for the generalized KdV equation with timedependent damping and dispersion, Commun. Nonlinear Sci. Numer. Simul., 2009, 14(9-10), 3503-3506. doi: 10.1016/j.cnsns.2008.09.026 [10] G. W. Bluman and J. D. Cole, Similarity methods for differential equations, Springer-Verlag, Berlin, 1974. [11] G. W. Bluman and S. Kumei, Symmetries and differential equations, SpringerVerlag, New York, 1989. [12] L. Bourdin, J. Cresson and I. Greff, A continuous/discrete fractional Noethers theorem, Commun. Nonlinear Sci. Numer. Simul., 2013, 18(4), 878-887. doi: 10.1016/j.cnsns.2012.09.003 [13] I. Colombaro, A. Giusti and F. Mainardi, Wave dispersion in the linearised fractional Korteweg-de Vries equation, arXiv preprint arXiv: 1704.02508, 2017. [14] A. Fokas and L. Luo, On the asymptotic integrability of a generalized Burgers equation, Contemp. Math., 1996, 200, 85-98. [15] G. S. F. Frederico and D. F. M. Torres, A formulation of Noether's theorem for fractional problems of the calculus of variations, J. Math. Anal. Appl., 2007, 334(2), 834-846. doi: 10.1016/j.jmaa.2007.01.013 [16] V. A. Galaktionov and S. R. Svirshchevskii, Exact solutions and invariant subspaces of nonlinear partial differential equations in mechanics and physics, CRC Press, 2006. [17] R. K. Gazizov, N. H. Ibragimov and S. Y. Lukashchuk, Nonlinear selfadjointness, conservation laws and exact solutions of time-fractional Kompaneets equations, Commun. Nonlinear Sci. Numer. Simul., 2015, 23(1), 153-163. [18] R. K. Gazizov, A. A. Kasatkin and S. Y. Lukashchuk, Continuous transformation groups of fractional differential equations, USATU, 2007, 9(3), 21. [19] R. K. Gazizov, A. A. Kasatkin and S. Y. Lukashchuk, Symmetry properties of fractional diffusion equations, Phys. Scr., 2009, 2009(T136), 014016. [20] V. Gerdjikov, G. Vilasi and A. B. Yanovski, Integrable hamiltonian hierarchies, Springer, Berlin, 2008. [21] A. Giusti, Dispersion relations for the time-fractional Cattaneo-Maxwell heat equation, J. Math. Phys., 2018, 59(1), 013506. doi: 10.1063/1.5001555 [22] M. S. Hashemi, Group analysis and exact solutions of the time fractional Fokker-Planck equation, Physica A, 2015, 417, 141-149. doi: 10.1016/j.physa.2014.09.043 [23] Q. Huang and R. Zhdanov, Symmetries and exact solutions of the time fractional Harry-Dym equation with Riemann-Liouville derivative, Physica A, 2014, 409, 110-118. doi: 10.1016/j.physa.2014.04.043 [24] N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl., 2007, 333(1), 311-328. doi: 10.1016/j.jmaa.2006.10.078 [25] M. Inc, A. Yusuf, A. I. Aliyu et al., Lie symmetry analysis and explicit solutions for the time fractional generalized Burgers-Huxley equation, Opt. Quantum.Electron., 2018, 50(2), 94. doi: 10.1007/s11082-018-1373-8 [26] M. Inc, A. Yusuf, A. I. Aliyu et al., Lie symmetry analysis, explicit solutions and conservation laws for the space-time fractional nonlinear evolution equations, Physica A, 2018, 496, 371-383. doi: 10.1016/j.physa.2017.12.119 [27] M. Inc, A. Yusuf, A. I. Aliyu et al., Time-fractional cahn-allen and timefractional Klein-Gordon equations: lie symmetry analysis, explicit solutions and convergence analysis, Physica A, 2018, 493, 94-106. doi: 10.1016/j.physa.2017.10.010 [28] H. Jafari, N. Kadkhoda and D. Baleanu, Fractional Lie group method of the time-fractional Boussinesq equation, Nonlinear Dyn., 2015, 81(3), 1569-1574. doi: 10.1007/s11071-015-2091-4 [29] G. F. Jefferson and J. Carminati, FracSym: Automated symbolic computation of Lie symmetries of fractional differential equations, Comput. Phys. Commun., 2014, 185(1), 430-441. doi: 10.1016/j.cpc.2013.09.019 [30] A. H. Kara and F. M. Mahomed, Noether-type symmetries and conservation laws via partial Lagrangians, Nonlinear Dyn., 2006, 45(3), 367-383. [31] B. Kaur and R. K. Gupta, Invariance properties, conservation laws, and soliton solutions of the time-fractional (2+1)-dimensional new coupled ZK system in magnetized dusty plasmas, Comput. Appl. Math., 2018, 37(5), 5981-6004. doi: 10.1007/s40314-018-0674-7 [32] B. Kaur and R. K. Gupta, Dispersion analysis and improved F-expansion method for space-time fractional differential equations, Nonlinear Dyn., 2019, 96(2), 837-852. doi: 10.1007/s11071-019-04825-w [33] B. Kaur and R. K. Gupta, Multiple types of exact solutions and conservation laws of new coupled (2+1)-dimensional Zakharov-Kuznetsov system with timedependent coefficients, Pramana −J. Phys., 2019, 93(4), 59. doi: 10.1007/s12043-019-1806-3 [34] B. Kaur and R. K. Gupta, Time fractional (2+1)-dimensional Wu-Zhang system: Dispersion analysis, similarity reductions, conservation laws, and exact solutions, Comput. Math. Appl., 2020, 79(4), 1031-1048. doi: 10.1016/j.camwa.2019.08.014 [35] T. Kawahara, Formation of saturated solitons in a nonlinear dispersive system with instability and dissipation, Phys. Rev. Lett., 1983, 51(5), 381. doi: 10.1103/PhysRevLett.51.381 [36] A. Kilbas, H. Srivastava and J. Trujillo, Theory and applications of fractional differential equations, Elsevier, Boston, 2006. [37] V. S. Kiryakova, Generalized fractional calculus and applications, Pitman Research Notes in Mathematics Series (Longman Scientific & Technical, Longman Group, UK), 1994. [38] R. A. Kraenkel, J. Pereira and E. de Rey Neto, Linearizability of the perturbed burgers equation, Phys. Rev. E, 1998, 58(2), 2526. doi: 10.1103/PhysRevE.58.2526 [39] N. A. Kudryashov and D. I. Sinelshchikov, Extended models of non-linear waves in liquid with gas bubbles, Int. J. Non Linear Mech., 2014, 63, 31-38. doi: 10.1016/j.ijnonlinmec.2014.03.011 [40] S. Y. Lukashchuk, Conservation laws for time-fractional subdiffusion and diffusion-wave equations, Nonlinear Dyn., 2015, 80(1-2), 791-802. doi: 10.1007/s11071-015-1906-7 [41] F. Mainardi, On signal velocity for anomalous dispersive waves, Il Nuovo Cimento B (1971-1996), 1983, 74(1), 52-58. doi: 10.1007/BF02721684 [42] F. Mainardi, Fractional calculus and waves in linear viscoelasticity: an introduction to mathematical models, World Scientific, London-Singapore, 2010. [43] A. B. Malinowska, A formulation of the fractional Noether-type theorem for multidimensional Lagrangians, Appl. Math. Lett., 2012, 25(11), 1941-1946. doi: 10.1016/j.aml.2012.03.006 [44] K. S. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley-Interscience, 1993. [45] E. Noether, Invariante variationsprobleme, Nachr. d. König. Gesellsch. d. Wiss.zu Göttingen, Math-phys. Klasse, 1918, 1(3), 235-257. [46] T. Odzijewicz, A. Malinowska and D. Torres, Noethers theorem for fractional variational problems of variable order, Cent. Eur. J. Phys., 2013, 11(6), 691-701. [47] K. B. Oldham and J. Spanier, The Fractional Calculus, vol. 111 of Mathematics in science and engineering, Academic Press, New York, London, 1974. [48] P. J. Olver, Applications of Lie groups to differential equations, 107, Graduate Texts in Mathematics. Springer-Verlag, Berlin, 1993. [49] I. Podlubny, Fractional Differential Equations, Academic press, 1999. [50] P. Prakash and R. Sahadevan, Lie symmetry analysis and exact solution of certain fractional ordinary differential equations, Nonlinear Dyn., 2017, 89(1), 305-319. doi: 10.1007/s11071-017-3455-8 [51] C. Qin, S. Tian, X. Wang and T. Zhang, Lie symmetries, conservation laws and explicit solutions for time fractional Rosenau-Haynam equation, Commun. Theor. Phys., 2017, 67(2), 157. doi: 10.1088/0253-6102/67/2/157 [52] W. Rudin, Principles of Mathematical Analysis, China Machine Press, Beijing, 2004. [53] W. Rui and X. Zhang, Lie symmetries and conservation laws for the time fractional Derrida-Lebowitz-Speer-Spohn equation, Commun. Nonlinear Sci. Numer. Simul., 2016, 34, 38-44. doi: 10.1016/j.cnsns.2015.10.004 [54] R. Sahadevan and T. Bakkyaraj, Invariant analysis of time fractional generalized Burgers and Korteweg-de Vries equations, J. Math. Anal. Appl., 2012, 393(2), 341-347. doi: 10.1016/j.jmaa.2012.04.006 [55] S. G. Samko, A. A. Kilbas, O. I. Marichev et al., Fractional integrals and derivatives, 1993, Theory and Applications, Gordon and Breach, Yverdon, 1993. [56] K. Singla and R. Gupta, On invariant analysis of space-time fractional nonlinear systems of partial differential equations. ii, J. Math. Phys., 2017, 58(5), 051503. doi: 10.1063/1.4982804 [57] K. Singla and R. K. Gupta, On invariant analysis of some time fractional nonlinear systems of partial differential equations. I, J. Math. Phys., 2016, 57(10), 101504. doi: 10.1063/1.4964937 [58] K. Singla and R. K. Gupta, Generalized Lie symmetry approach for fractional order systems of differential equations. Ⅲ, J. Math. Phys., 2017, 58(6), 061501. doi: 10.1063/1.4984307 [59] G. Wang, A. H. Kara and K. Fakhar, Symmetry analysis and conservation laws for the class of time-fractional nonlinear dispersive equation, Nonlinear Dyn., 2015, 82(1-2), 281-287. doi: 10.1007/s11071-015-2156-4 [60] G. Wang and T. Xu, Invariant analysis and exact solutions of nonlinear time fractional Sharma-Tasso-Olver equation by Lie group analysis, Nonlinear Dyn., 2014, 76(1), 571-580. doi: 10.1007/s11071-013-1150-y [61] A. M. Wazwaz, Partial differential equations and solitary waves theory, 2009. [62] A. M. Wazwaz, Burgers hierarchy: Multiple kink solutions and multiple singular kink solutions, J. Franklin Inst., 2010, 347(3), 618-626. doi: 10.1016/j.jfranklin.2010.01.003 [63] G. Whitham, Linear and Nonlinear Waves, John Wiley and Sons, New York(1974), Wiley, New York, 1974. [64] E. Yaşsar, Y. Yıldırım and C. M. Khalique, Lie symmetry analysis, conservation laws and exact solutions of the seventh-order time fractional Sawada-Kotera- Ito equation, Results Phys., 2016, 6, 322-328. doi: 10.1016/j.rinp.2016.06.003 [65] F. You and T. Xia, The integrable couplings of the generalized coupled Burgers hierarchy and its Hamiltonian structures, Chaos Solitons Fractals, 2008, 36(4), 953-960. doi: 10.1016/j.chaos.2006.07.029 [66] S. Zhang and H. Zhang, Fractional sub-equation method and its applications to nonlinear fractional PDEs, Phys. Lett. A, 2011, 375(7), 1069-1073. doi: 10.1016/j.physleta.2011.01.029 -
-
Figures(9)
Export File
Citation
Bikramjeet Kaur, R.K. Gupta. DISPERSION AND FRACTIONAL LIE GROUP ANALYSIS OF TIME FRACTIONAL EQUATION FROM BURGERS HIERARCHY[J]. Journal of Applied Analysis & Computation, 2021, 11(1): 1-22. doi: 10.11948/20180152
Format
Content
DownLoad:
- Figure 1. Variation of phase and group velocities with k at α = 0:75.
- Figure 2. Variation of phase and group velocities with k at α = 0:5. Here real part of phase and group velocities are vanishes and velocities are purely imaginary functions of wave number k.
- Figure 3. Variation of phase and group velocities with α having µ = 1, k = 1.
- Figure 4. 3D plot of the solution (4.6) with a0 = 1, a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = −4, a6 = 0:17, a7 = 0:76, µ = 2, α = 0:5, n = 0 to 7.
- Figure 5. 2D plot of the solution (4.6) with a0 = 1, a1 = 1, a2 = 1, a3 = 1, a4 = 1, a5 = −4, a6 = 0:17, a7 = 0:76, µ = 2, α = 0:5, t = 1, n = 0 to 7
- Figure 6. 3D plot of the solution (4.6) with a0 = 1, a1 = 0:5, a2 = 0:25, a3 = 0:125, a4 = 2, a5 = −2:41, a6 = 0:53, µ = 2, α = 0:75, n = 0 to 6.
- Figure 7. 2D plot of the solution (4.6) with a0 = 1, a1 = 0:5, a2 = 0:25, a3 = 0:125, a4 = 2, a5 = −2:41, a6 = 0:53, µ = 2, α = 0:75, t = 1, n = 0 to 6
- Figure 8. 3D plot of the solution (4.6) with a0 = 1, a1 = 0:5, a2 = 0:25, a3 = 0:125, a4 = 2, a5 = −2:41, a6 = 0:53, µ = 2, α = 0:9, n = 0 to 5.
- Figure 9. 2D plot of the solution (4.6) with a0 = 1, a1 = 0:5, a2 = 0:25, a3 = 0:125, a4 = 2, a5 = −2:41, a6 = 0:53, µ = 2, α = 0:9, t = 1, n = 0 to 5