Symmetry Analysis and Conservation Laws of the Boundary Value Problems for Time-Fractional Generalized Burgers' Differential Equation

Abstract

Many physical phenomena in nature can be described or modeled via a differential equation or a system of differential equations. In this work, we restrict our attention to research a solution of fractional nonlinear generalized Burgers' differential equations.  Thereby we find some exact solutions for the nonlinear generalized Burgers' differential equation with a fractional derivative, which has domain as $\mathbb{R}^2\times\mathbb{R}^+$. Here we use the Lie groups method. After applying the Lie groups to the boundary value problem we get the partial differential equations on the domain $\mathbb{R}^2$ with reduced boundary and initial conditions. Also, we find conservation laws for the nonlinear generalized Burgers' differential equation.

Keywords

• Boundary value problem,Conservation laws,Generalized Burgers' equation,Lie groups method,Riemann--Liouville derivative

References

1. [1] C. S. Gardner, J. M. Greene, M. D. Kruskal, R. M. Miura, Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19 (1967), 1095–1097.
2. [2] R. Hirota, J. Satsuma, A variety of nonlinear network equations generated from the B¨acklund transformation for the Tota lattice, Suppl. Prog. Theor. Phys., 59 (1976), 64–100.
3. [3] G. W. Bluman, S. C. Anco, Symmetry and integration methods for differential equations, 154 Appl. Math. Sci., Springer-Verlag, New York, 2002.
4. [4] P. Olver, Applications of Lie Groups to Differential Equations, Springer Science, Germany, 2012.
5. [5] P. Clarkson, M. Kruskal, New similarity reductions of the Boussinesq equation, J. Math. Phys., 30(10) (1989), 2201–2213.
6. [6] P. Clarkson, New similarity reductions for the modified Boussinesq equation, J. Phys. A: Gen., 22 (1989), 2355–2367.
7. [7] R. K. Gazizov, A. A. Kasatkin, S. Y. Lukashchuk, Continuous transformation groups of fractional differential equations, Vestn. USATU, 9 (2007), 125–135.
8. [8] C. M. Khalique, K. R. Adem, Exact solutions of the (2+1)-dimensional Zakharov-Kuznetsov modified equal width equation using Lie group analysis, Math. Comp. Modelling, 54 (2011), 184–189.

9. [9] S. S. Ray, Invariant analysis and conservation laws for the time fractional (2+1)-dimensional Zakharov􀀀Kuznetsov modified equal width equation using Lie group analysis, Comput. Math. Appl., 76 (2018), 2110–2118
10. [10] N. Heymans, I. Podlubny, Physical interpretation of initial conditions for fractional differential equations with Riemann–Liouville fractional derivatives, Rheol. Acta, 45(5) (2006), 765–771.
11. [11] C. Li, D. Qian, Y. Q. Chen, On Riemann–Liouville and Caputo derivatives, Discrete Dyn. Nat. Soc., 15 (2011), Article ID 562494.
12. [12] P. Hydon, Symmetry Methods for Differential Equations: A Beginner’s Guide, Cambridge University press., UK, 2000.
13. [13] G. Iskandarova, D. Kaya, Symmetry solution on fractional equation, J. Optim. Control: Theories Appl., 7(3) (2017) 255–259.
14. [14] D. Kaya, G. Iskandarova, Lie group analysis for a time-fractional nonlinear generalized KdV differential equation, Turk. J. Math., 43(3) (2019), 1263-1275.
15. [15] N. M. Ivanova, C. Sophocleous, R. Tracin, Lie group analysis of two-dimensional variable􀀀coefficient Burgers equation, Z. Angew. Math. Phys., 61(5) (2010), 793􀀀809.
16. [16] M. Abd-el-Malek, A. Amin, Lie group method for solving the generalized Burgers’, Burgers’􀀀KdV and KdV equations with time􀀀dependent Kiryakiable coefficients, J. Symmetry, 7 (2015), 1816–1830.
17. [17] A. Yokus, M. Yavuz, Novel comparison of numerical and analytical methods for fractional Burger-Fisher equation, Discrete Contin. Dyn. Syst., (2020), (in press).
18. [18] R. Sinuvasan, K. M. Tamizhmani, P. G. L. Leach, Algebraic resolution of the Burgers equation with a forcing term, Pramana – J. Phys. 88(5) (2017), 74 pages.
19. [19] I. Podlubny, Fractional Differential Equations: An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications, Academic Press, San Diego, 1999.
20. [20] N. Ibragimov, Lie group analysis classical heritage, ALGA Publications Blekinge Institute of Technology Karlskrona, Sweden, 2004.
21. [21] N. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, 1 CRC Press, Boca Raton, 1994.
22. [22] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley–Interscience, New York, 1993.
23. [23] K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
24. [24] N. Ibragimov, A new conservation theorem, J Math. Anal. Appl., 333(1) (2007), 311–328.
25. [25] N. Ibragimov, Nonlinear self-adjointness and conservation laws, J. Phys A: Math. Gen., 44(43) (2011), 4109–4112.
26. [26] Z. Xiao, L. Wei, Symmetry analysis conservation laws of a time fractional fifth-order Sawada–Kotera equation, J. Appl. Anal. Comput., 7 (2017), 1275–1284.
27. [27] S. Y. Lukashchuk, Conservation laws for time-fractional subdiffusion and diffusion-wave equations, Nonlinear Dyn., 80(1-2) (2015), 791–802.
28. [28] R. K. Gazizov, N. H. Ibragimov, S. Y. Lukashchuk, Conlinear self-adjointness, conservation laws and exat solution of fractional Kompaneets equations, Commun. Nonlinear SCI, 23(1) (2015), 153–163.
29. [29] G. W. Bluman, S. Kumei, Symmetries and Differential Equations, Berlin etc., Springer-Verlag, 1989.
30. [30] R. Cherniha, S. Kovalenko, Lie symmetry of a class of nonlinear boundary value problems with free boundaries, Banach Center Publ., 93 (2011), 73–82.
31. [31] R. Cherniha, S. Kovalenko, Lie symmetries of nonlinear boundary value problems, Commun. Nonlinear SCI, 17 (2012), 71–84.