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

Year 2019, , 139 - 147, 20.12.2019

Gulistan Iskenderoglu , Dogan Kaya

https://doi.org/10.33401/fujma.598107

Cited By: 2

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

Supporting Institution

Istanbul Commerce University

Project Number

22-2018/34

References

• [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] 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] G. W. Bluman, S. C. Anco, Symmetry and integration methods for differential equations, 154 Appl. Math. Sci., Springer-Verlag, New York, 2002.
• [4] P. Olver, Applications of Lie Groups to Differential Equations, Springer Science, Germany, 2012.
• [5] P. Clarkson, M. Kruskal, New similarity reductions of the Boussinesq equation, J. Math. Phys., 30(10) (1989), 2201–2213.
• [6] P. Clarkson, New similarity reductions for the modified Boussinesq equation, J. Phys. A: Gen., 22 (1989), 2355–2367.
• [7] R. K. Gazizov, A. A. Kasatkin, S. Y. Lukashchuk, Continuous transformation groups of fractional differential equations, Vestn. USATU, 9 (2007), 125–135.
• [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] 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] 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] C. Li, D. Qian, Y. Q. Chen, On Riemann–Liouville and Caputo derivatives, Discrete Dyn. Nat. Soc., 15 (2011), Article ID 562494.
• [12] P. Hydon, Symmetry Methods for Differential Equations: A Beginner’s Guide, Cambridge University press., UK, 2000.
• [13] G. Iskandarova, D. Kaya, Symmetry solution on fractional equation, J. Optim. Control: Theories Appl., 7(3) (2017) 255–259.
• [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] 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] 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] A. Yokus, M. Yavuz, Novel comparison of numerical and analytical methods for fractional Burger-Fisher equation, Discrete Contin. Dyn. Syst., (2020), (in press).
• [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] 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] N. Ibragimov, Lie group analysis classical heritage, ALGA Publications Blekinge Institute of Technology Karlskrona, Sweden, 2004.
• [21] N. Ibragimov, CRC Handbook of Lie Group Analysis of Differential Equations, 1 CRC Press, Boca Raton, 1994.
• [22] K. S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, Wiley–Interscience, New York, 1993.
• [23] K. B. Oldham, J. Spanier, The Fractional Calculus, Academic Press, New York, 1974.
• [24] N. Ibragimov, A new conservation theorem, J Math. Anal. Appl., 333(1) (2007), 311–328.
• [25] N. Ibragimov, Nonlinear self-adjointness and conservation laws, J. Phys A: Math. Gen., 44(43) (2011), 4109–4112.
• [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] S. Y. Lukashchuk, Conservation laws for time-fractional subdiffusion and diffusion-wave equations, Nonlinear Dyn., 80(1-2) (2015), 791–802.
• [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] G. W. Bluman, S. Kumei, Symmetries and Differential Equations, Berlin etc., Springer-Verlag, 1989.
• [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] R. Cherniha, S. Kovalenko, Lie symmetries of nonlinear boundary value problems, Commun. Nonlinear SCI, 17 (2012), 71–84.