Dym equation: group analysis and conservation laws

Document Type : Original Article


Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood, Semnan, Iran


In this paper group-invariant properties of the Dym equation are studied. Lie symmetries are given and some group-invariant solutions are found with the use of similarity variables obtained from these operators. Conservation laws are computed via three methods. Direct method for construction of conservation laws is introduced by the concept of multipliers and Euler-Lagrange operator. Next, the non-linearly self-adjointness of the considered PDE is stated. Then, the modified Noether’s theorem is used for finding conservation laws. Finally, the third method is established via the Hereman-Pole method by using the evolutionary form of the equation.


Main Subjects

[1] D. Baleanu, K. Diethelm, E. Scalas, and J. J. Trujillo, Fractional calculus, vol. 3 of Series on Complexity, Nonlinearity and Chaos, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2012. Models and numerical methods.
[2] G. W. Bluman and S. C. Anco, Symmetry and integration methods for differential equations, vol. 154 of Applied Mathematical Sciences, Springer-Verlag, New York, 2002.
[3] G. W. Bluman, A. F. Cheviakov, and S. C. Anco, Applications of symmetry methods to partial differential equations, vol. 168 of Applied Mathematical Sciences, Springer, New York, 2010.
[4] E. F. Doungmo Goufo, S. Kumar, and S. B. Mugisha, Similarities in a fifth-order evolution equation with and with no singular kernel, Chaos Solitons Fractals, 130 (2020), pp. 109467, 7.
[5] F. Gesztesy and K. Unterkofler, Isospectral deformations for Sturm-Liouville and Dirac-type operators and associated nonlinear evolution equations, Rep. Math. Phys., 31 (1992), pp. 113–137.
[6] B. Ghanbari, S. Kumar, and R. Kumar, A study of behaviour for immune and tumor cells in immuno[1]genetic tumour model with non-singular fractional derivative, Chaos Solitons Fractals, 133 (2020), pp. 109619, 11.
[7] N. Habibi, E. Lashkarian, E. Dastranj, and S. R. Hejazi, Lie symmetry analysis, conservation laws and numerical approximations of time-fractional Fokker-Planck equations for special stochastic process in foreign exchange markets, Phys. A, 513 (2019), pp. 750–766.
[8] S. R. Hejazi and E. Lashkarian, Lie group analysis and conservation laws for the time-fractional third order KdV-type equation with a small perturbation parameter, J. Geom. Phys., 157 (2020), pp. 103830, 10.
[9] S. R. Hejazi, E. Saberi, and F. Mohammadizadeh, Anisotropic non-linear time-fractional diffusion equation with a source term: classification via Lie point symmetries, analytic solutions and numerical simulation, Appl. Math. Comput., 391 (2021), pp. Paper No. 125652, 21.
[10] P. E. Hydon, Symmetry methods for differential equations, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2000. A beginner’s guide.
[11] N. Ibragimov, Nonlinear self-adjointness in constructing conservation laws, in Archives of ALGA, vol. 7/8, 2010, pp. 1–99.
[12] A. H. Kara and F. M. Mahomed, Noether-type symmetries and conservation laws via partial Lagrangians, Nonlinear Dynam., 45 (2006), pp. 367–383.
[13] M. Kruskal, Nonlinear wave equations, in Dynamical systems, theory and applications (Rencontres, Battelle Res. Inst., Seattle, Wash., 1974), 1975, pp. 310–354. Lecture Notes in Phys., Vol. 38.
[14] S. Kumar, A new analytical modelling for fractional telegraph equation via Laplace transform, Appl. Math. Model., 38 (2014), pp. 3154–3163.
[15] S. Kumar, A. Ahmadian, R. Kumar, D. Kumar, J. Singh, D. Baleanu, and M. Salimi, An efficient numerical method for fractional sir epidemic model of infectious disease by using bernstein wavelets, Mathematics, 8 (2020), p. 558.
[16] S. Kumar, S. Ghosh, R. Kumar, and M. Jleli, A fractional model for population dynamics of two interacting species by using spectral and Hermite wavelets methods, Numer. Methods Partial Differential Equations, 37 (2021), pp. 1652–1672.
[17] S. Kumar, S. Ghosh, B. Samet, and E. F. D. Goufo, An analysis for heat equations arises in diffusion process using new Yang-Abdel-Aty-Cattani fractional operator, Math. Methods Appl. Sci., 43 (2020), pp. 6062– 6080.
[18] S. Kumar, R. Kumar, R. P. Agarwal, and B. Samet, A study of fractional Lotka-Volterra popula[1]tion model using Haar wavelet and Adams-Bashforth-Moulton methods, Math. Methods Appl. Sci., 43 (2020), pp. 5564–5578.
[19] S. Kumar, R. Kumar, C. Cattani, and B. Samet, Chaotic behaviour of fractional predator-prey dynamical system, Chaos Solitons Fractals, 135 (2020), pp. 109811, 12.
[20] S. Kumar, R. Kumar, M. S. Osman, and B. Samet, A wavelet based numerical scheme for fractional order SEIR epidemic of measles by using Genocchi polynomials, Numer. Methods Partial Differential Equations, 37 (2021), pp. 1250–1268.
[21] E. Lashkarian and S. R. Hejazi, Polynomial and non-polynomial solutions set for wave equation using Lie point symmetries, Comput. Methods Differ. Equ., 4 (2016), pp. 298–308.
[22] E. Lashkarian, E. Saberi, and S. Reza Hejazi, Symmetry reductions and exact solutions for a class of nonlinear PDEs, Asian-Eur. J. Math., 9 (2016), pp. 1650061, 11.
[23] F. Mohammadizadeh, S. Rashidi, and S. R. Hejazi, Space-time fractional Klein-Gordon equation: symmetry analysis, conservation laws and numerical approximations, Math. Comput. Simulation, 188 (2021), pp. 476–497.
[24] R. Mokhtari, Exact solutions of the Harry-Dym equation, Commun. Theor. Phys. (Beijing), 55 (2011), pp. 204–208.
[25] A. Naderifard, S. R. Hejazi, and E. Dastranj, Symmetry properties, conservation laws and exact solutions of time-fractional irrigation equation, Waves Random Complex Media, 29 (2019), pp. 178–194.
[26] M. Nadjafikhah and P. Kabi-Nejad, Approximate symmetries of the harry dym equation, International Scholarly Research Notices, 2013 (2013).
[27] P. J. Olver, Applications of Lie groups to differential equations, vol. 107 of Graduate Texts in Mathematics, Springer-Verlag, New York, second ed., 1993.
[28]__ , Equivalence, invariants, and symmetry, Cambridge University Press, Cambridge, 1995.
[29] S. Rashidi and S. Reza Hejazi, Symmetry properties, similarity reduction and exact solutions of fractional Boussinesq equation, Int. J. Geom. Methods Mod. Phys., 14 (2017), pp. 1750083, 15.
[30] S. Reza Hejazi and S. Rashidi, Symmetries, conservation laws and exact solutions of the time-fractional diffusivity equation via Riemann-Liouville and Caputo derivatives, Waves Random Complex Media, 31 (2021), pp. 690–711.
[31] E. Saberi and S. Reza Hejazi, A comparison of conservation laws of the Boussinesq system, Kragujevac J. Math., 43 (2019), pp. 173–200.
[32] E. Saberi, S. Reza Hejazi, and A. Motamednezhad, Lie symmetry analysis, conservation laws and similarity reductions of Newell-Whitehead-Segel equation of fractional order, J. Geom. Phys., 135 (2019), pp. 116– 128.
[33] P. Veeresha, D. Prakasha, and S. Kumar, A fractional model for propagation of classical optical solitons by using nonsingular derivative, Mathematical Methods in the Applied Sciences, (2020).
[34] G. Wang, A. Kara, E. Buhe, and K. Fakhar, Group analysis and conservation laws of a coupled system of partial differential equations describing the carbon nanotubes conveying fluid, Romanian Journal in physics, 60 (2015), pp. 952–960.
[35] G. Wang, A. H. Kara, and K. Fakhar, Symmetry analysis and conservation laws for the class of timefractional nonlinear dispersive equation, Nonlinear Dynam., 82 (2015), pp. 281