Non-classical symmetry and new exact solutions of the Kudryashov-Sinelshchikov and modified KdV-ZK equations

Document Type : Original Article

Authors

Department of Mathematics, Payame Noor University, PO BOX 19395-3697, Tehran, Iran

Abstract

‎In this paper‎, ‎by applying the non-classical symmetry method‎, ‎non-classical symmetries of the Kodryashov-Sinleschikov (K-S) and‎ ‎modified Korteweg-de Vries-Zaharov-Kuznetsov (mKdV-ZK) equations are‎ ‎obtained‎. ‎Apart from classical symmetries‎, ‎this theory can be‎ ‎effective in finding a few other solutions for a system of PDEs and‎ ‎ODEs‎. ‎Also‎, ‎non-classical symmetries of a system of PDEs can be‎ ‎applied to reduce the number of independent variables‎. ‎By adding the‎
‎invariance surface condition to the assumed equations and applying‎ ‎the classical symmetry method for them‎, ‎non-classical symmetries are‎ ‎calculated‎. ‎Finally‎, ‎some of the group invariant solutions and the‎ ‎similarity reduced equations associated to non-classical symmetry‎ ‎are obtained‎.

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References

[1] A. Akgul, M. Inc, E. Karatas, and D. Baleanu ¨ , Numerical solutions of fractional differential equations of Lane-Emden type by an accurate technique, Adv. Difference Equ., (2015), pp. 2015:220, 12.
[2] R. Bakhshandeh-Chamazkoti, Abelian Lie symmetry algebras of two-dimensional quasilinear evolution equations, Math. Methods Appl. Sci., 46 (2023), pp. 867–878.
[3] R. Bakhshandeh Chamazkoti and M. Alipour, Lie symmetry classification and numerical analysis of KdV equation with power-law nonlinearity, Math. Rep. (Bucur.), 22(72) (2020), pp. 163–176.
[4] S. Bilal, I. Ali Shah, A. Akgul, M. Tas¸tan Tekin, T. Botmart, E. Sayed Yousef, and I. Yahia ¨ , A comprehensive mathematical structuring of magnetically effected sutterby fluid flow immersed in dually stratified medium under boundary layer approximations over a linearly stretched surface, Alex. Eng. J., 61 (2022), pp. 11889–11898.
[5] N. Bˆıla and J. Niesen ˇ , On a new procedure for finding nonclassical symmetries, J. Symb. Comput., 38 (2004), pp. 1523–1533.
[6] G. W. Bluman and J. D. Cole, The general similarity solution of the heat equation, J. math. mech., 18 (1969), pp. 1025–1042.
[7] G. Cai, Y. Wang, and F. Zhang, Nonclassical symmetries and group invariant solutions of burgers-fisher equations, World J. Model. Simul., 3 (2007), pp. 305–309.
[8] P. A. Clarkson and E. L. Mansfield, Algorithms for the nonclassical method of symmetry reductions, SIAM J. Appl. Math., 54 (1994), pp. 1693–1719.
[9] F. Demontis, Exact solutions of the modified Korteweg-de Vries equation, Theoret. and Math. Phys., 168 (2011), pp. 886–897. Russian version appears in Teoret. Mat. Fiz. 168 (2011), no. 1, 35–48.
[10] J. Fang, M. Nadeem, M. Habib, and A. Akgul¨ , Numerical investigation of nonlinear shock wave equations with fractional order in propagating disturbance, Symmetry, 14 (2022), p. 1179.
[11] S. R. Hejazi and A. Naderifard, Dym equation: Group analysis and conservation laws, AUT J. Math. Com., 3 (2022), pp. 17–26.
[12] M. Inc and A. Akgul¨ , Classifications of soliton solutions of the generalized benjamin-bona-mahony equation with power-law nonlinearity, J. Adv. Phys., 7 (2018), pp. 130–134.
[13] M. S. Iqbal, M. W. Yasin, N. Ahmed, A. Akgul, M. Rafiq, and A. Raza ¨ , Numerical simulations of nonlinear stochastic Newell-Whitehead-Segel equation and its measurable properties, J. Comput. Appl. Math., 418 (2023), pp. Paper No. 114618, 16.
[14] M. Jafari and R. Darvazebanzade, Approximate symmetry group analysis and similarity reductions of the perturbed mKdV-KS equation, Comput. Methods Differ. Equ., 11 (2023), pp. 175–182.
[15] M. Jafari, A. Zaeim, and A. Tanhaeivash, Symmetry group analysis and conservation laws of the potential modified KdV equation using the scaling method, Int. J. Geom. Methods Mod. Phys., 19 (2022), pp. Paper No. 2250098, 14.
[16] N. A. Kudryashov and D. I. Sinelshchikov, Nonlinear evolution equation for describing waves in a viscoelastic tube, Commun. Nonlinear Sci. Numer. Simul., 16 (2011), pp. 2390–2396.
[17] S. Lie, Integration of a class of linear partial differential equations by means of definite integrals, in Lie group analysis: Classical heritage. Translated by Nail H. Ibragimov, Elena D. Ishmakova, Roza M. Yakushina, Karlskrona: ALGA, Blekinge Institute of Technology, 2004, pp. 65–102.
[18] M. Nadjafikhah, Lie group analysis for short pulse equation, AUT J. Math. Com., 1 (2020), pp. 223–227.
[19] M. Nadjafikhah and M. Jafari, Computation of partially invariant solutions for the einstein walker man[1]ifolds’ identifying equations, Commun. Nonlinear Sci. Numer. Simul., 18 (2013), pp. 3317–3324.
[20] , Some general new Einstein Walker manifolds, Adv. Math. Phys., (2013), pp. Art. ID 591852, 8.
[21] M. Nadjafikhah and V. Shirvani-Sh, Lie symmetry analysis of Kudryashov-Sinelshchikov equation, Math. Probl. Eng., (2011), pp. Art. ID 457697, 9.
[22] P. J. Olver, Applications of Lie groups to differential equations, vol. 107 of Graduate Texts in Mathematics, Springer-Verlag, New York, 1986.
[23] M. Partohaghighi, A. Akgul, L. Guran, and M.-F. Bota ¨ , Novel mathematical modelling of plateletpoor plasma arising in a blood coagulation system with the fractional caputo–fabrizio derivative, Symmetry, 14 (2022), p. 1128.
[24] Z. A. Qureshi, S. Bilal, U. Khan, A. Akgul, M. Sultana, T. Botmart, H. Y. Zahran, and I. S. ¨ Yahia, Mathematical analysis about influence of lorentz force and interfacial nano layers on nanofluids flow through orthogonal porous surfaces with injection of swcnts, Alex. Eng. J., 61 (2022), pp. 12925–12941.
[25] G. Sadiq, A. Ali, S. Ahmad, K. Nonlaopon, and A. Akgul¨ , Bright soliton behaviours of fractal fractional nonlinear good boussinesq equation with nonsingular kernels, Symmetry, 14 (2022), p. 2113.
[26] N. A. Shah, I. Ahmed, K. K. Asogwa, A. A. Zafar, W. Weera, and A. Akgul¨ , Numerical study of a nonlinear fractional chaotic Chua’s circuit, AIMS Math., 8 (2023), pp. 1636–1655.
[27] A. Shahzad, M. Imran, M. Tahir, S. A. Khan, A. Akgul, S. Abdullaev, C. Park, H. Y. Zahran, ¨ and I. S. Yahia, Brownian motion and thermophoretic diffusion impact on darcy-forchheimer flow of biocon[1]vective micropolar nanofluid between double disks with cattaneo-christov heat flux, Alex. Eng. J., 62 (2023), pp. 1–15.
AUT Journal of Mathematics and Computing
Volume 4, Issue 2
March 2023
Pages 195-203

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