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Mathematical Biosciences and Engineering

2022, Volume 19, Issue 7: 6962-6984. doi: 10.3934/mbe.2022328
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On reducing and finding solutions of nonlinear evolutionary equations via generalized symmetry of ordinary differential equations

  • AGH University of Science and Technology, Faculty of Applied Mathematics, aleja Adama Mickiewicza 30, 30-059 Kraków, Poland

  • Academic Editor: Masood Khalique
  • Received: 30 December 2021 Revised: 26 April 2022 Accepted: 03 May 2022 Published: 10 May 2022

  • We study symmetry reductions of nonlinear partial differential equations that can be used for describing diffusion processes in heterogeneous medium. We find ansatzes reducing these equations to systems of ordinary differential equations. The ansatzes are constructed using generalized symmetries of second-order ordinary differential equations. The method applied gives the possibility to find exact solutions which cannot be obtained by virtue of the classical Lie method. Such solutions are constructed for nonlinear diffusion equations that are invariant with respect to one-parameter and two-parameter Lie groups of point transformations. We prove a theorem relating the property of invariance of a found solution to the dimension of the Lie algebra admitted by the corresponding equation. We also show that the method is applicable to non-evolutionary partial differential equations and ordinary differential equations.

    Citation: Ivan Tsyfra, Wojciech Rzeszut. On reducing and finding solutions of nonlinear evolutionary equations via generalized symmetry of ordinary differential equations[J]. Mathematical Biosciences and Engineering, 2022, 19(7): 6962-6984. doi: 10.3934/mbe.2022328

    Related Papers:

  • We study symmetry reductions of nonlinear partial differential equations that can be used for describing diffusion processes in heterogeneous medium. We find ansatzes reducing these equations to systems of ordinary differential equations. The ansatzes are constructed using generalized symmetries of second-order ordinary differential equations. The method applied gives the possibility to find exact solutions which cannot be obtained by virtue of the classical Lie method. Such solutions are constructed for nonlinear diffusion equations that are invariant with respect to one-parameter and two-parameter Lie groups of point transformations. We prove a theorem relating the property of invariance of a found solution to the dimension of the Lie algebra admitted by the corresponding equation. We also show that the method is applicable to non-evolutionary partial differential equations and ordinary differential equations.


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    [1] P. J. Olver, Applications of Lie Groups to Differential Equations, 2$^{nd}$ edition, Springer-Verlag, New York, 1993. https://doi.org/10.1007/978-1-4684-0274-2
    [2] G. Bluman, J. D. Cole, The general similarity solution of the heat equation, Indiana Univ. Math. J., 18 (1969), 1025–1042. https://doi.org/10.1512/iumj.1969.18.18074 doi: 10.1512/iumj.1969.18.18074
    [3] P. J. Olver, P. Rosenau, The construction of special solutions to partial differential equations, Phys. Lett. A, 114 (1986), 107–112. https://doi.org/10.1016/0375-9601(86)90534-7 doi: 10.1016/0375-9601(86)90534-7
    [4] P. J. Olver, P. Rosenau, Group-invariant solutions of differential equations, SIAM J. Appl. Math., 47 (1987), 263–278. https://doi.org/10.1137/0147018 doi: 10.1137/0147018
    [5] W. I. Fushchych, I. M. Tsyfra, On a reduction and solutions of nonlinear wave equations with broken symmetry, J. Phys. A: Math. Gen., 20 (1987), L45–L48. https://doi.org/10.1088/0305-4470/20/2/001 doi: 10.1088/0305-4470/20/2/001
    [6] R. O. Popovych, Reduction operators of linear second-order parabolic equations, J. Phys. A: Math. Theor., 41 (2008), 185202. https://doi.org/10.1088/1751-8113/41/18/185202 doi: 10.1088/1751-8113/41/18/185202
    [7] M. Kunzinger, R. O. Popovych, Singular reduction operators in two dimensions, J. Phys. A: Math. Theor., 41 (2008), 505201. https://doi.org/10.1088/1751-8113/41/50/505201 doi: 10.1088/1751-8113/41/50/505201
    [8] V. M. Boyko, Reduction of differential equations to algebraic ones, Proceedings of the National Academy of Sciences of the Ukraine, 3 (2014), 7–12. https://doi.org/10.15407/dopovidi2014.03.007 doi: 10.15407/dopovidi2014.03.007
    [9] V. M. Boyko, M. Kunzinger, R. O. Popovych, Singular reduction modules of differential equations, J. Math. Phys., 57 (2016), 101503. https://doi.org/10.1063/1.4965227 doi: 10.1063/1.4965227
    [10] A. S. Fokas, Q. M. Liu, Nonlinear interaction of traveling waves of non-integrable equations, Phys. Rev. Lett., 72 (1994), 3293–3296. https://doi.org/10.1103/PhysRevLett.72.3293 doi: 10.1103/PhysRevLett.72.3293
    [11] R. Z. Zhdanov, Conditional Lie–Bäcklund symmetry and reduction of evolution equations, J. Phys. A: Math. Gen., 28 (1995), 3841–3850. https://doi.org/10.1088/0305-4470/28/13/027 doi: 10.1088/0305-4470/28/13/027
    [12] Q. Changzheng, Exact solutions to nonlinear diffusion equations obtained by a generalized conditional symmetry method, IMA J. Appl. Math., 62 (1999), 283–302. https://doi.org/10.1093/imamat/62.3.283 doi: 10.1093/imamat/62.3.283
    [13] M. Kunzinger, R. O. Popovych, Generalized conditional symmetries of evolution equations, J. Math. Anal. Appl., 379 (2011), 444–460. https://doi.org/10.1016/j.jmaa.2011.01.027 doi: 10.1016/j.jmaa.2011.01.027
    [14] P. J. Olver, Direct reduction and differential constraints, Proc. R. Soc. A, 444 (1994), 509–523. https://doi.org/10.1098/rspa.1994.0035 doi: 10.1098/rspa.1994.0035
    [15] S. R. Svirshchevskii, Lie–Bäcklund symmetries of linear ODEs and generalized separation of variables in nonlinear equations, Phys. Lett. A, 199 (1995), 344–348. https://doi.org/10.1016/0375-9601(95)00136-Q doi: 10.1016/0375-9601(95)00136-Q
    [16] N. Kamran, R. Milson, P. J. Olver, Invariant modules and the reduction of nonlinear partial differential equations to dynamical systems, Adv. Math., 156 (2000), 286–319. https://doi.org/10.1006/aima.2000.1953 doi: 10.1006/aima.2000.1953
    [17] I. M. Tsyfra, Symmetry reduction of nonlinear differential equations, Proc. Inst. Math. NAS Ukr., 50 (2004), 266–270. Available from: https://www.imath.kiev.ua/snmp2003/Proceedings/tsyfra.pdf.
    [18] I. M. Tsyfra, Symmetry reduction of partial differential equations with two independent variables, Trans. Inst. Math. NAS Ukr., 3(2) (2006), 308–314. Available from: https://trim.imath.kiev.ua/index.php/trim/article/view/475 (in Ukrainian).
    [19] I. M. Tsyfra, Symmetry reduction of evolution and wave type equations, AIP Conf. Proc., 1307 (2010), 175–180. https://doi.org/10.1063/1.3527414 doi: 10.1063/1.3527414
    [20] I. M. Tsyfra, T. Czyżycki, Nonpoint symmetry and reduction of nonlinear evolution and wave type equations, Abstr. Appl. Anal., 2015 (2015). https://doi.org/10.1155/2015/181275
    [21] I. M. Tsyfra, W. Rzeszut, Lie–Bäcklund symmetry reduction of nonlinear and non-evolution equations, Trans. Inst. Math. NAS Ukr., 16 (2019), 174–180. Available from: https://trim.imath.kiev.ua/index.php/trim/article/view/374
    [22] O. O. Vaneeva, R. O. Popovych, C. Sophocleous, Extended group analysis of variable coefficient reaction-diffusion equations with exponential nonlinearities, J. Math. Anal. Appl., 396 (2012), 225–242. https://doi.org/10.1016/j.jmaa.2012.05.084 doi: 10.1016/j.jmaa.2012.05.084
    [23] O. O. Vaneeva, R. O. Popovych, C. Sophocleous, Reduction operators of variable coefficient semilinear diffusion equations with an exponential source, Proceedings of 5th Workshop "Group Analysis of Differential Equations and Integrable Systems", preprint, arXiv: 1010.2046.
    [24] N. M. Ivanova, R. O. Popovych, C. Sophocleous, Group analysis of variable coefficient diffusion-convection equations. I. Enhanced group classification, Lobachevskii J. Math., 31 (2010), 100–122. https://doi.org/10.1134/S1995080210020034 doi: 10.1134/S1995080210020034
    [25] Z. Zhao, L. He, Lie symmetry, nonlocal symmetry analysis, and interaction of solutions of a (2+1)-dimensional KdV–mKdV equation, Theor. Math. Phys., 206 (2021), 142–162. https://doi.org/10.1134/S0040577921020033 doi: 10.1134/S0040577921020033
    [26] Z. Zhao, Conservation laws and nonlocally related systems of the Hunter–Saxton equation for liquid crystal, Anal. Math. Phys., 9 (2019), 2311–2327. https://doi.org/10.1007/s13324-019-00337-3 doi: 10.1007/s13324-019-00337-3
    [27] J. D. Murray, Mathematical Biology: I. An Introduction, 3$^{rd}$ edition, Springer, New York, 2002.
    [28] G. W. Bluman, A. F. Cheviakov, S. C. Anco, Applications of Symmetry Methods to Partial Differential Equations, Springer, New York, 2010.
    [29] I. Tsyfra, On classical symmetries of ordinary differential equations related to stationary integrable partial differential equations, Opusc. Math., 41 (2021), 685–699. https://doi.org/10.7494/OpMath.2021.41.5.685 doi: 10.7494/OpMath.2021.41.5.685
    [30] A. V. Mikhailov, V. S. Novikov, J. P. Wang, Partially integrable nonlinear equations with one higher symmetry, J. Phys. A: Math. Gen., 38 (2005), L337–L341. https://doi.org/10.1088/0305-4470/38/20/L02 doi: 10.1088/0305-4470/38/20/L02
    [31] P. Grindrod, The Theory and Applications of Reaction-Diffusion Equations, 2$^{nd}$ edition, Oxford University Press, 1996.
    [32] N. Shigesada, K. Kawasaki, E. Teramoto, Spatial segregation of interacting species, J. Theor. Biol., 79 (1979), 83–99. https://doi.org/10.1016/0022-5193(79)90258-3 doi: 10.1016/0022-5193(79)90258-3
    [33] N. F. Britton, Essential Mathematical Biology, Springer, Berlin, 2003. https://doi.org/10.1007/978-1-4471-0049-2
    [34] D. J. Arrigo, J. M. Hill, Nonclassical symmetries for nonlinear diffusion and absorption, Stud. Appl. Math., 94 (1995), 21–39. https://doi.org/10.1002/sapm199594121 doi: 10.1002/sapm199594121
    [35] V. P. Gavrilov, N. V. Klepikova, H. C. Rodean, Trial of nonlinear diffusion as a model of turbulent diffusion, Atmos. Environ., 29 (1995), 2317–2322. https://doi.org/10.1016/1352-2310(95)00148-R doi: 10.1016/1352-2310(95)00148-R
    [36] R. E. Grundy, Large time solution of an inhomogeneous nonlinear diffusion equation, Proc. R. Soc. A, 386 (1983), 347–372. https://doi.org/10.1098/rspa.1983.0040 doi: 10.1098/rspa.1983.0040
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