Optimal Algebra and Invariant Solutions for Chazy’s Equation
Main Article Content
Keywords
Chazy’s equation, Lie group symmetries, optimal algebra, Optimal system, invariant solutions
Abstract
We characterized the invariant solutions for Chazy’s equation using the generators of the optimal algebra, which was obtained using Lie group symmetries for the equation.
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References
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[2] E. Noether, “Invariante variationsprobleme. nachrichten der königlichen gessellschaft der wissenschaften. mathematisch-physikalishe klasse 2, 235–257,” Transport Theory and Statistical Physics, pp. 183–207, 1918.
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[7] E. Alimirzaluo, M. Nadjafikhah, and J. Manafian, “Some new exact solutions of $(3+1)$-dimensional burgers system via lie symmetry analysis,” 2021.
[8] A. Paliathanasis, “Lie symmetry analysis and one-dimensional optimal system for the generalized 2+1 kadomtsev-petviashvili equation,” Physica Scripta, vol. 95, no. 5, p. 055223, 2020.
[9] H. Lu and Y. Zhang, “Lie symmetry analysis, exact solutions, conservation laws and bäcklund transformations of the gibbons-tsarev equation,” Symmetry, vol. 12, no. 8, p. 1378, 2020.
[10] S.-F. Tian, “Lie symmetry analysis, conservation laws and solitary wave solutions to a fourth-order nonlinear generalized boussinesq water wave equation,” Applied Mathematics Letters, vol. 100, p. 106056, 2020.
[11] M. R. Ali and R. Sadat, “Lie symmetry analysis, new group invariant for the (3+1)-dimensional and variable coefficients for liquids with gas bubbles models,” Chinese Journal of Physics, 2021.
[12] L. Rosenhead, Laminar Boundary Layers. Clarendon Press, (1963).
[13] P. J. Olver, “Applications of Lie groups to differential equations,” Gradua-te texts in Mathematics, vol. 107 (1993), 1993.
[14] H. Schlichting, “Laminare strahlausbreitung,” Z. Angew. Math. Mech., vol. 13, pp. 260–263, 1933. https://doi.org/10.1002/zamm.19330130403
[15] W. G. Bickley, “The plane jet,” Phil. Mag., vol. 23, pp. 727–721, 1937.
[16] H. B. Squire, 50 fahre Grenzschichtforschung, vol. (1955).
[17] M. B. Glauert, “The wall jet,” J. Fluid Mech., vol. 1, p. 625–643, 1956.
[18] N. Riley, “Asymptotic expansions in radial jets,” J.Math. and Phys., vol. 41, pp. 132–146, 1962. https://doi.org/10.1002/sapm1962411132
[19] L. J. Crane, “Flow past a stretching plate,” Z. Angew. Math. Mech., vol. 21, p. 645–647, 1970. https://doi.org/10.1007/BF01587695
[20] N. H. Ibragimov and M. C. Nucci, “Integration of third order ordinary differential equations by Lie’s method: Equations admitting three-dimensional Lie algebras,” Lie Groups and their applications, vol. 1, pp. 49–64, 1994.
[21] F. M. Mahomed, “Symmetry group classification of ordinary differential equations: Survey of some results,” Mathematical Methods in the Applied Sciences, vol. 30, p. 1995–2012, 2007. https://doi.org/10.1002/mma.934
[22] J. Chazy, “Sur les equations differentielles dont l’integrale generale est uniforme et admet des singelarities essentielles mobiles,” C. R. Acad. Sc. Paris, vol. 149, p. 563–565, 1909.
[23] ——, “Sur les equations differentielles dont l’integrale generale possede une coupure essentielle mobile,” C. R. Acad. Sc. Paris, vol. 150, p. 456–458, 1910.
[24] ——, “Sur les equations differentielles du troisieme ordre et d’ordre superieur dont l’integrale generale a ses points critiques fixes,” Acta Math., vol. 34, p. 317–385, 1911.
[25] P. A. Clarkson and P. J. Olver, “Symmetry and the Chazy equation,” J. Diff. Eq., vol. 124, p. 225–246, 1996. https://www-users.math.umn.edu/ ~olver/s_/chazy.pdf
[2] E. Noether, “Invariante variationsprobleme. nachrichten der königlichen gessellschaft der wissenschaften. mathematisch-physikalishe klasse 2, 235–257,” Transport Theory and Statistical Physics, pp. 183–207, 1918.
[3] N. H. Ibragimov, CRC handbook of Lie group analysis of differential equations. CRC press, 1995, vol. 3.
[4] A. Paliathanasis and P. Leach, “Symmetries and singularities of the szekeres system,” Physics Letters A, vol. 381, no. 15, p. 1277–1280, Apr 2017.
[5] A. Ghose-Choudhury, P. Guha, A. Paliathanasis, and P. G. L. Leach, “Noetherian symmetries of noncentral forces with drag term,” International Journal of Geometric Methods in Modern Physics, vol. 14, no. 02, p. 1750018, Jan 2017.
[6] W. Hu, Z. Wang, Y. Zhao, and Z. Deng, “Symmetry breaking of infinitedimensional dynamic system,” Applied Mathematics Letters, vol. 103, p. 106207, 2020.
[7] E. Alimirzaluo, M. Nadjafikhah, and J. Manafian, “Some new exact solutions of $(3+1)$-dimensional burgers system via lie symmetry analysis,” 2021.
[8] A. Paliathanasis, “Lie symmetry analysis and one-dimensional optimal system for the generalized 2+1 kadomtsev-petviashvili equation,” Physica Scripta, vol. 95, no. 5, p. 055223, 2020.
[9] H. Lu and Y. Zhang, “Lie symmetry analysis, exact solutions, conservation laws and bäcklund transformations of the gibbons-tsarev equation,” Symmetry, vol. 12, no. 8, p. 1378, 2020.
[10] S.-F. Tian, “Lie symmetry analysis, conservation laws and solitary wave solutions to a fourth-order nonlinear generalized boussinesq water wave equation,” Applied Mathematics Letters, vol. 100, p. 106056, 2020.
[11] M. R. Ali and R. Sadat, “Lie symmetry analysis, new group invariant for the (3+1)-dimensional and variable coefficients for liquids with gas bubbles models,” Chinese Journal of Physics, 2021.
[12] L. Rosenhead, Laminar Boundary Layers. Clarendon Press, (1963).
[13] P. J. Olver, “Applications of Lie groups to differential equations,” Gradua-te texts in Mathematics, vol. 107 (1993), 1993.
[14] H. Schlichting, “Laminare strahlausbreitung,” Z. Angew. Math. Mech., vol. 13, pp. 260–263, 1933. https://doi.org/10.1002/zamm.19330130403
[15] W. G. Bickley, “The plane jet,” Phil. Mag., vol. 23, pp. 727–721, 1937.
[16] H. B. Squire, 50 fahre Grenzschichtforschung, vol. (1955).
[17] M. B. Glauert, “The wall jet,” J. Fluid Mech., vol. 1, p. 625–643, 1956.
[18] N. Riley, “Asymptotic expansions in radial jets,” J.Math. and Phys., vol. 41, pp. 132–146, 1962. https://doi.org/10.1002/sapm1962411132
[19] L. J. Crane, “Flow past a stretching plate,” Z. Angew. Math. Mech., vol. 21, p. 645–647, 1970. https://doi.org/10.1007/BF01587695
[20] N. H. Ibragimov and M. C. Nucci, “Integration of third order ordinary differential equations by Lie’s method: Equations admitting three-dimensional Lie algebras,” Lie Groups and their applications, vol. 1, pp. 49–64, 1994.
[21] F. M. Mahomed, “Symmetry group classification of ordinary differential equations: Survey of some results,” Mathematical Methods in the Applied Sciences, vol. 30, p. 1995–2012, 2007. https://doi.org/10.1002/mma.934
[22] J. Chazy, “Sur les equations differentielles dont l’integrale generale est uniforme et admet des singelarities essentielles mobiles,” C. R. Acad. Sc. Paris, vol. 149, p. 563–565, 1909.
[23] ——, “Sur les equations differentielles dont l’integrale generale possede une coupure essentielle mobile,” C. R. Acad. Sc. Paris, vol. 150, p. 456–458, 1910.
[24] ——, “Sur les equations differentielles du troisieme ordre et d’ordre superieur dont l’integrale generale a ses points critiques fixes,” Acta Math., vol. 34, p. 317–385, 1911.
[25] P. A. Clarkson and P. J. Olver, “Symmetry and the Chazy equation,” J. Diff. Eq., vol. 124, p. 225–246, 1996. https://www-users.math.umn.edu/ ~olver/s_/chazy.pdf
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Published
May 12, 2021
Article Details
How to Cite
Loaiza, G., Acevedo-Agudelo, Y., & Londoño-Duque, O. (2021). Optimal Algebra and Invariant Solutions for Chazy’s Equation. Ingeniería Y Ciencia, 17(33), 7–21. https://doi.org/10.17230/ingciencia.17.33.1
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