INTEGRATION OF A THIRD-ORDER ODE VIA ANALYTICAL AND GEOMETRICAL METHODS
DOI:
https://doi.org/10.52754/16948645_2023_1_262Ачкыч сөздөр:
Lie symmetries, first-order symmetries, first integrals, involutive distributions.Аннотация
Analytical and geometrical methods are applied to integrate an ordinary differential equation of third order. The main objective is to compare both approaches and show the possibilities that each one of them offers in the integration process of the considered equation, specially when not only Lie point symmetries but also generalized symmetries are involved. The analytical method of order reduction by using a generalized symmetry provides the general solution of the equation but in terms of a primitive that cannot be explicitly evaluated. On the other hand, the application of geometrical tools previously reported in the recent literature leads to two functionally independent first integrals of the equation without any kind of integration. In order to complete the integration of the given third-order equation, a third independent first integral arises by quadrature as the primitive of a closed differential one-form. From these first integrals, the expression of the general solution of the equation can be expressed in parametric form and in terms of elementary functions.
Библиографиялык шилтемелер
P. J. Olver. Applications of Lie groups to differential equations. Graduate Texts in Mathematics. Springer US, 2nd edition, 1986. DOI: https://doi.org/10.1007/978-1-4684-0274-2
H. Stephani and M. MacCallum. Differential equations: their solution using symmetries. Cambridge University Press, 1989. DOI: https://doi.org/10.1017/CBO9780511599941
G. W. Bluman and S. C. Anco. Symmetry and integration methods for differential equations. Applied Mathematical Science. Springer New York, 2nd edition, 2002.
P. E. Hydon. Symmetry Methods for Differential Equations: A Beginner’s Guide. Cambridge Texts in Applied Mathematics. Cambridge University Press, 2000. DOI: https://doi.org/10.1017/CBO9780511623967
L.V. Ovsiannikov, editor. Group Analysis of Differential Equations. Academic Press, 1982. DOI: https://doi.org/10.1016/B978-0-12-531680-4.50012-5
P. E. Hydon. Self-invariant first-order symmetries. Journal of Nonlinear Mathematical Physics, 11(2):233–242, 2004. DOI: https://doi.org/10.2991/jnmp.2004.11.2.8
E. Pucci and G. Saccomandi. First-order symmetries and solutions by reduction of partial differential equations. Journal of Physics A: Mathematical and General, 27(1):177–184, 1994. DOI: https://doi.org/10.1088/0305-4470/27/1/011
P. Basarab-Horwath. Integrability by quadratures for systems of involutive vector fields. Ukrainian Math. Zh., 43:1330–1337, 1991. DOI: https://doi.org/10.1007/BF01061807
G. Prince and J. Sherring. Geometric aspects of reduction of order. Trans. Amer. Math. Soc., 334 (1):433–453, 1992. DOI: https://doi.org/10.1090/S0002-9947-1992-1149125-6
M. A. Barco and G. E. Prince. Solvable symmetry structures in differential form applications. Acta Applicandae Mathematica, 66:89–121, 2001. DOI: https://doi.org/10.1023/A:1010609817442
T. Hartl and C. Athorne. Solvable structures and hidden symmetries. Journal of Physics A: Mathematical and General, 27(10):34–63, 1994. DOI: https://doi.org/10.1088/0305-4470/27/10/022
D. C. Ferraioli and P. Morando. Local and nonlocal solvable structures in the reduction of odes. Journal of Physics A: Mathematical and Theoretical, 42(3):035210, 2008. DOI: https://doi.org/10.1088/1751-8113/42/3/035210
P. Morando. Reduction by μ–symmetries and σ–symmetries: a Frobenius approach. Journal of Nonlinear Mathematical Physics, 22(1):47–59, 2015. DOI: https://doi.org/10.1080/14029251.2015.996439
A. Kushner, V. Lychagin and V. Rubtsov. First-order geometry and nonlinear differential equations. Encyclopedia of Mathematics and its Applications. Cambridge University Press, 2007.
F. Warner. Foundations of differentiable manifolds and Lie groups. Graduate Texts in Mathematics. Springer, 1971.
S. Morita. Geometry of differential forms. Translations of mathematical monographs, Iwanami series in modern mathematics 201. American Mathematical Society, 2001. DOI: https://doi.org/10.1090/mmono/201
A. W. Knapp. Lie groups beyond an introduction. Progress in Mathematics. Birkhäuser Boston, 2002.
J. M. Lee. Introduction to smooth manifolds. Graduate Texts in Mathematics 218. Springer New York, 2003. DOI: https://doi.org/10.1007/978-0-387-21752-9
Жүктөөлөр
Жарыяланды
Кандай шилтеме берүү керек
Саны (чыгарылыш)
Бөлүм
Лицензия
Copyright (c) 2023 Ош мамлекеттик университетинин Жарчысы. Математика. Физика. Техника
