Skip to main content
SpringerLink
Log in

Systems of Vector Fields for the Integration of Ordinary Differential Equations

  • Conference paper
  • First Online:
Recent Advances in Differential Equations and Control Theory

Part of the book series: SEMA SIMAI Springer Series ((ICIAM2019SSSS,volume 9))

  • 258 Accesses

Abstract

In this work, we investigate different classes of vector fields that can be used to find exact solutions of ordinary differential equations. The presented approaches are based on the integrability by quadrature via solvable structures associated with integrable distributions. The methods are specially relevant for equations that lack Lie point symmetries or whose symmetry algebra is nonsolvable, because in such cases the classical Lie procedure cannot be applied to solve the equations by quadrature.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abraham-Shrauner, B.: Hidden symmetries and nonlocal group generators for ordinary differential equations. IMA J. Appl. Math. 56(3), 235–252 (1996)

    Article  MathSciNet  Google Scholar 

  2. Barco, M.A., Prince, G.E.: Solvable symmetry structures in differential form applications. Acta Appl. Math. 66(1), 89–121 (2001)

    Article  MathSciNet  Google Scholar 

  3. Basarab-Horwath, P.: Integrability by quadratures for systems of involutive vector fields. Ukr. Math. J. 43(10), 1236–1242 (1991)

    Article  MathSciNet  Google Scholar 

  4. Bluman, G.W., Anco, S.C.: Symmetry and Integration Methods for Differential Equations. Springer, New York (2002)

    MATH  Google Scholar 

  5. Cicogna, G., Gaeta, G., Morando, P.: On the relation between standard and μ-symmetries for PDEs. J. Phys. A Math. Gen. 37, 9467–9486 (2004)

    Article  MathSciNet  Google Scholar 

  6. Cicogna, G., Gaeta, G., Walcher, S.: A generalization of λ-symmetry reduction for systems of ODEs: σ-symmetries. J. Phys. A Math. Theor. 45(35), 355205–355234 (2012)

    Article  MathSciNet  Google Scholar 

  7. Clarkson, P.A., Olver, P.J.: Symmetry and the Chazy equation. J. Diff. Eqns. 214, 225–246 (1996)

    Article  MathSciNet  Google Scholar 

  8. Gaeta, G.: Twisted symmetries of differential equations. J. Nonlinear Math. Phys. 16, 107–136 (2009)

    Article  MathSciNet  Google Scholar 

  9. González-Gascón, F., González-López, A.: Newtonian systems of differential equations, integrable via quadratures, with trivial group of point symmetries. Phys. Lett. A 129, 153–156 (1988)

    Article  MathSciNet  Google Scholar 

  10. González-López, A.: Symmetry and integrability by quadratures of ordinary differential equations. Phys. Lett. A. 133(4–5), 190–194 (1988)

    Article  MathSciNet  Google Scholar 

  11. Govinder, K.S., Leach, P.G.L.: On the determination of non-local symmetries. J. Phys. A Math. Gen. 28(18), 5349–5359 (1995)

    Article  MathSciNet  Google Scholar 

  12. Harko, T., Lobo, F.S.N., Mak M.K.: A class of exact solutions of the Liénard-type ordinary nonlinear differential equation. J. Eng. Math. 89, 193–205 (2014)

    Article  Google Scholar 

  13. Hartl, T., Athorne, C.: Solvable structures and hidden symmetries. J. Phys. A Math. Gen. 27, 3463–3474 (1994)

    Article  MathSciNet  Google Scholar 

  14. Ibrahimov, N.H.: Elementary Lie Group Analysis and Ordinary Differential Equations. Wiley, Chichester (1999)

    Google Scholar 

  15. Kamran, N., Olver, P.J., González-López, A.: Lie algebras of the vector fields in the real plane. Proc. Lond. Math. Soc. 64, 339–68 (1992)

    MathSciNet  MATH  Google Scholar 

  16. Kudryashov, N.A., Sinelshchikov, D.I.: On the criteria for integrability of the Liénard equation. Appl. Math. Lett. 57, 114–120 (2016)

    Article  MathSciNet  Google Scholar 

  17. Kudryashov, N.A., Sinelshchikov, D.I.: On the integrability conditions for a family of Liénard-type equations. Regul. Chaotic Dyn. 21, 548–555 (2016)

    Article  MathSciNet  Google Scholar 

  18. Kudryashov, N.A., Sinelshchikov, D.I.: On connections of the Liénard equation with some equations of Painlevé–Gambier type. J. Math. Appl. 449, 1570–1580 (2017)

    MathSciNet  MATH  Google Scholar 

  19. Liénard, A.: Étudie des oscillations entreténues. Rev. Génerale LÉlectricité. 23, 901–912 (1928)

    Google Scholar 

  20. Liénard, A.: Étudie des oscillations entreténues. Rev. Génerale LÉlectricité. 23, 946–954 (1928)

    Google Scholar 

  21. Mgaga, T.C., Govinder, K.S.: On the linearization of some second-order ODEs via contact transformations. J. Phys. A Math. Theor. 44, 015203–015210 (2011)

    Article  MathSciNet  Google Scholar 

  22. Morando, P., Muriel, C., Ruiz, A.: Generalized solvable structures and first integrals for ODEs admitting an \(\mathfrak {sl}(2,\mathbb {R})\) symmetry algebra. J. Nonlinear Math. Phys. 26(2), 188–201 (2019)

    Google Scholar 

  23. Muriel, C., Romero, J.L.: New methods of reduction for ordinary differential equations IMA J. Appl. Math. 66(2), 111–25 (2001)

    MathSciNet  MATH  Google Scholar 

  24. Muriel, C., Romero, J.L.: First integrals, integrating factors and λ-symmetries of second-order differential equations. J. Phys. A Math. Theor. 42, 365207–365224 (2009)

    Article  MathSciNet  Google Scholar 

  25. Muriel, C., Romero, J.L., Ruiz, A.: Integration methods for equations without enough Lie point symmetries. AIP Conf. Proc. 2153, 020013–020021 (2018)

    Article  Google Scholar 

  26. Olver, P.: Applications of Lie Groups to Differential Equations, 2nd edn. Springer, New York (1993)

    Book  Google Scholar 

  27. Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W.: NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)

    MATH  Google Scholar 

  28. Ruiz, A., Muriel, C.: Solvable structures associated to the nonsolvable symmetry algebra \(\mathfrak {sl}(2,\mathbb {R})\). SIGMA 077, 18 (2016)

    Google Scholar 

  29. Ruiz, A., Muriel, C.: First integrals and parametric solutions of third-order ODEs with Lie symmetry algebra isomorphic to \(\mathfrak {sl}(2,\mathbb {R})\). J. Phys. A Math. Theor. 50, 205201–205222 (2017)

    Google Scholar 

  30. Ruiz, A., Muriel, C.: On the integrability of Liénard I-type equations via λ-symmetries and solvable structures. Appl. Math. Comput. 339, 888–898 (2018)

    MathSciNet  MATH  Google Scholar 

  31. Sardanashvily, G.: Advanced Differential Geometry for Theoreticians. Lap Lambert Academic, Riga (2013)

    Google Scholar 

  32. Stephani, H.: Differential Equations: Their Solution Using Symmetries. Cambridge University Press, Cambridge (1989)

    MATH  Google Scholar 

  33. Zartsev, V.F., Polyanin, A.D.: Handbook of Exact Solutions for Ordinary Differential Equations. Chapman and Hall/CRC, Boca Raton (2002)

    Book  Google Scholar 

Download references

Acknowledgements

The authors acknowledge the financial support from FEDER—Ministerio de Ciencia, Innovación y Universidades—Agencia Estatal de Investigación of Spain, by means of the project PGC2018-101514-B-I00, and from Junta de Andalucía to the research group FQM–377.

A. Ruiz also acknowledges the financial support from the Plan Propio de Investigación of the University of Cádiz (MV2019-341).

Author information

Authors and Affiliations

  1. Departamento de Matemáticas, Escuela de Ingenierías Marina, Náutica y Radioelectrónica, Universidad de Cádiz, Puerto Real, Spain

    A. Ruiz

  2. Departamento de Matemáticas, Facultad de Ciencias, Universidad de Cádiz, Puerto Real, Spain

    C. Muriel

Authors
  1. A. Ruiz
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Muriel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Ruiz .

Editor information

Editors and Affiliations

  1. Departamento Matemáticas, University of Cádiz, Puerto Real, Spain

    Concepción Muriel

  2. Departamento Matemáticas, University of Cádiz, Puerto Real, Spain

    Carmen Pérez-Martinez

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Ruiz, A., Muriel, C. (2021). Systems of Vector Fields for the Integration of Ordinary Differential Equations. In: Muriel, C., Pérez-Martinez, C. (eds) Recent Advances in Differential Equations and Control Theory. SEMA SIMAI Springer Series(), vol 9. Springer, Cham. https://doi.org/10.1007/978-3-030-61875-9_6

Download citation

Publish with us

Policies and ethics

Search

Navigation