Table of Contents Author Guidelines Submit a Manuscript
Advances in Mathematical Physics
Volume 2018 (2018), Article ID 7590847, 8 pages
https://doi.org/10.1155/2018/7590847
Research Article

Burgers’ Equations in the Riemannian Geometry Associated with First-Order Differential Equations

1Department of Physics, Ege University, 35040 İzmir, Turkey
2Department of Mathematics, Akdeniz University, 07058 Antalya, Turkey

Correspondence should be addressed to Z. Ok Bayrakdar

Received 12 October 2017; Revised 13 December 2017; Accepted 21 December 2017; Published 8 February 2018

Academic Editor: Boris G. Konopelchenko

Copyright © 2018 Z. Ok Bayrakdar and T. Bayrakdar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. T. Witelski and M. Bowen, Methods of Mathemtical Modelling Continious System and Differential Equations, Springer, Switzerland, 2015. View at MathSciNet
  2. P. J. Olver, Introduction to Partial Differential Equations, Springer, Switzerland, 2014. View at MathSciNet
  3. J. M. Burgers, “A mathematical model illustrating the theory of turbulence,” in Advances in Applied Mechanics, pp. 171–199, Academic Press, Inc., New York, N. Y., 1948. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. M. Nadjafikhah, “Lie symmetries of inviscid Burgers' equation,” Advances in Applied Clifford Algebras (AACA), vol. 19, no. 1, pp. 101–112, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. H. Liu, J. Li, and Q. Zhang, “Lie symmetry analysis and exact explicit solutions for general Burgers' equation,” Journal of Computational and Applied Mathematics, vol. 228, no. 1, pp. 1–9, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. I. L. Freire, “A note on 'Lie symmetries of inviscid Burgers equation',” Advances in Applied Clifford Algebras (AACA), vol. 22, no. 2, pp. 297–300, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  7. M. A. Abdulwahhab, A. H. Bokhari, A. H. Kara, and F. D. Zaman, “On the Lie point symmetry analysis and solutions of the inviscid Burgers equation,” Pramana—Journal of Physics, vol. 77, no. 3, pp. 407–414, 2011. View at Publisher · View at Google Scholar · View at Scopus
  8. I. L. Freire, “New conservation laws for inviscid Burgers equation,” Computational & Applied Mathematics, vol. 31, no. 3, pp. 559–567, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  9. C. S. Rao and M. K. Yadav, “On the solution of a nonhomogeneous Burgers equation,” International Journal of Nonlinear Science, vol. 10, no. 2, pp. 141–145, 2010. View at Google Scholar · View at MathSciNet
  10. O. E. Barndorff-Nielsen and N. N. Leonenko, “Burgers' turbulence problem with linear or quadratic external potential,” Journal of Applied Probability, vol. 42, no. 2, pp. 550–565, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. S. Kar, S. K. Banik, and D. S. Ray, “Exact solutions of Fisher and Burgers equations with finite transport memory,” Journal of Physics A: Mathematical and General, vol. 36, no. 11, pp. 2771–2780, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. S. Buyukasik and O. K. Pashaev, “Exact solutions of forced Burgers equations with time variable coefficients,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 7, pp. 1635–1651, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  13. D. R. Nelson, “Population dynamics and Burgers' equation,” Physica A: Statistical Mechanics and its Applications, vol. 274, no. 1-2, pp. 85–90, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. H. Liu, X. Xin, Z. Wang, and X. Liu, “Bäcklund transformation classification, integrability and exact solutions to the generalized Burgers'-KdV equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 44, pp. 11–18, 2017. View at Publisher · View at Google Scholar · View at MathSciNet
  15. B. Khesin and R. Wendt, The Geometry of Infinite-Dimensional Groups, Sipringer-Verlag, Berlin Heidelberg, 2009. View at MathSciNet
  16. A. Tresse, “Sur les invariants differentiels des groupes continus de transformations,” Acta Mathematica, vol. 18, no. 1, pp. 1–3, 1894. View at Publisher · View at Google Scholar · View at MathSciNet
  17. M. A. Tresse, Determination des Invariantes Ponctuels de l’Equation Differentielle du Second Ordre y''=ω(x,y,y'), Hirzel, Leipzig, 1896.
  18. S. Lie, “Classification und Integration von gewöhnlichen Differentialgleichungen zwischenxy, die eine Gruppe von Transformationen gestatten,” Mathematische Annalen, vol. 32, no. 2, pp. 213–281, 1888. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  19. E. Cartan, “Sur les variétés à connexion projective,” Bull. Soc. Math. France, vol. 52, pp. 205–241, 1924. View at Google Scholar · View at MathSciNet
  20. N. Kamran, K. G. Lamb, and W. F. Shadwick, “The Local Equivalence Problem For d2y/dx2 = F(x, y, dy/dx) And The Painlevé Transcendents,” Journal of Differential Geometry, vol. 22, no. 2, pp. 139–150, 1985. View at Publisher · View at Google Scholar · View at MathSciNet
  21. C. Grissom, G. Thompson, and G. Wilkens, “Linearization of second order ordinary differential equations via Cartan's equivalence method,” Journal of Differential Equations, vol. 77, no. 1, pp. 1–15, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. M. E. Fels, “The equivalence problem for systems of second-order ordinary differential equations,” Proceedings of the London Mathematical Society, vol. 3-71, no. 1, pp. 221–240, 1995. View at Publisher · View at Google Scholar · View at Scopus
  23. M. Crampin and D. J. Saunders, “Cartan's concept of duality for second-order ordinary differential equations,” Journal of Geometry and Physics, vol. 54, no. 2, pp. 146–172, 2005. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. S. Frittelli, C. Kozameh, and E. T. Newman, “Differential geometry from differential equations,” Communications in Mathematical Physics, vol. 223, no. 2, pp. 383–408, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  25. E. T. Newman and P. Nurowski, “Projective connections associated with second-order {ODE}s,” Classical and Quantum Gravity, vol. 20, no. 11, pp. 2325–2335, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. M. Crampin, E. Marnez, and W. Sarlet, “Linear connections for systems of second-order ordinary differential equations,” Annales de l’ I. H. P. section A, vol. 65, no. 2, pp. 223–249, 1996. View at Google Scholar · View at MathSciNet
  27. M. Eastwood and V. Matveev, “Metric Connections in Projective Differential Geometry,” in Symmetries and Overdetermined Systems of Partial Differential Equations, vol. 144 of IMA Vol. Math. Appl., pp. 339–350, Springer, New York, NY, USA, 2008. View at Publisher · View at Google Scholar · View at MathSciNet
  28. E. Cartan, Riemannian Geometry in an Orthogonal Frame, World Scientific Publishing Co., Singapore, 2001. View at MathSciNet
  29. T. A. Ivey and J. M. Landsberg, Cartan for Beginners: Differential Geometry via Moving Frames and Exterior Differential Systems, American Mathematical Society, Providence, RI, 2003. View at MathSciNet
  30. S. Morita, Geometry of Differential Forms, volume 201 of Translations of Mathematical Monographs, AMS Providence, RI, 2001. View at MathSciNet
  31. P. J. Vassiliou, “Introduction: Geometric Approaches to Differential Equations,” in Geometric approaches to differential equations (Canberra, 1995), vol. 15 of Austral. Math. Soc. Lect. Ser., pp. 1–15, Cambridge University Press, Cambridge, UK, 2000. View at Google Scholar · View at MathSciNet
  32. R. L. Bryant, P. A. Griffiths, S. S. Chern, R. B. Gardner, and H. L. Goldschmidt, Exterior Differential Systems, Springer, Berlin, Germany, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  33. D. Krupka and D. Saunders, Eds., Handbook of Global Analysis, Elsevier, Oxford, UK, 2008. View at MathSciNet
  34. D. J. Korteweg and G. de Vries, “XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves,” Philosophical Magazine, vol. 39, no. 240, pp. 422–443, 1895. View at Publisher · View at Google Scholar
  35. Y. Aminov, The Geometry of Submanifolds, Gordon and Breach Science Publishers, Netherlands, 2001. View at MathSciNet