Table of Contents Author Guidelines Submit a Manuscript
Mathematical Problems in Engineering
Volume 2015, Article ID 721637, 8 pages
http://dx.doi.org/10.1155/2015/721637
Research Article

Closed-Form Solutions of the Thomas-Fermi in Heavy Atoms and the Langmuir-Blodgett in Current Flow ODEs in Mathematical Physics

School of Applied Mathematical and Physical Sciences, Department of Mechanics, Laboratory of Testing and Materials, National Technical University of Athens, Zographou Campus, Theocaris Building, 5 Heroes of Polytechniou Avenue, 157-73 Athens, Greece

Received 25 March 2015; Revised 2 July 2015; Accepted 7 July 2015

Academic Editor: Hang Xu

Copyright © 2015 Efstathios E. Theotokoglou et al. 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. I. Langmuir and K. B. Blodgett, “Currents limited by space charge between coaxial cylinders,” Physical Review, vol. 22, no. 4, pp. 347–356, 1923. View at Publisher · View at Google Scholar · View at Scopus
  2. H. T. Davis, Introduction to Nonlinear Differential and Integral Equations, Dover, New York, NY, USA, 1962. View at MathSciNet
  3. E. Kamke, Differentialgleichungen I, Gewöhnliche Differentialgleichungen and II, Partielle Differentialgleichungen, Akademische Verlagsgesellschaft, Leipzig, Germany, 1962.
  4. A. D. Polyanin and V. F. Zaitsev, Handbook of Exact Solutions for Ordinary Differential Equations, CRC Press, Boca Raton, Fla, USA, 1999.
  5. K. Horváth, J. N. Fairchild, K. Kaczmarski, and G. Guiochon, “Martin-Synge algorithm for the solution of equilibrium-dispersive model of liquid chromatography,” Journal of Chromatography A, vol. 1217, no. 52, pp. 8127–8135, 2010. View at Publisher · View at Google Scholar · View at Scopus
  6. Y. Shang and X. Zheng, “The first-integral method and abundant explicit exact solutions to the zakharov equations,” Journal of Applied Mathematics, vol. 2012, Article ID 818345, 16 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus
  7. D. B. Graham and I. H. Cairns, “Constraints on the formation and structure of langmuir eigenmodes in the solar wind,” Physical Review Letters, vol. 111, no. 12, Article ID 121101, 2013. View at Publisher · View at Google Scholar · View at Scopus
  8. L. H. Thomas, “The calculation of atomic fields,” Mathematical Proceedings of the Cambridge Philosophical Society, vol. 23, no. 5, pp. 542–548, 1927. View at Publisher · View at Google Scholar
  9. E. Fermi, “Statistical method of investigating electrons in atoms,” Zeitschrift für Physik, vol. 48, pp. 73–79, 1927. View at Google Scholar
  10. T. Tsurumi and M. Wadati, “Dynamics of magnetically trapped boson-fermion mixtures,” Journal of the Physical Society of Japan, vol. 69, no. 1, pp. 97–103, 2000. View at Publisher · View at Google Scholar · View at Scopus
  11. S. Liao, “An explicit analytic solution to the Thomas-Fermi equation,” Applied Mathematics and Computation, vol. 144, no. 2-3, pp. 495–506, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. S. Abbasbandy and C. Bervillier, “Analytic continuation of Taylor series and the boundary value problems of some nonlinear ordinary differential equations,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 2178–2199, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  13. K. Ourabah and M. Tribeche, “Relativistic formulation of the generalized nonextensive Thomas-Fermi model,” Physica A, vol. 393, pp. 470–474, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. D. E. Panayotounakos, “Exact analytic solutions of unsolvable classes of first and second order nonlinear ODEs (part I: Abel's equations),” Applied Mathematics Letters, vol. 18, no. 2, pp. 155–162, 2005. View at Publisher · View at Google Scholar · View at Scopus
  15. D. E. Panayotounakos, N. B. Sotiropoulos, A. B. Sotiropoulou, and N. D. Panayotounakou, “Exact analytic solutions of nonlinear boundary value problems in fluid mechanics (Blasius equations),” Journal of Mathematical Physics, vol. 46, no. 3, Article ID 033101, 2005. View at Publisher · View at Google Scholar · View at Scopus
  16. D. E. Panayotounakos, E. E. Theotokoglou, and M. P. Markakis, “Exact analytic solutions for the unforded damped duffing nonlinear oscillator,” Comptes Rendus Mechanics Journal, vol. 334, pp. 311–316, 2006. View at Google Scholar
  17. V. Bush and S. H. Caldwell, “Thomas-Fermi equation solution by the differential analyzer,” Physical Review, vol. 38, no. 10, pp. 1898–1901, 1931. View at Publisher · View at Google Scholar · View at Scopus
  18. R. P. Feynman, N. Metropolis, and E. Teller, “Equations of state of elements based on the generalized fermi-thomas theory,” Physical Review, vol. 75, no. 10, pp. 1561–1573, 1949. View at Publisher · View at Google Scholar · View at Scopus
  19. C. A. Coulson and N. H. March, “Momenta in atoms using the Thomas-Fermi method,” Proceedings of the Physical Society Section A, vol. 63, no. 4, pp. 367–374, 1950. View at Publisher · View at Google Scholar · View at Scopus
  20. S. Kobayashi, T. Matsukuma, S. Nagai, and K. Umeda, “Accurate value of the initial slope of the ordinary TF function,” Journal of the Physical Society of Japan, vol. 10, no. 9, pp. 759–762, 1955. View at Publisher · View at Google Scholar · View at Scopus
  21. S. Kobayashi, “Some coefficients of the series expansion of the TFD function,” Journal of the Physical Society of Japan, vol. 10, no. 9, pp. 824–825, 1955. View at Google Scholar · View at MathSciNet
  22. S. A. Sommerfeld, “Integrazione Asintotica dell'Equazione Differenziali di Thomas-Fermi,” Accademia dei Lincei, Atti-Rendiconte, vol. 4, pp. 85–110, 1932. View at Google Scholar