Abstract
A two-point boundary value problem is considered for the Emden–Fowler equation, which is a singular nonlinear ordinary differential equation of the second order. Assuming that the exponent in the coefficient of the nonlinear term is rational, new parametric representations are obtained for the solution of the boundary value problem on the half-line and on the interval. For the problem on the half-line, a new efficient formula is given for the first term of the well-known Coulson–March expansion of the solution in a neighborhood of infinity, and generalizations of this representation and its analogues for the inverse of the solution are obtained. For the Thomas–Fermi model of a multielectron atom and a positively charged ion, highly efficient computational algorithms are constructed that solve the problem for an atom (that is, the boundary value problem on the half-line) and find the derivative of this solution with any prescribed accuracy at an arbitrary point of the half-line. The results are based on an analytic property of a special Abel equation of the second kind to which the original Emden–Fowler equation reduces, to be precise, the property of partially passing a modified Painlevé test at a nodal singular point.
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REFERENCES
E. Fermi, “Un metodo statistico per la determinazione di alcune prioprieta dell’atomo,” Rend. Accad. Naz. Lincei 6, 602–607 (1927).
L. H. Thomas, “The calculations of atomic fields,” Proc. Cambridge Philos. Soc., No. 23, 542–598 (1927).
L. D. Landau and E. M. Lifshitz, Quantum Mechanics: Non-Relativistic Theory (Butterworth-Heinemann, Oxford, 1977; Nauka, Moscow, 1989).
N. H. March, “The Fermi–Thomas theory,” Theory of the Inhomogeneous Electron Gas, Ed. by S. Lunqvist and N. H. March (Plenum, New York, 1983), pp. 9–85.
R. Bellman, Stability Theory of Differential Equations (McGraw-Hill, New York, 1953).
D. Sansone, Equazioni differenziali nel campo reale (Nicola Zanichelli, Bologna, 1948).
A. Sommerfeld, “Integrazione asintotica dell’equazione differenziale di Fermi–Thomas,” Rend. R. Accad. Lincei 15, 293–308 (1932).
S. Flugge, Practical Quantum Mechanics (Springer-Verlag, Berlin, 1971).
E. Hille, “Some aspects of the Fermi–Thomas equation,” J. Anal. Math. 23, 147–170 (1970).
E. B. Baker, “The application of the Fermi–Thomas statistical model to the calculation of potential distribution in positive ions,” Phys. Rev. 36, 630–647 (1930).
C. A. Coulson and N. H. March, “Momenta in atoms using the Fermi–Thomas method,” Proc. Phys. Soc. Sect. A 63 (4), 367–367 (1950).
A. L. Dyshko, M. P. Carpentier, N. B. Konyukhova, and P. M. Lima, “Singular problems for Emden–Fowler-type second-order nonlinear ordinary differential equations,” Comput. Math. Math. Phys. 41 (4), 557–580 (2001).
V. Bush and S. H. Caldwell, “Fermi–Thomas equation solution by the differential analyzer,” Phys. Rev. 38 (10), 1898–1902 (1931).
C. Miranda, “Teoremi e metodi per l’integrazione numerica dell' equazione differenziale di Fermi,” Mem. R. Acc. Italia, No. 5, 285–322 (1934).
J. C. Slater and H. M. Krutter, “The Fermi–Thomas method for metals,” Phys. Rev. 47 (7), 559–568 (1935).
R. P. Feynman, N. Metropolis, and E. Teller, “Equations of state of elements based on the generalized Fermi–Thomas theory,” Phys. Rev. 75 (10), 1561–1573 (1949).
S. Kobayashi, T. Matsukuma, S. Nagai, and K. Umeda, “Accurate value of the initial slope of the ordinary TF function,” J. Phys. Soc. Jpn. 10, 759–762 (1955).
H. Krutter, “Numerical integration of the Thomas–Fermi equation from zero to infinity,” J. Comput. Phys. 47 (2), 308–312 (1982).
R. Bellman, “Dynamic programming and the variational solution of the Fermi–Thomas equation,” J. Phys. Soc. Jpn. 12, 1049 (1957).
T. Ikebe and T. Kato, “Application of variational method to the Fermi–Thomas equation,” J. Phys. Soc. Jpn. 12 (2), 201–203 (1957).
R. V. Ramnath, “A new analytical approximation for the Fermi–Thomas model in atomic physics,” J. Math. Anal. Appl. 31 (2), 285–296 (1970).
I. M. Torrens, Interatomic Potentials (Academic, New York, 1972).
N. Anderson and A. M. Arthurs, “Variational solutions of the Fermi–Thomas equation,” Q. Appl. Math. 39, 127–129 (1981–1982).
M. Desaix, D. Anderson, and M. Lisak, “Variational approach to the Fermi–Thomas equation,” Eur. J. Phys. 25, 699–705 (2004).
M. Oulne, “Variation and series approach to the Fermi–Thomas equation,” Appl. Math. Comput. 218 (2), 303–307 (2011).
R. C. Flagg, C. D. Luning, and W. L. Perry, “Implementation of new iterative techniques for solutions of Thomas–Fermi and Emden–Fowler equations,” J. Comput. Phys. 38, 396–405 (1980).
C. M. Bender, K. A. Milton, S. S. Pinsky, and L. M. Simmons, “A new perturbative approach to nonlinear problems,” J. Math. Phys. 30 (7), 1447–1455 (1989).
K. Tu, “Analytic solution to the Fermi–Thomas and Fermi–Thomas–Dirac–Weizsäcker equations,” J. Math. Phys. 32, 2250–2253 (1991).
N. A. Zaitsev, I. V. Matyushkin, and D. V. Shamonov, “Numerical solution of the Fermi–Thomas equation for the centrally symmetric atom,” Russ. Microelectron. 33 (5), 303–309 (2004).
A. J. MacLeod, “Chebyshev series solution of the Fermi–Thomas equation,” Comput. Phys. Commun. 67 (3), 389–391 (1992).
K. Parand and M. Shahini, “Rational Chebyshev pseudospectral approach for solving Fermi–Thomas equation,” Phys. Lett. A 373 (2), 210–213 (2009).
K. Parand, K. Rabiei, and M. Delkhosh, “An efficient numerical method for solving nonlinear Thomas–Fermi equation,” Acta Univ. Sapientiae, Math. 10 (1), 134–151 (2018).
J. P. Boyd, “Rational Chebyshev series for the Fermi–Thomas function: Endpoint singularities and spectral methods,” J. Comput. Appl. Math. 244, 90–101 (2013).
J. C. Mason, “Rational approximations to the ordinary Fermi–Thomas function and its derivative,” Proc. Phys. Soc. 84 (3), 357 (1964).
S. Abbasbandy and C. Bervillier, “Analytic continuation of Taylor series and the boundary value problems of some nonlinear ordinary differential equations,” Appl. Math. Comput. 218, 2178–2199 (2011).
L. N. Epele, H. Fanchiotti, C. A. García Canal, and J. A. Ponciano, “Padé approximant approach to the Fermi–Thomas problem,” Phys. Rev. A 60, 280–283 (1999).
S. Liao, “An explicit analytic solution to the Fermi–Thomas equation,” Appl. Math. Comput. 144, 495–506 (2003).
G. I. Plindov and S. K. Pogrebnaya, “The analytical solution of the Fermi–Thomas equation for a neutral atom,” J. Phys. B 20, 547–550 (1987).
F. M. Fernandez and J. F. Ogilvie, “Approximate solutions to the Fermi–Thomas equation,” Phys. Rev. A 42 (1), 149–154 (1990).
S. Esposito, “Majorana solution of the Fermi–Thomas equation,” Am. J. Phys. 70 (8), 852–856 (2002).
S. Esposito, “Majorana transformation for differential equations,” Int. J. Theor. Phys. 41 (12), 2417–2426 (2002).
S. R. Finch, “Mathematical constants II,” Encyclopedia of Mathematics and Its Applications (Cambridge Univ. Press, Cambridge, 2018).
D. E. Panayotounakos and D. C. Kravvaritisb, “Exact analytic solutions of the Abel, Emden–Fowler, and generalized Emden–Fowler nonlinear ODEs,” Nonlinear Anal. Real World Appl. 7 (4), 634–650 (2006).
D. E. Panayotounakos and N. Sotiropoulos, “Exact analytic solutions of unsolvable classes of first- and second-order nonlinear ODEs (Part II: Emden–Fowler and relative equations),” Appl. Math. Lett. 18 (4), 367–374 (2005).
E. E. Theotokoglou, T. I. Zarmpoutis, and I. H. Stampouloglou, “Closed-form solutions of the Fermi–Thomas in heavy atoms and the Langmuir–Blodgett in current flow ODEs in mathematical physics,” Math. Probl. Eng. 2015, Article ID 721637 (2015).
S. I. Bezrodnykh and V. I. Vlasov, “The boundary value problem for the simulation of physical fields in a semiconductor diode,” Comput. Math. Math. Phys. 44 (12), 2112–2142 (2004).
S. V. Pikulin, “The behavior of solutions to a special Abel equation of the second kind near a nodal singular point,” Comput. Math. Math. Phys. 58 (12), 1948–1966 (2018).
R. M. Conte and M. Musette, The Painlevé Handbook (Springer Science+Business Media, Dordrecht, 2008).
E. Hille, Ordinary Differential Equations in the Complex Domain (Wiley, New York, 1976).
V. P. Varin, “A solution of the Blasius problem,” Comput. Math. Math. Phys. 54 (6), 1025–1036 (2014).
V. P. Varin, “Asymptotic expansion of Crocco solution and the Blasius constant,” Comput. Math. Math. Phys. 58 (4), 517–528 (2018).
C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).
S. V. Pikulin, “Traveling-wave solutions of the Kolmogorov–Petrovskii–Piskunov equation,” Comput. Math. Math. Phys. 58 (2), 230–237 (2018).
S. V. Pikulin, “On intermediate asymptotic modes in certain combustion models,” Tavrich. Vestn. Inf. Mat., No. 3 (36), 55–72 (2017).
S. V. Pikulin, “On travelling-wave solutions of a nonlinear parabolic equation,” Vestn. Samar. Gos. Univ. Estestv. Ser., No. 6 (128), 110–116 (2015).
V. V. Golubev, Lectures on the Analytical Theory of Differential Equations (Gostekhizdat, Moscow, 1950) [in Russian].
E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw-Hill, New York, 1955).
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Pikulin, S.V. The Thomas–Fermi Problem and Solutions of the Emden–Fowler Equation. Comput. Math. and Math. Phys. 59, 1292–1313 (2019). https://doi.org/10.1134/S096554251908013X
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DOI: https://doi.org/10.1134/S096554251908013X