Abstract.
This paper shows how to build an analytical solution for a differential equation of arbitrary order and with variable coefficients. It proofs that the most known approximated solutions for such a problem can be derived from the analytical expression presented in the paper. The formalism can be easily extended to the infinite dimensional case such as the quantum time-dependent Hamiltonian problem and used for the eigenvalues problems.
Article PDF
Similar content being viewed by others
References
A.L. Fetter, J.D. Walecka, Quantum Theory of Many-Particle Systems (Dover Publications, New York, 2003)
S. Weinberg, Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity (John Wiley & Sons, 1972)
N.G. Van Kampen, Stochastic Processes in Physics and Chemistry, (Elsevier, Amsterdam, 2007)
E.W. Montroll, B.J. West, On an enriched collection of stochastic processes, in Fluctuation Phenomena, edited by E.W. Montroll, J.L. Lebowitz, Studies in Statistical Mechanics, Vol. VII (North-Holland, Amsterdam, 1979)
M. Bianucci, R. Mannella, P. Grigolini, B.J. West, Int. J. Mod. Phys. B 8, 1191 (1994)
M. Bianucci, R. Mannella, P. Grigolini, B.J. West, Int. J. Mod. Phys. B 8, 1211 (1994)
M. Bianucci, R. Mannella, P. Grigolini, B.J. West, Int. J. Mod. Phys. B 8, 1225 (1994)
R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)
B.J. West, Fractional Calculus View of Complexity: Tomorrow’s Science (CRC Press, 2016)
H.-C. Kim, M.-H. Lee, J.-Y. Ji, J.K. Kim, Phys. Rev. A 53, 3767 (1996)
J.M. Cerveró, J.D. Lejarreta, Europhys. Lett. 45, 6 (1999)
M. Fernández Guasti, A. Gil-Villegas, Phys. Lett. A 292, 243 (2002)
I.A. Pedrosa, Alexandre Rosas, Phys. Rev. Lett. 103, 010402 (2009)
M. Eshghi, R. Sever, S.M. Ikhdair, Eur. Phys. J. Plus 131, 223 (2016)
G. Peano, Math. Ann. 32, 450 (1888)
W. Magnus, Commun. Pure Appl. Math. 7, 649 (1954)
S. Blanes, F. Casas, J.A. Oteo, J. Ros, Phys. Rep. 470, 151 (2009)
J. Kevorkian, J.D. Cole, Multiple Scale and Singular Perturbation Methods (Springer, 1996)
W.C. Brenke, Bull. Am. Math. Soc. 36, 77 (1930)
A.F. Nikiforov, V.B. Uvarov, Special Functions of Mathematical Physics. A Unified Introduction with Applications (Springer, 1988)
A. Turbiner, J. Math. Phys. 33, 3989 (1992)
G.G. Gundersen, E.M. Steinbart, S. Wang, Trans. Am. Math. Soc. 350, 1225 (1998)
N. Saad, R.L. Hall, H. Ciftci, J. Phys. A: Math. Gen. 39, 13445 (2006)
M. Bologna, J. Phys. A: Math. Theor. 43, 375203 (2010)
V.E. Kruglov, Differ. Equ. 47, 20 (2011)
E. Nelson, J. Math. Phys. 5, 332 (1964)
H.F. Baker, Proc. London Math. Soc. 3, 24 (1905)
J.E. Campbell, Proc. London Math. Soc. 29, 14 (1898)
F. Hausdorff, Leipziger Ber. 58, 19 (1906)
V. Braginsky, F. Ya Khalili, Quantum Measurement (Cambridge University Press, Cambridge, 1992)
P.A.M. Dirac, The Principles of Quantum Mechanics (Oxford University Press, 1981)
M. Born, V.A. Fock, Z. Phys. A 51, 165 (1928)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Bologna, M. Exact analytical approach to differential equations with variable coefficients. Eur. Phys. J. Plus 131, 386 (2016). https://doi.org/10.1140/epjp/i2016-16386-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjp/i2016-16386-9