Abstract
In this chapter we consider various classes of dth-order (dāā„ā2) linear homogeneous equations \(\displaystyle{ y^{(d)} + a_{ 1}(t)y^{(d-1)} +\,\ldots \, +a_{ d}(t)y = 0, }\)
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
F.V. Atkinson, The asymptotic solution of second-order differential equations. Ann. Mat. Pura Appl. 37, 347ā378 (1954)
O. Blumenthal, Ćber asymptotische Integration linearer Differentialgleichungen mit Anwendung auf eine asymptotische Theorie der Kugelfunktionen. Arch. Math. Phys. (Leipzig) 19, 136ā174 (1912)
S. Bodine, D.A. Lutz, Asymptotic integration of nonoscillatory differential equations: a unified approach. J. Dyn. Control Syst. 17, 329ā358 (2011)
M. Bohner, S. SteviÄ, Trenchās perturbation theorem for dynamic equations. Discrete Dyn. Nat. Soc., Art. ID 75672, 11 pp. (2007)
M. Bohner, S. SteviÄ, Linear perturbations of a nonoscillatory second-order dynamic equation. J. Differ. Equ. Appl. 15, 1211ā1221 (2009)
V.S. Burd, P. Nesterov, Parametric resonance in adiabatic oscillators. Results Math. 58, 1ā15 (2010)
J.S. Cassell, The asymptotic integration of some oscillatory differential equations. Q. J. Math. Oxford Ser. (2) 33, 281ā296 (1982)
S.Z. Chen, Asymptotic integrations of nonoscillatory second order differential equations. Trans. Am. Math. Soc. 327, 853ā865 (1991)
N. Chernyavskaya, L. Shuster, Necessary and sufficient conditions for the solvability of a problem of Hartman and Wintner. Proc. Am. Math. Soc. 125, 3213ā3228 (1997)
M. Christ, A. Kiselev, Absolutely continuous spectrum for one-dimensional Schrƶdinger operators with slowly decaying potentials: some optimal results. J. Am. Math. Soc. 11, 771ā797 (1998)
M. Christ, A. Kiselev, WKB asymptotic behavior of almost all generalized eigenfunctions for one-dimensional Schrƶdinger operators with slowly decaying potentials. J. Funct. Anal. 179, 426ā447 (2001)
W.A. Coppel, Stability and Asymptotic Behavior of Differential Equations (D. C. Heath and Co., Boston, 1965)
S.A. Denisov, A. Kiselev, Spectral properties of Schrƶdinger operators with decaying potentials. Spectral theory and mathematical physics: a Festschrift in honor of Barry Simonās 60th birthday. Proc. Symp. Pure Math. 76(Part 2), 565ā589 (2007)
M.S.P. Eastham, The Liouville-Green asymptotic theory for second-order differential equations: a new approach and some extensions, in Ordinary Differential Equations and Operators (Dundee, 1982), ed. by W.N. Everitt, R.T. Lewis. Lecture Notes in Mathematics, vol. 1032 (Springer, Berlin, 1983), pp. 110ā122
M.S.P. Eastham, The Asymptotic Solution of Linear Differential Systems. Applications of the Levinson Theorem (Oxford University Press, Oxford, 1989)
G. Green, On the motion of waves in a variable canal of small depth and width. Trans. Camb. Philos. Soc. 6, 457ā462 (1837)
M. GreguÅ”, Third Order Linear Differential Equations. Translated from the Slovak by J. DraveckĆ½. Mathematics and Its Applications (East European Series), vol. 22 (D. Reidel Publishing Co., Dordrecht, 1987)
W.A. Harris, D.A. Lutz, On the asymptotic integration of linear differential systems. J. Math. Anal. Appl. 48, 1ā16 (1974)
W.A. Harris, D.A. Lutz, Asymptotic integration of adiabatic oscillators. J. Math. Anal. Appl. 51(1), 76ā93 (1975)
W.A. Harris, D.A. Lutz, A unified theory of asymptotic integration. J. Math. Anal. Appl. 57(3), 571ā586 (1977)
P. Hartman, Ordinary Differential Equations (BirkhƤuser, Boston, 1982)
P. Hartman, A. Wintner, Asymptotic integrations of linear differential equations. Am. J. Math. 77, 45ā86, 404, 932 (1955)
R.B. Kelman, N.K. Madsen, Stable motions of the linear adiabatic oscillator. J. Math. Anal. Appl. 21, 458ā465 (1968)
A. Kiselev, Absolutely continuous spectrum of one-dimensional Schrƶdinger operators and Jacobi matrices with slowly decreasing potentials. Commun. Math. Phys. 179, 377ā400 (1996)
A. Kiselev, Examples of potentials and l p estimates. Private E-mail Communication (2009)
J. Liouville, Second mĆ©moire sur le dĆ©veloppement des fonctions ou parties de fonctions en sĆ©ries dont les divers termes sont assujĆ©tis Ć satisfaire Ć une mĆŖme Ć©quation diffĆ©rentielle du second ordre, contenant un paramĆØtre variable. J. Math. Pures Appl. 2, 16ā35 (1837)
F.W.J. Olver, Error bounds for the LiouvilleāGreen (or WKB) approximation. Proc. Camb. Philos. Soc. 57, 790ā810 (1961)
F.W.J. Olver, Asymptotics and Special Functions. Computer Science and Applied Mathematics (Academic, New York/London, 1974)
O. Perron, Ćber lineare Differentialgleichungen, bei denen die unabhƤngige Variable reell ist (Erste Mitteilung). J. Reine Angew. Math. 142, 254ā270 (1913)
O. Perron, Ćber lineare Differentialgleichungen, bei denen die unabhƤngige Variable reell ist (Zweite Mitteilung). J. Reine Angew. Math. 143, 29ā50 (1913)
G.W. Pfeiffer, Asymptotic solutions of y ā² ā² ā² + qy ā² + ryā=ā0. J. Differ. Equ. 11, 145ā155 (1972)
H. PoincarĆ©, Sur les equations linĆ©aires aux diffĆ©rentielles ordinaires et aux diffĆ©rences finies. Am. J. Math 7(3), 203ā258 (1885)
J. Rovder, Asymptotic behaviour of solutions of the differential equation of the fourth order. Math. Slovaca 30, 379ā392 (1980)
J. Å imÅ”a, Asymptotic integration of a second order ordinary differential equation. Proc. Am. Math. Soc. 101, 96ā100 (1987)
D.R. Smith, LiouvilleāGreen approximations via the Riccati transformation. J. Math. Anal. Appl. 116, 147ā165 (1986)
D.R. Smith, Decoupling of recessive and nonrecessive solutions for a second-order system. J. Differ. Equ. 68, 383ā399 (1987)
R. Spigler, M. Vianello, WKB-type approximations for second-order differential equations in Cā algebras. Proc. Am. Math. Soc. 124, 1763ā1771 (1996)
R. Spigler, M. Vianello, LiouvilleāGreen asymptotics for almost-diagonal second-order matrix differential equations. Asymptot. Anal. 48, 267ā294 (2006)
R. Spigler, M. Vianello, LiouvilleāGreen asymptotic approximations for a class of matrix differential equations and semi-discretized partial differential equations. J. Math. Anal. Appl. 325, 69ā89 (2007)
S.A. Stepin, Asymptotic integration of second-order nonoscillatory differential equations (Russian). Dokl. Akad. Nauk 434, 315ā318 (2010); Translation in Dokl. Math. 82(2), 751ā754 (2010)
J.G. Taylor, Improved error bounds for the Liouville-Green (or WKB) approximation. J. Math. Anal. Appl. 85, 79ā89 (1982)
W. Trench, On the asymptotic behavior of solutions of second order linear differential equations. Proc. Am. Math. Soc. 14, 12ā14 (1963)
W. Trench, Extensions of a theorem of Wintner on systems with asymptotically constant solutions. Trans. Am. Math. Soc. 293, 477ā483 (1986)
W. Trench, Linear perturbations of a nonoscillatory second order equation. Proc. Am. Math. Soc. 97, 423ā428 (1986)
W. Trench, Linear perturbations of a nonoscillatory second order differential equation. II. Proc. Am. Math. Soc. 131, 1415ā1422 (2003)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
Ā© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Bodine, S., Lutz, D.A. (2015). Applications to Classes of Scalar Linear Differential Equations. In: Asymptotic Integration of Differential and Difference Equations. Lecture Notes in Mathematics, vol 2129. Springer, Cham. https://doi.org/10.1007/978-3-319-18248-3_8
Download citation
DOI: https://doi.org/10.1007/978-3-319-18248-3_8
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-18247-6
Online ISBN: 978-3-319-18248-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)