Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-19T05:54:59.391Z Has data issue: false hasContentIssue false
  • English
  • Français

On the asymptotic expansions of solutions of an nth order linear differential equation

Published online by Cambridge University Press:  14 November 2011

R. B. Paris
R. B. Paris
Affiliation:
Association Euratom — CEA sur la Fusion, Département de Physique du Plasma et de la Fusion Contrôlée, Centre d'Etudes Nucléaires, 92260 Fontenay-aux-Roses, France
Get access Rights & Permissions [Opens in a new window]

Synopsis

The asymptotic expansions of solutions of a class of linear ordinary differential equations of arbitrary order n are investigated for large values of the independent variable z in the complex plane. Solutions are expressed in terms of Mellin-Barnes integrals and their asymptotic expansions are subsequently determined by means of the asymptotic theory of integral functions of the hypergeometric type. Three classes of solutions are considered: (i) solutions whose behaviour is either exponentially large or algebraic for |z|→∞ in different sectors of the z-plane, (ii) solutions which are even and odd functions of z when the order n of the differential equation is even and (iii) solutions which are exponentially damped as |z|→∞ in a certain sector of the z-plane.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 85 , Issue 1-2 , 1980 , pp. 15 - 57
Copyright
Copyright © Royal Society of Edinburgh 1980

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Abramowitz, M. and Stegun, I. (Eds). Handbook of Mathematical Functions (New York: Dover, 1965).Google Scholar
2Bakhoom, N. G.. Asymptotic expansions of the function Proc. London Math. Soc. 35 (1933), 83–100.CrossRefGoogle Scholar
3Barnes, E. W.. The asymptotic expansion of integral functions defined by generalised hypergeometric series. Proc. London Math. Soc. 5 (1907), 59–116.CrossRefGoogle Scholar
4Braaksma, B. L. J.. Asymptotic expansions and analytic continuations for a class of Barnes integrals. Compositie Math. 15 (1964), 239–341.Google Scholar
5Dagazian, R. Y. and Paris, R. B.. Stationary convective-like modes in a plasma slab with magnetic shear. Phys. Fluids 20 (1977), 917–927.CrossRefGoogle Scholar
6Delerue, P.. Sur l'utilisation des fonctions hyperbesséliennes à la résolution d'une équation différentielle. C. R. Acad. Sci. Paris 240 (1950), 912–914.Google Scholar
7Fidone, I. and Paris, R. B.. Lower hybrid wave conversion for a parabolic density profile. Pfiys. Fluids 17 (1974), 1921–1922.Google Scholar
8Ford, W. B.. The Asymptotic Developments of Functions defined by Maclaurin Series. (New York: Chelsea, 1961).Google Scholar
9Forsyth, A. R.. Theory of Differential Equations (New York: Dover, 1959).Google Scholar
10Gill, A. E. and Smith, R. K.. On similarity solutions of the differential equation Ψzzzz + Ψx = 0. Math. Proc. Cambridge Philos. Soc. 67 (1970), 163–171.CrossRefGoogle Scholar
11Hughes, H. K.. On the asymptotic expansions of entire functions defined by Maclaurin series. Bull. Amer. Math. Soc. 50 (1944), 425–430.CrossRefGoogle Scholar
12Kamke, E.. Differentialgleichungen (New York: Chelsea, 1948).Google Scholar
13Langer, R. E.. The solutions of the differential equation v‴ + λ2zv′ + 3μλ2v = 0. Duke Math. J. 22 (1955), 525–541.CrossRefGoogle Scholar
14Newsom, C. V.. On the character of certain entire functions in distant portions of the plane. Amer. J. Math. 60 (1938), 561–572.CrossRefGoogle Scholar
15Olver, F. W. J.. Error bounds for asymptotic expansions, with an application to cylinder functions of large argument. In Asymptotic solutions of Differential Equations and their Applications (Ed. Wilcox, C. H.), 163–183 (New York: Wiley, 1964).Google Scholar
16Paris, R. B.. The asymptotic behaviour of solutions of the differential equation u IV + [z 2u II+azu I + bu] = 0. Proc. Roy. Soc. London Ser. A 346 (1975), 171–207.Google Scholar
17Paris, R. B.. The asymptotic expansions of integral functions of the hypergeometric type. Assoc. Euratom-CEA Repori EUR-CEA FC 959 (1978).Google Scholar
18Paris, R. B. and Wood, A. D.. The asymptotic expansion of solutions of the differential equation u IV + λ2[(z 2 + c)u II + azu I + bu] = 0 for large |z|. Philos. Trans. Roy. Soc. London Ser. A 293 (1979), 511–533.Google Scholar
19Saxton, R. A.. An Integral Representation Solution for a Class of Higher Order Linear Ordinary Differential Equations (M.Sc. Dissertation, Cranfield Inst. of Tech., 1978); R. A. Saxton and A. D. Wood. Asymptotics of integral representation solutions for a class of higher order linear ordinary differential equations, submitted for publication.Google Scholar
20Scheffé, H.. Linear differential equations with two-term recurrence formulas. J. Math, and Phys. 21 (1942), 240–249.CrossRefGoogle Scholar
21Slater, L. J.. Generalised Hypergeometric Functions (Cambridge Univ. Press, 1966).Google Scholar
22Spitzer, S.. Integration der linearen Differentialgleichung y (n) = y 2″ + Bxy′ + Cy. Math. Ann. 3 (1871), 453–455.CrossRefGoogle Scholar
23Watson, G. N.. Theory of Bessel Functions (Cambridge Univ. Press, 1944).Google Scholar
24Whittaker, E. T. and Watson, G. N.. Modern Analysis (Cambridge Univ. Press, 1965).Google Scholar
25Wood, A. D.. Asymptotic Methods for the Determination of Deficiency Indices (Ph.D. Dissertation, Univ. St Andrews, 1968).Google Scholar
26Wood, A. D.. Deficiency indices of some fourth order differential operators. J. London Math. Soc. 3 (1971), 96–100.CrossRefGoogle Scholar
27Wright, E. M.. The asymptotic expansion of the generalised hypergeometric function. J. London Math. Soc. 10 (1935), 286–293. (corrigendum ibid. 27, 256).CrossRefGoogle Scholar
28Wright, E. M.. The asymptotic expansion of integral functions defined by Taylor series. Philos. Trans. Roy. Soc. Lond. Ser. A 238 (1940), 423–451.Google Scholar
29Wright, E. M.. A recursion formula for the coefficients in an asymptotic expansion. Proc. Glasgow Math. Assoc. 4 (1958), 38–41.CrossRefGoogle Scholar
30Wrinch, D.. A generalised hypergeometric function with n parameters. Philos. Mag. 41 (1921), 174–186.CrossRefGoogle Scholar