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Applied Mathematics and Mechanics

, Volume 12, Issue 9, pp 907–926 | Cite as

Extended airy function and differential equations with n-turning points

  • Zhang Ju-ling
Article

Abstract

This paper studies a second order linear ordinary differential equation with n-turning points
$$\frac{{d^2 y}}{{dx^2 }} + [\lambda ^2 q_1 (x) + q_2 (x)]y = 0$$
Where q1(x)=(x-μ1) (x-μ2) ... (x-μm) f(x), f(x)≠0, and λ is a large parameter.

The formal uniformly valid asymptotic solution of the equation is obtained based on the analysis of the three points by means of the matched method. By the work a method is developed and the applicability of this method to the n-turning points is demonstrated.

Key words

Uniformly valid asymptotic solution turning point 

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References

  1. [1]
    Pu Fu-quan, The problem about uniformly valid asymptotic solutions of a differential equation with turning point of second order,Journal of Tsinghua University,20, 3 (1980), 103–109. (in Chinese)Google Scholar
  2. [2]
    Guo Dun-ren,The Equation of Mathematical Physics People's Education Press (1979). (in Chinese)Google Scholar
  3. [3]
    Olver, F.W.J.,Asymptotics and Special Function (1974).Google Scholar
  4. [4]
    Nayfeh, A.H.,Perturbation Methods (1973).Google Scholar
  5. [5]
    Sibuya, Y.,Global Theory of a Second Order Linear Ordinary Differential Equatin with a Polynomial Coefficient, North-Holland Publishing Company (1975).Google Scholar

Copyright information

© Shanghai University of Technology (SUT) 1991

Authors and Affiliations

  • Zhang Ju-ling
    • 1
  1. 1.Sichnan Institute of Light Industry and Chemical TechnologyZigong

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