On a Singular Perturbation Problem

  • K.-C. Ng
  • R. Wong
Part of the ISNM International Series of Numerical Mathematics book series (ISNM, volume 119)


A uniformly valid asymptotic approximation is constructed for the solution to the initial value problem
$$\ddot v + \varepsilon t\dot v + v = 0, v(0) = 0, \dot v(0) = 1$$
, as ε → 0. From this, it is deduced that if εt → 0 then
$$v(t,\varepsilon ) \sim {e^{ - \varepsilon {t^2}/4}}\sin t$$
, and if εt → ∞ then
$$v(t,\varepsilon ) \sim \left( {\sin \frac{\pi }{{2\varepsilon }}} \right){e^{ - 1/2\varepsilon }}\frac{{\sqrt 2 }}{{{{(\varepsilon t)}^{1/\varepsilon }}}}$$


Exact Solution Asymptotic Formula Airy Function Perturbation Series Secular Term 
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Copyright information

© Birkhäuser 1994

Authors and Affiliations

  • K.-C. Ng
    • 1
  • R. Wong
    • 2
  1. 1.Department of MathematicsTamkang UniversityTaipeiTaiwan, ROC
  2. 2.Department of Applied MathematicsUniversity of ManitobaWinnipegCanada

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