Singularly Perturbed Linear Ordinary Differential Equations with Turning Points

2015 ◽  
pp. 239-290
Author(s):  
S. M. Bauer ◽  
S. B. Filippov ◽  
A. L. Smirnov ◽  
P. E. Tovstik ◽  
R. Vaillancourt
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Turning Points ◽  
Singularly Perturbed ◽  
Linear Ordinary Differential Equations

Asymptotics beyond all orders and Stokes lines in nonlinear differential-difference equations

2001 ◽  
Vol 12 (4) ◽  
pp. 433-463 ◽  
Author(s):  
J. R. KING ◽  
S. J. CHAPMAN
Keyword(s):  
Asymptotic Expansion ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Difference Equations ◽  
Singularly Perturbed ◽  
Time Dependent ◽  
Stokes Line ◽  
Linear Ordinary Differential Equations ◽  
Stokes Lines ◽  
Asymptotics Beyond All Orders

A technique for calculating exponentially small terms beyond all orders in singularly perturbed difference equations is presented. The approach is based on the application of a WKBJ-type ansatz to the late terms in the naive asymptotic expansion and the identification of Stokes lines, and is closely related to the well-known Stokes line smoothing phenomenon in linear ordinary differential equations. The method is illustrated by application to examples and the results extended to time-dependent differential-difference problems.

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