Asymptotic solutions of some pseudodifferential equations and dynamical systems with small dispersion

2001 ◽  
pp. 133-157
Keyword(s):  
Dynamical Systems ◽  
Asymptotic Solutions ◽  
Pseudodifferential Equations ◽  
Small Dispersion

UNIFORMLY ASYMPTOTIC SOLUTIONS FOR PSEUDODIFFERENTIAL EQUATIONS WITH SINGULAR INTEGRAL OPERATORS

2001 ◽  
Vol 09 (02) ◽  
pp. 495-513 ◽  
Author(s):  
A. HANYGA ◽  
M. SEREDYŃSKA
Keyword(s):  
Asymptotic Solution ◽  
Frequency Domain ◽  
Singular Integral ◽  
Hyperbolic Equations ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Asymptotic Solutions ◽  
Convolution Operators ◽  
Pseudodifferential Equations

Uniformly asymptotic frequency-domain solutions for a class of hyperbolic equations with singular convolution operators are derived. Asymptotic solutions for this class of equations involve additional parameters — called attenuation parameters — which control the smoothing of the wavefield at the wavefront. At caustics the ray amplitudes have a singularity associated with vanishing of ray spreading and with divergence of an integral controlling the rate of exponential amplitude decay. Both problems are resolved by applying a generalized Kravtsov–Ludwig formula derived in this paper. A different asymptotic solution is constructed in the case of separation of dispersion and focusing effects.

scholarly journals Global-in-Time Asymptotic Solutions to Kolmogorov-Feller-Type Parabolic Pseudodifferential Equations with Small Parameter—Forward- and Backward-in-Time Motion

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
V. G. Danilov
Keyword(s):  
Cauchy Problem ◽  
Small Parameter ◽  
Asymptotic Solutions ◽  
Pseudodifferential Equations ◽  
Time Motion

We discuss the construction of solutions to the inverse Cauchy problem by using characteristics.

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