Systems of elliptic pseudodifferential equations in a halfspace

2008 ◽  
pp. 165-216
Keyword(s):  
Pseudodifferential Equations

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.

Nonclassical semilinear boundary value problem for parabolic pseudodifferential equations in Sobolev spaces

2006 ◽  
Vol 74 (3) ◽  
pp. 874-877
Author(s):  
Yu. V. Egorov ◽  
Nguyen Minh Chuong ◽  
Dang Anh Tuan
Keyword(s):  
Boundary Value Problem ◽  
Sobolev Spaces ◽  
Boundary Value ◽  
Pseudodifferential Equations

Efficient asymptotics of solutions to the Cauchy problem with localized initial data for linear systems of differential and pseudodifferential equations

2021 ◽  
Vol 76 (5) ◽  
pp. 745-819
Author(s):  
S. Yu. Dobrokhotov ◽  
V. E. Nazaikinskii ◽  
A. I. Shafarevich
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Water Waves ◽  
Theory Of Elasticity ◽  
Extensive Literature ◽  
Pseudodifferential Equations ◽  
Tsunami Waves ◽  
The Cauchy Problem ◽  
Linear Differential ◽  
Localized Initial Data

Abstract We say that the initial data in the Cauchy problem are localized if they are given by functions concentrated in a neighbourhood of a submanifold of positive codimension, and the size of this neighbourhood depends on a small parameter and tends to zero together with the parameter. Although the solutions of linear differential and pseudodifferential equations with localized initial data constitute a relatively narrow subclass of the set of all solutions, they are very important from the point of view of physical applications. Such solutions, which arise in many branches of mathematical physics, describe the propagation of perturbations of various natural phenomena (tsunami waves caused by an underwater earthquake, electromagnetic waves emitted by antennas, etc.), and there is extensive literature devoted to such solutions (including the study of their asymptotic behaviour). It is natural to say that an asymptotics is efficient when it makes it possible to examine the problem quickly enough with relatively few computations. The notion of efficiency depends on the available computational tools and has changed significantly with the advent of Wolfram Mathematica, Matlab, and similar computing systems, which provide fundamentally new possibilities for the operational implementation and visualization of mathematical constructions, but which also impose new requirements on the construction of the asymptotics. We give an overview of modern methods for constructing efficient asymptotics in problems with localized initial data. The class of equations and systems under consideration includes the Schrödinger and Dirac equations, the Maxwell equations, the linearized gasdynamic and hydrodynamic equations, the equations of the linear theory of surface water waves, the equations of the theory of elasticity, the acoustic equations, and so on. Bibliography: 109 titles.

Sign in / Sign up

Export Citation Format

Share Document