Properties of Fundamental Solutions, Theorems on Integral Representations of Solutions and Correct Solvability of the Cauchy Problem for Ultraparabolic Kolmogorov-Type Equations with Two Groups of Space Variables of Degeneration

2021 ◽  
Author(s):  
S. D. Ivasyshen ◽  
I. P. Medynsky
Keyword(s):  
Cauchy Problem ◽  
Fundamental Solutions ◽  
Integral Representations ◽  
Kolmogorov Type ◽  
Correct Solvability ◽  
The Cauchy Problem

scholarly journals On regularization of the Cauchy problem for elliptic systems in weighted Sobolev spaces

2019 ◽  
Vol 27 (6) ◽  
pp. 815-834
Author(s):  
Yulia Shefer ◽  
Alexander Shlapunov
Keyword(s):  
Cauchy Problem ◽  
Sobolev Spaces ◽  
Integral Formula ◽  
Weighted Sobolev Spaces ◽  
Elliptic Systems ◽  
Fundamental Solutions ◽  
Type Integral ◽  
The Cauchy Problem ◽  
Ill Posed ◽  
Neumann Type

AbstractWe consider the ill-posed Cauchy problem in a bounded domain{\mathcal{D}}of{\mathbb{R}^{n}}for an elliptic differential operator{\mathcal{A}(x,\partial)}with data on a relatively open subsetSof the boundary{\partial\mathcal{D}}. We do it in weighted Sobolev spaces{H^{s,\gamma}(\mathcal{D})}containing the elements with prescribed smoothness{s\in\mathbb{N}}and growth near{\partial S}in{\mathcal{D}}, controlled by a real number γ. More precisely, using proper (left) fundamental solutions of{\mathcal{A}(x,\partial)}, we obtain a Green-type integral formula for functions from{H^{s,\gamma}(\mathcal{D})}. Then a Neumann-type series, constructed with the use of iterations of the (bounded) integral operators applied to the data, gives a solution to the Cauchy problem in{H^{s,\gamma}(\mathcal{D})}whenever this solution exists.

scholarly journals On a numerical solution of the Cauchy problem for the Laplace equation by the fundamental solutions method

PAMM ◽  
2007 ◽  
Vol 7 (1) ◽  
pp. 2040035-2040036
Author(s):  
Takashi Ohe ◽  
Katsu Yamatani ◽  
Kohzaburo Ohnaka
Keyword(s):  
Cauchy Problem ◽  
Numerical Solution ◽  
Laplace Equation ◽  
Fundamental Solutions ◽  
The Cauchy Problem
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