Enhanced and restored signals as a generalized solution for shock filter models. I: Existence and uniqueness result of the Cauchy problem
type: Item
instance of: scholarly article
title: Enhanced and restored signals as a generalized solution for shock filter models. I: Existence and uniqueness result of the Cauchy problem
zbMATH Open document ID: 1023.35024
DOI: 10.1016/S0022-247X(02)00713-8
author: L. Remaki
author: Mohamed Cheriet
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-05-25T00:00:00Z
review text: In this paper the authors propose generalized one- and two-dimensional shock models for signal enhancement and restoration. In the one-dimensional case the following quasilinear partial differential equation is proposed as shock filter model \[ u_{t}+a(x)F(u_{x^{2}},u_{x})\partial _{x}f(u)=0\text{ \;in \;}\mathbb{R}\times \mathbb{R}^{+}, \tag{1} \] and in the two-dimensional case \[ u_{t}+a_{1}(x,y)F_{1}(\Delta u,u_{x})\partial _{x}f_{1}(u)+a_{2}(x,y)F_{2}(\Delta u,u_{y})\partial _{y}f_{2}(u)=0\text{ \;in \;}\mathbb{R}^{2}\times \mathbb{R}^{+}, \tag{2} \] where the functions \(a,a_{1}\) and \(a_{2}\) are bounded and measurable and the functions \(f,f_{1},f_{2}\) and \(F,F_{1},F_{2}\) are smooth. The Cauchy problems for equations (1) and (2) are studied within the framework of Colombeau generalized functions. These Cauchy problems are rewritten as regularized partial differential equations. This interpretation consists of a regularization, with an adequate choice of mollifiers, of space derivatives. Then the authors prove theorems of existence and uniqueness of Colombeau generalized global solutions for these regularized partial differential equations.
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 46F30
zbMATH DE Number: 1915742
zbMATH Keywords: generalized functions
zbMATH Keywords: adequate choice of mollifiers
zbMATH Keywords: numerical schemes
zbMATH Keywords: signals enhancement and restoration
reviewed by: Chikh Bouzar
MaRDI profile type: Publication
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full work available at URL: s0022-247x(02)00713-8
OpenAlex ID: W2114101185
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description: scientific article; zbMATH DE number 1915742