Regularization of the Ill-Posed Cauchy Problem for Matrix Factorizations of the Helmholtz Equation on the Plane

In this paper, we present an explicit formula for the approximate solution of the Cauchy problem for the matrix factorizations of the Helmholtz equation in a bounded domain on the plane. Our formula for an approximate solution also includes the construction of a family of fundamental solutions for t...

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Main Authors:	Davron Aslonqulovich Juraev, Samad Noeiaghdam
Format:	Article
Language:	English
Published:	MDPI AG 2021-05-01
Series:	Axioms
Subjects:	cauchy problem regularization factorization regular solution fundamental solution
Online Access:	https://www.mdpi.com/2075-1680/10/2/82

id	doaj-6b3efe5f72f04d569f10e0f7c32e1f18
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spelling	doaj-6b3efe5f72f04d569f10e0f7c32e1f182021-05-31T23:06:18ZengMDPI AGAxioms2075-16802021-05-0110828210.3390/axioms10020082Regularization of the Ill-Posed Cauchy Problem for Matrix Factorizations of the Helmholtz Equation on the PlaneDavron Aslonqulovich Juraev0Samad Noeiaghdam1Higher Military Aviation School of the Republic of Uzbekistan, Karshi City 180100, UzbekistanIndustrial Mathematics Laboratory, Baikal School of BRICS, Irkutsk National Research Technical University, 664074 Irkutsk, RussiaIn this paper, we present an explicit formula for the approximate solution of the Cauchy problem for the matrix factorizations of the Helmholtz equation in a bounded domain on the plane. Our formula for an approximate solution also includes the construction of a family of fundamental solutions for the Helmholtz operator on the plane. This family is parameterized by function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>(</mo><mi>w</mi><mo>)</mo></mrow></semantics></math></inline-formula> which depends on the space dimension. In this paper, based on the results of previous works, the better results can be obtained by choosing the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>(</mo><mi>w</mi><mo>)</mo></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2075-1680/10/2/82cauchy problemregularizationfactorizationregular solutionfundamental solution
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Davron Aslonqulovich Juraev Samad Noeiaghdam
spellingShingle	Davron Aslonqulovich Juraev Samad Noeiaghdam Regularization of the Ill-Posed Cauchy Problem for Matrix Factorizations of the Helmholtz Equation on the Plane Axioms cauchy problem regularization factorization regular solution fundamental solution
author_facet	Davron Aslonqulovich Juraev Samad Noeiaghdam
author_sort	Davron Aslonqulovich Juraev
title	Regularization of the Ill-Posed Cauchy Problem for Matrix Factorizations of the Helmholtz Equation on the Plane
title_short	Regularization of the Ill-Posed Cauchy Problem for Matrix Factorizations of the Helmholtz Equation on the Plane
title_full	Regularization of the Ill-Posed Cauchy Problem for Matrix Factorizations of the Helmholtz Equation on the Plane
title_fullStr	Regularization of the Ill-Posed Cauchy Problem for Matrix Factorizations of the Helmholtz Equation on the Plane
title_full_unstemmed	Regularization of the Ill-Posed Cauchy Problem for Matrix Factorizations of the Helmholtz Equation on the Plane
title_sort	regularization of the ill-posed cauchy problem for matrix factorizations of the helmholtz equation on the plane
publisher	MDPI AG
series	Axioms
issn	2075-1680
publishDate	2021-05-01
description	In this paper, we present an explicit formula for the approximate solution of the Cauchy problem for the matrix factorizations of the Helmholtz equation in a bounded domain on the plane. Our formula for an approximate solution also includes the construction of a family of fundamental solutions for the Helmholtz operator on the plane. This family is parameterized by function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>(</mo><mi>w</mi><mo>)</mo></mrow></semantics></math></inline-formula> which depends on the space dimension. In this paper, based on the results of previous works, the better results can be obtained by choosing the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>K</mi><mo>(</mo><mi>w</mi><mo>)</mo></mrow></semantics></math></inline-formula>.
topic	cauchy problem regularization factorization regular solution fundamental solution
url	https://www.mdpi.com/2075-1680/10/2/82
work_keys_str_mv	AT davronaslonqulovichjuraev regularizationoftheillposedcauchyproblemformatrixfactorizationsofthehelmholtzequationontheplane AT samadnoeiaghdam regularizationoftheillposedcauchyproblemformatrixfactorizationsofthehelmholtzequationontheplane
_version_	1721418294865428480

Regularization of the Ill-Posed Cauchy Problem for Matrix Factorizations of the Helmholtz Equation on the Plane

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