Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay

In this paper, approaches to the numerical recovering of the initial condition in the inverse problem for a nonlinear singularly perturbed reaction–diffusion–advection equation are considered. The feature of the formulation of the inverse problem is the use of additional information about the value...

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Main Authors:	Dmitry Lukyanenko, Tatyana Yeleskina, Igor Prigorniy, Temur Isaev, Andrey Borzunov, Maxim Shishlenin
Format:	Article
Language:	English
Published:	MDPI AG 2021-02-01
Series:	Mathematics
Subjects:	inverse problem of recovering the initial condition reaction–diffusion–advection equation inverse problem with data on the reaction front position
Online Access:	https://www.mdpi.com/2227-7390/9/4/342

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spelling	doaj-7dcace8410d44985ac722f972dfefbe82021-02-10T00:04:12ZengMDPI AGMathematics2227-73902021-02-01934234210.3390/math9040342Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time DelayDmitry Lukyanenko0Tatyana Yeleskina1Igor Prigorniy2Temur Isaev3Andrey Borzunov4Maxim Shishlenin5Department of Mathematics, Faculty of Physics, Lomonosov Moscow State University, 119991 Moscow, RussiaFaculty of Physics, Lomonosov Moscow State University, Baku Branch, Baku 1143, AzerbaijanDepartment of Mathematics, Faculty of Physics, Lomonosov Moscow State University, 119991 Moscow, RussiaDepartment of Mathematics, Faculty of Physics, Lomonosov Moscow State University, 119991 Moscow, RussiaDepartment of Mathematics, Faculty of Physics, Lomonosov Moscow State University, 119991 Moscow, RussiaInstitute of Computational Mathematics and Mathematical Geophysics of SB RAS, 630090 Novosibirsk, RussiaIn this paper, approaches to the numerical recovering of the initial condition in the inverse problem for a nonlinear singularly perturbed reaction–diffusion–advection equation are considered. The feature of the formulation of the inverse problem is the use of additional information about the value of the solution of the equation at the known position of a reaction front, measured experimentally with a delay relative to the initial moment of time. In this case, for the numerical solution of the inverse problem, the gradient method of minimizing the cost functional is applied. In the case when only the position of the reaction front is known, the method of deep machine learning is applied. Numerical experiments demonstrated the possibility of solving such kinds of considered inverse problems.https://www.mdpi.com/2227-7390/9/4/342inverse problem of recovering the initial conditionreaction–diffusion–advection equationinverse problem with data on the reaction front position
collection	DOAJ
language	English
format	Article
sources	DOAJ
author	Dmitry Lukyanenko Tatyana Yeleskina Igor Prigorniy Temur Isaev Andrey Borzunov Maxim Shishlenin
spellingShingle	Dmitry Lukyanenko Tatyana Yeleskina Igor Prigorniy Temur Isaev Andrey Borzunov Maxim Shishlenin Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay Mathematics inverse problem of recovering the initial condition reaction–diffusion–advection equation inverse problem with data on the reaction front position
author_facet	Dmitry Lukyanenko Tatyana Yeleskina Igor Prigorniy Temur Isaev Andrey Borzunov Maxim Shishlenin
author_sort	Dmitry Lukyanenko
title	Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay
title_short	Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay
title_full	Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay
title_fullStr	Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay
title_full_unstemmed	Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay
title_sort	inverse problem of recovering the initial condition for a nonlinear equation of the reaction–diffusion–advection type by data given on the position of a reaction front with a time delay
publisher	MDPI AG
series	Mathematics
issn	2227-7390
publishDate	2021-02-01
description	In this paper, approaches to the numerical recovering of the initial condition in the inverse problem for a nonlinear singularly perturbed reaction–diffusion–advection equation are considered. The feature of the formulation of the inverse problem is the use of additional information about the value of the solution of the equation at the known position of a reaction front, measured experimentally with a delay relative to the initial moment of time. In this case, for the numerical solution of the inverse problem, the gradient method of minimizing the cost functional is applied. In the case when only the position of the reaction front is known, the method of deep machine learning is applied. Numerical experiments demonstrated the possibility of solving such kinds of considered inverse problems.
topic	inverse problem of recovering the initial condition reaction–diffusion–advection equation inverse problem with data on the reaction front position
url	https://www.mdpi.com/2227-7390/9/4/342
work_keys_str_mv	AT dmitrylukyanenko inverseproblemofrecoveringtheinitialconditionforanonlinearequationofthereactiondiffusionadvectiontypebydatagivenonthepositionofareactionfrontwithatimedelay AT tatyanayeleskina inverseproblemofrecoveringtheinitialconditionforanonlinearequationofthereactiondiffusionadvectiontypebydatagivenonthepositionofareactionfrontwithatimedelay AT igorprigorniy inverseproblemofrecoveringtheinitialconditionforanonlinearequationofthereactiondiffusionadvectiontypebydatagivenonthepositionofareactionfrontwithatimedelay AT temurisaev inverseproblemofrecoveringtheinitialconditionforanonlinearequationofthereactiondiffusionadvectiontypebydatagivenonthepositionofareactionfrontwithatimedelay AT andreyborzunov inverseproblemofrecoveringtheinitialconditionforanonlinearequationofthereactiondiffusionadvectiontypebydatagivenonthepositionofareactionfrontwithatimedelay AT maximshishlenin inverseproblemofrecoveringtheinitialconditionforanonlinearequationofthereactiondiffusionadvectiontypebydatagivenonthepositionofareactionfrontwithatimedelay
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Inverse Problem of Recovering the Initial Condition for a Nonlinear Equation of the Reaction–Diffusion–Advection Type by Data Given on the Position of a Reaction Front with a Time Delay

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