Existence and stability of periodic contrast structure in reaction-advection-diffusion equation with discontinuous reactive and convective terms

Wu, Xiao; Ni, Mingkang

doi:10.1016/j.cnsns.2020.105457

Cited by 14 publications

(3 citation statements)

References 39 publications

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“…This paper discusses the inverse problem of numerical recovering of the initial condition for a nonlinear singularly perturbed reaction-diffusion-advection equation with data on the position of a reaction front, measured in an experiment with a delay relative to the initial time. Problems for equations of this type arise in gas dynamics [1], chemical kinetics [2][3][4][5][6], nonlinear wave theory [7], biophysics [8][9][10][11][12], medicine [13][14][15][16], ecology [17][18][19] and other fields of science [20]. A feature of this type of problem is the presence of multiscale processes.…”

Section: Introductionmentioning

confidence: 99%

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

et al. 2021

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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.

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Section: Introductionmentioning

confidence: 99%

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

et al. 2021

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show abstract

“…Problems for nonlinear singularly perturbed reaction-diffusion-advection equations arise in gas dynamics [1], combustion theory [2], chemical kinetics [3][4][5][6][7][8][9][10], nonlinear wave theory [11], biophysics [12][13][14][15][16], medicine [17][18][19][20], ecology [21][22][23][24][25], finance [26] and other fields of science [27]. A specific feature of problems of this type is the presence of processes of different scales.…”

Section: Introductionmentioning

confidence: 99%

Inverse Problem for an Equation of the Reaction-Diffusion-Advection Type with Data on the Position of a Reaction Front: Features of the Solution in the Case of a Nonlinear Integral Equation in a Reduced Statement

et al. 2021

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The paper considers the features of numerical reconstruction of the advection coefficient when solving the coefficient inverse problem for a nonlinear singularly perturbed equation of the reaction-diffusion-advection type. Information on the position of a reaction front is used as data of the inverse problem. An important question arises: is it possible to obtain a mathematical connection between the unknown coefficient and the data of the inverse problem? The methods of asymptotic analysis of the direct problem help to solve this question. But the reduced statement of the inverse problem obtained by the methods of asymptotic analysis contains a nonlinear integral equation for the unknown coefficient. The features of its solution are discussed. Numerical experiments demonstrate the possibility of solving problems of such class using the proposed methods.

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“…Nonlinear singularly perturbed reaction-diffusion-advection equations arise when solving the problems in gas dynamics [1], combustion theory [2], chemical kinetics [3][4][5][6][7][8][9], nonlinear wave theory [10], biophysics [11][12][13][14][15], medicine [16][17][18][19], ecology [20][21][22], finance [23], and other areas of science [24]. Inverse problems for the equation of this type often arise when solving various applied problems and consist of recovering some coefficient in the equation.…”

Section: Introductionmentioning

confidence: 99%

On Some Features of the Numerical Solving of Coefficient Inverse Problems for an Equation of the Reaction-Diffusion-Advection-Type with Data on the Position of a Reaction Front

et al. 2021

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The work continues a series of articles devoted to the peculiarities of solving coefficient inverse problems for nonlinear singularly perturbed equations of the reaction-diffusion-advection-type with data on the position of the reaction front. In this paper, we place the emphasis on some problems of the numerical solving process. One of the approaches to solving inverse problems of the class under consideration is the use of methods of asymptotic analysis. These methods, under certain conditions, make it possible to construct the so-called reduced formulation of the inverse problem. Usually, a differential equation in this formulation has a lower dimension/order with respect to the differential equation, which is included in the full statement of the inverse problem. In this paper, we consider an example that leads to a reduced formulation of the problem, the solving of which is no less a time-consuming procedure in comparison with the numerical solving of the problem in the full statement. In particular, to obtain an approximate numerical solution, one has to use the methods of the numerical diagnostics of the solution’s blow-up. Thus, it is demonstrated that the possibility of constructing a reduced formulation of the inverse problem does not guarantee its more efficient solving. Moreover, the possibility of constructing a reduced formulation of the problem does not guarantee the existence of an approximate solution that is qualitatively comparable to the true one. In previous works of the authors, it was shown that an acceptable approximate solution can be obtained only for sufficiently small values of the singular parameter included in the full statement of the problem. However, the question of how to proceed if the singular parameter is not small enough remains open. The work also gives an answer to this question.

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Existence and stability of periodic contrast structure in reaction-advection-diffusion equation with discontinuous reactive and convective terms

Cited by 14 publications

References 39 publications

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

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

Inverse Problem for an Equation of the Reaction-Diffusion-Advection Type with Data on the Position of a Reaction Front: Features of the Solution in the Case of a Nonlinear Integral Equation in a Reduced Statement

On Some Features of the Numerical Solving of Coefficient Inverse Problems for an Equation of the Reaction-Diffusion-Advection-Type with Data on the Position of a Reaction Front

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