Some features of solving an inverse backward problem for a generalized Burgers’ equation

Lukyanenko, D. V.; Prigorniy, Igor V.; Shishlenin, Maxim A.

doi:10.1515/jiip-2020-0078

Cited by 14 publications

(11 citation statements)

References 46 publications

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“…(t) we will call as "transition point" ("t.p."). Note that it is known from [50] that for the problem (1) the expressions for ϕ l (x) and ϕ r (x) can be written out explicitly:…”

Section: Statement Of the Inverse Problem And A Gradient Methods Of Itmentioning

confidence: 99%

“…The function x = x t.p. (t), which describes the position of the reaction front, can be found as a solution to, for example, the following functional equation [50]:…”

Section: Statement Of the Inverse Problem And A Gradient Methods Of Itmentioning

confidence: 99%

“…Remark 1. Note that the features of solving the retrospective inverse problem (an inverse backward problem) for (1) were considered in [50].…”

Section: Statement Of the Inverse Problem And A Gradient Methods Of Itmentioning

confidence: 99%

“…In the proposed formulation, it is impossible to extract a priori information about the solution of the inverse problem using the methods of asymptotic analysis [2,47], because the data of the inverse problem do not contain information about the reaction front at all times, including the initial time. This significantly distinguishes this work from the previous works of the authors [48][49][50]. Now we consider the formulation of the inverse problem, in which the requirement for the presence of an experimentally distinguishable reaction front at the initial moment of time is not required.…”

Section: Introductionmentioning

confidence: 99%

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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

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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“…(t) we will call as "transition point" ("t.p."). Note that it is known from [50] that for the problem (1) the expressions for ϕ l (x) and ϕ r (x) can be written out explicitly:…”

Section: Statement Of the Inverse Problem And A Gradient Methods Of Itmentioning

confidence: 99%

“…The function x = x t.p. (t), which describes the position of the reaction front, can be found as a solution to, for example, the following functional equation [50]:…”

Section: Statement Of the Inverse Problem And A Gradient Methods Of Itmentioning

confidence: 99%

“…Remark 1. Note that the features of solving the retrospective inverse problem (an inverse backward problem) for (1) were considered in [50].…”

Section: Statement Of the Inverse Problem And A Gradient Methods Of Itmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

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

Self Cite

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“…This work is a continuation of [36,[38][39][40][41]. In these works, questions were considered about the possibility of applying the methods of asymptotic analysis [3,[42][43][44] to recover some coefficients in inverse problems for nonlinear singularly perturbed reaction-diffusionadvection equations with data on the position of a reaction front.…”

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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Multidimensional thermal structures in the singularly perturbed stationary models of heat and mass transfer with a nonlinear thermal diffusion coefficient

Давыдова

Zakharova

2022

Journal of Computational and Applied Mathematics

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Some features of solving an inverse backward problem for a generalized Burgers’ equation

Cited by 14 publications

References 46 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

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

Multidimensional thermal structures in the singularly perturbed stationary models of heat and mass transfer with a nonlinear thermal diffusion coefficient

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