scholarly journals 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 ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 342
Author(s):  
Dmitry Lukyanenko ◽  
Tatyana Yeleskina ◽  
Igor Prigorniy ◽  
Temur Isaev ◽  
Andrey Borzunov ◽  
...  

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.

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2894
Author(s):  
Raul Argun ◽  
Alexandr Gorbachev ◽  
Dmitry Lukyanenko ◽  
Maxim Shishlenin

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.


2020 ◽  
Vol 28 (5) ◽  
pp. 641-649
Author(s):  
Dmitry V. Lukyanenko ◽  
Igor V. Prigorniy ◽  
Maxim A. Shishlenin

AbstractIn this paper, we consider an inverse backward problem for a nonlinear singularly perturbed parabolic equation of the Burgers’ type. We demonstrate how a method of asymptotic analysis of the direct problem allows developing a rather simple algorithm for solving the inverse problem in comparison with minimization of the cost functional. Numerical experiments demonstrate the effectiveness of this approach.


Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2342
Author(s):  
Raul Argun ◽  
Alexandr Gorbachev ◽  
Natalia Levashova ◽  
Dmitry Lukyanenko

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.


Author(s):  
Д.В. Лукьяненко ◽  
А.А. Мельникова

Продемонстрированы возможности методов асимптотического анализа в применении к решению коэффициентной обратной задачи для системы нелинейных сингулярно возмущенных уравнений типа реакциядиффузия с кубической нелинейностью. Рассматриваемая в статье задача для системы уравнений в частных производных сводится к гораздо более простой для численного исследования системе алгебраических уравнений, которая связывает данные обратной задачи (информацию о положении фронта реакции во времени) с коэффициентом, который необходимо восстановить. Численные эксперименты подтверждают эффективность предложенного подхода. The capabilities of asymptotic analysis methods for solving a coefficient inverse problem for a system of nonlinear singularly perturbed equations of reactiondiffusion type with cubic nonlinearity are shown. The problem considered for a system of partial differential equations is reduced to a system of algebraic equations that is much simpler for a numerical study and relates the data of the inverse problem (the information on the position of the reaction front in time) with the coefficient to be recovered. Numerical results confirm the efficiency of the proposed approach.


2003 ◽  
Vol 3 (3) ◽  
pp. 417-423 ◽  
Author(s):  
Torsten Linss ◽  
Niall Madden

AbstractWe consider a central difference scheme for the numerical solution of a system of coupled reaction-diffusion equations. We show that the scheme is almost second-order convergent, uniformly in the perturbation parameter. We present the results of numerical experiments to confirm our theoretical results.


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