scholarly journals 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

Mathematics ◽  
2021 ◽  
Vol 9 (18) ◽  
pp. 2342
Author(s):  
Raul Argun ◽  
Alexandr Gorbachev ◽  
Natalia Levashova ◽  
Dmitry Lukyanenko
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Asymptotic Analysis ◽  
Reaction Front ◽  
Nonlinear Integral Equation ◽  
Reaction Diffusion ◽  
Unknown Coefficient ◽  
Singularly Perturbed Equation ◽  
Numerical Reconstruction ◽  
Type Information

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.

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 ◽  
...  
Keyword(s):  
Inverse Problem ◽  
Reaction Front ◽  
Numerical Experiments ◽  
Reaction Diffusion ◽  
Singularly Perturbed ◽  
Initial Moment ◽  
Advection Equation ◽  
Initial Condition ◽  
Additional Information ◽  
The Cost

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.

scholarly journals Application of asymptotic analysis methods for solving a coefficient inverse problem for a system of nonlinear singularly perturbed reaction-diffusion equations with cubic nonlinearity

2019 ◽  
pp. 363-377
Author(s):  
Д.В. Лукьяненко ◽  
А.А. Мельникова
Keyword(s):  
Inverse Problem ◽  
Asymptotic Analysis ◽  
Numerical Study ◽  
Reaction Diffusion ◽  
Reaction Diffusion Equations ◽  
Singularly Perturbed ◽  
Algebraic Equations ◽  
Cubic Nonlinearity ◽  
Coefficient Inverse Problem ◽  
Analysis Methods

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.

Fast numerical method of solving 3D coefficient inverse problem for wave equation with integral data

2018 ◽  
Vol 26 (4) ◽  
pp. 477-492 ◽  
Author(s):  
Anatoly B. Bakushinsky ◽  
Alexander S. Leonov
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Wave Equation ◽  
Wave Field ◽  
A Priori ◽  
Three Dimensional ◽  
Simulated Data ◽  
Three Dimensions ◽  
Unknown Coefficient ◽  
Computational Resources

Abstract An inverse coefficient problem for time-dependent wave equation in three dimensions is under consideration. We are looking for a spatially varying coefficient of this equation knowing special time integrals of the wave field in an observation domain. The inverse problem has applications to the reconstruction of the refractive index of an inhomogeneous medium, as well as to acoustic sounding, medical imaging, etc. In the article, a new linear three-dimensional Fredholm integral equation of the first kind is introduced from which it is possible to find the unknown coefficient. We present and substantiate a numerical algorithm for solving this integral equation. The algorithm does not require significant computational resources and a long solution time. It is based on the use of fast Fourier transform under some a priori assumptions about unknown coefficient and observation region of the wave field. Typical results of solving this three-dimensional inverse problem on a personal computer for simulated data demonstrate high capabilities of the proposed algorithm.

Asymptotic analysis of the American call option with dividends

2002 ◽  
Vol 13 (6) ◽  
pp. 587-616 ◽  
Author(s):  
CHARLES KNESSL
Keyword(s):  
Integral Equation ◽  
Asymptotic Analysis ◽  
Nonlinear Integral Equation ◽  
Perturbation Methods ◽  
Asset Price ◽  
Call Option ◽  
Optimal Exercise Boundary ◽  
Exercise Boundary ◽  
Expiration Time

We consider an American call option and let C(S, T0) be the price of an option corresponding to asset price S at some time T0 prior to the expiration time TF . We analyze C(S, T0) in various asymptotic limits. These include situations where the interest and dividend rates are large or small, compared to the volatility of the asset. We also analyze the optimal exercise boundary for the option. We use perturbation methods to analyze either the PDE that C(S, T0) satisfies, or a nonlinear integral equation that is satisfied by the optimal exercise boundary.

scholarly journals 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

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2894
Author(s):  
Raul Argun ◽  
Alexandr Gorbachev ◽  
Dmitry Lukyanenko ◽  
Maxim Shishlenin
Keyword(s):  
Differential Equation ◽  
Inverse Problem ◽  
Approximate Solution ◽  
Inverse Problems ◽  
Reaction Front ◽  
Blow Up ◽  
Reaction Diffusion ◽  
Singularly Perturbed ◽  
Singularly Perturbed Equations ◽  
Coefficient Inverse Problems

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.

On existence result of a class of nonlinear integral equation

Filomat ◽  
2017 ◽  
Vol 31 (11) ◽  
pp. 3593-3597
Author(s):  
Ravindra Bisht
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Existence And Uniqueness ◽  
Existence Result ◽  
Nonlinear Integral Equation ◽  
Continuous Functions ◽  
Concave Functions ◽  
Real Numbers ◽  
Fixed Point Techniques ◽  
Uniqueness Of A Solution

Combining the approaches of functionals associated with h-concave functions and fixed point techniques, we study the existence and uniqueness of a solution for a class of nonlinear integral equation: x(t) = g1(t)-g2(t) + ? ?t,0 V1(t,s)h1(s,x(s))ds + ? ?T,0 V2(t,s)h2(s,x(s))ds; where C([0,T];R) denotes the space of all continuous functions on [0,T] equipped with the uniform metric and t?[0,T], ?,? are real numbers, g1, g2 ? C([0, T],R) and V1(t,s), V2(t,s), h1(t,s), h2(t,s) are continuous real-valued functions in [0,T]xR.

Asymptotic analysis of Complex Automata models for reaction–diffusion systems

2009 ◽  
Vol 59 (8) ◽  
pp. 2023-2034 ◽  
Author(s):  
Alfonso Caiazzo ◽  
Jean-Luc Falcone ◽  
Bastien Chopard ◽  
Alfons G. Hoekstra
Keyword(s):  
Asymptotic Analysis ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Diffusion Systems

scholarly journals Fejér monotonicity and fixed point theorems with applications to a nonlinear integral equation in complex valued Banach spaces

2020 ◽  
Vol 21 (1) ◽  
pp. 135
Author(s):  
Godwin Amechi Okeke ◽  
Mujahid Abbas
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Banach Spaces ◽  
Nonlinear Integral Equation ◽  
Metric Spaces ◽  
Banach Algebras ◽  
Cone Metric Spaces ◽  
Nonlinear Operators ◽  
Complex Valued ◽  
Cone Metric

It is our purpose in this paper to prove some fixed point results and Fej´er monotonicity of some faster fixed point iterative sequences generated by some nonlinear operators satisfying rational inequality in complex valued Banach spaces. We prove that results in complex valued Banach spaces are valid in cone metric spaces with Banach algebras. Furthermore, we apply our results in solving certain mixed type VolterraFredholm functional nonlinear integral equation in complex valued Banach spaces.

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