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
Abstract
:1. Introduction
2. Statement of the Inverse Problem
3. Construction of the Reduced Statement of the Inverse Problem Using the Asymptotic Analysis Methods
- (A1)
- Let the initial function of the problem (1) have the form of a formed front with a large gradient in the vicinity of the point . Let the initial and boundary conditions be consistent to continuity: , .
- (A2)
- Let the inequalities , be satisfied everywhere on the segment .
4. Numerical Algorithm for Solving the Inverse Problem in the Reduced Statement
- It is assumed that we know the grid values , , the experimentally measured function on the grid of the time variable .
- Smooth the function given by a set of grid values , , using the cubic smoothing spline . The spline minimizes the functionalThis method is well known and implemented in many software packages, so we will not describe its numerical implementation in this article. The value of the smoothing parameter p must be consistent with the error of the input data . For example, the parameter p can be selected based on the generalized residual principle [46]:Next, we will redefine .
- Calculate the grid values of , , which are derivatives of the smoothed function , according to formulas with the second order of accuracy:As a result, we will define the function as a set of its grid values , , on the grid of temporary variable . On the other hand, each moment of time corresponds to the value , which determines the position of the reaction front at this moment of time (here we proceed from the assumption of the monotonicity of the function ). Thus, we can define the function as a set of its grid values , , already on a non-uniform grid of spatial variable with nodes , [39].
- It is assumed that a uniform grid is introduced with respect to the spatial variable . Let us determine the values of , , by interpolating the function of one variable , given by its grid values , , on a grid with nodes , . Here , . (see Figure 3).
- Let us write the Equation (14) for all grid nodes by spatial variable. However, this can be done only for those nodes of the grid in which the experimental observation of the reaction front movement was carried out, i.e., in nodes with indices . Only at these nodes we know the grid values of . Thus we getHere .Since the system (15) contains equations, we can determine from it no more than unknowns, namely only for . The system (15) can be rewritten in the following form, using only the specified grid values :This nonlinear system can be rewritten in the following form
- The problem of determining the vector of unknown Q is reduced to the search for the extremal of the functional
5. Numerical Experiments
6. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Conflicts of Interest
References
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Argun, R.; Gorbachev, A.; Levashova, N.; Lukyanenko, D. 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, 9, 2342. https://doi.org/10.3390/math9182342
Argun R, Gorbachev A, Levashova N, Lukyanenko D. 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; 9(18):2342. https://doi.org/10.3390/math9182342
Chicago/Turabian StyleArgun, Raul, Alexandr Gorbachev, Natalia Levashova, and Dmitry Lukyanenko. 2021. "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 9, no. 18: 2342. https://doi.org/10.3390/math9182342
APA StyleArgun, R., Gorbachev, A., Levashova, N., & Lukyanenko, D. (2021). 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, 9(18), 2342. https://doi.org/10.3390/math9182342