Abstract
We consider an identification problem for a stationary nonlinear convection–diffusion–reaction equation in which the reaction coefficient depends nonlinearly on the concentration of the substance. This problem is reduced to an inverse extremal problem by an optimization method. The solvability of the boundary value problem and the extremal problem is proved. In the case that the reaction coefficient is quadratic, when the equation acquires cubic nonlinearity, we deduce an optimality system. Analyzing it, we establish some estimates of the local stability of solutions to the extremal problem under small perturbations both of the quality functional and the given velocity vector which occurs multiplicatively in the convection–diffusion–reaction equation.
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References
A. N. Tikhonov and V. Ya. Arsenin, Solutions of Ill-Posed Problems (Nauka, Moscow, 1986; Halsted Press, New York, 1977).
G. I. Marchuk, Mathematical Models in Environmental Problems (Nauka, Moscow, 1982; North-Holland, Amsterdam, 1986).
O. M. Alifanov, E. A. Artyukhin, and C. V. Rumyantsev, ExtremeMethods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems (Nauka, Moscow, 1988; Begell House, New York, 1995).
K. Ito and K. Kunish, “Estimation of the Convection Coefficient in Elliptic Equations,” Inverse Problems 14, 995–1013 (1997).
V. I. Agoshkov, F. P. Minuk, A. S. Rusakov, and V. B. Zalesny. “Study and Solution of Identification Problems for Nonstationary 2D-and 3D-Convection-Diffusion-Reaction,” Russian J. Numer. Anal. Math. Model. 20, 19–43 (2005).
G. V. Alekseev and E. A. Adomavichus, “Theoretical Analysis of Inverse Extremal Problems of Admixture Diffusion in Viscous Fluid,” J. Inverse Ill-Posed Probl. 9, 435–468 (2001).
G. V. Alekseev, “Inverse Extremal Problems for Stationary Equations inMass Transfer Theory,” Zh. Vychisl. Mat. Mat. Fiz. 42 (3), 380–394 (2002) [Comp. Math. Math. Phys. 42 (3), 363–376 (2002)].
G. V. Alekseev and E. A. Kalinina, “Identification of the Lowest Coefficient of a Stationary Convection-Diffusion-Reaction Equation,” Sibirsk. Zh. Industr. Mat. 10 (29), 3–16 (2007).
G. V. Alekseev, O. V. Soboleva, and D. A. Tereshko, “Identification Problems for a Steady-State Model of Mass Transfer,” Prikl. Mekh. Tekhn. Fiz. No. 4, 24–35 (2008). [J. Appl. Mech. Tech. Phys. 49 (4), 537–547 (2008)].
G. V. Alekseev and D. A. Tereshko, Analysis and Optimization in Hydrodynamics of a Viscous Fluid (Dal’nauka, Vladivostok, 2008) [in Russian].
G. V. Alekseev, I. S. Vakhitov, and O. V. Soboleva, “Stability Estimates in Identification Problems for the Convection-Diffusion-Reaction Equation,” Zh. Vychisl. Mat. Mat. Fiz. 52 (12), 2190–2205 (2012) [Comp. Math. Math. Phys. 52 (12), 1635-1649 (2012)].
V. V. Penenko, “Variational Methods of Data Assimilation and Inverse Problems for Studying the Atmosphere, Ocean, and Environment,” Sibirsk. Zh. Vychisl. Mat. 2 (4), 421–434 (2009) [Numer. Anal. Appl. 2 (4), 341-351 (2009)].
E. V. Dement’eva, E. D. Karepova, and V. V. Shaidurov, “Reconstruction of a Boundary Function by Some Given Observations for a Problem of Surface Wave Propagation in a Water Area with Free Boundary,” Sibirsk. Zh. Industr. Mat. 10 (29), 3–16 (2007).
A. I. Korotkii and D. A. Kovtunov, “Boundary Regime Reconstruction in an Inverse Problem of Heat Convection of aHigh-Viscosity Fluid,” Trudy Inst. Mat. Mekh. Ural. Otdel. Ross. Akad. Nauk 12 (2), 88–97 (2006).
A. I. Korotkii and D. A. Kovtunov, “Optimal Boundary Control of a System Describing Heat Convection,” Trudy Inst. Mat. Mekh. Ural. Otdel. Ross. Akad. Nauk 12 (2), 88–97 (2006).
A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications (Nauchn. Kniga, Novosibirsk, 1999; AMS, Boston, MA, 2000).
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Original Russian Text © G.V. Alekseev, R.V. Brizitskii, Zh.Yu. Saritskaya, 2016, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2016, Vol. XIX, No. 2, pp. 3–16.
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Alekseev, G.V., Brizitskii, R.V. & Saritskaya, Z.Y. Stability estimates of solutions to extremal problems for a nonlinear convection-diffusion-reaction equation. J. Appl. Ind. Math. 10, 155–167 (2016). https://doi.org/10.1134/S1990478916020010
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DOI: https://doi.org/10.1134/S1990478916020010