Geodesic
On the solution of an inverse problem simulating two-dimensional motion of a viscous fluid
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 9 (2016) no. 4, pp. 5-16 Cet article a éte moissonné depuis la source Math-Net.Ru

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An inverse initial boundary value problem for a linear parabolic equation that arises as a result of mathematical modelling of 2D creeping motion of viscous liquid in a flat channel is considered. The unknown function of time is added in the right part of equation and can be found from additional condition of integral overdetermination. This problem has two different integral identities, permitting to obtain a priori estimates of solutions in uniform metric and to proof the uniqueness theorem. Under some restrictions on input data the solution is constructed as a series in the special basis. For this purpose the problem is reduced by differentiation with respect to the spatial variable to a direct non-classic problem with two integral conditions instead of ordinary ones. The new problem is solved by separation of variables, which allows one to find the unknown functions in the form of rapidly converging series. Another method for solving the initial problem is to reduce the problem to the loaded equation and to state the first initial boundary value problem for this equation. In its turn, this problem is reduced to one-dimensional in time Volterra operator equation with a special kernel. It is proved that it has a series solution. Some auxiliary formulas which are useful for the numerical solution of this equation by the Laplace transform are obtained. Sufficient conditions under which the solution with increasing time converges to steady regime by exponential law are established.
Keywords: inverse problem; a priori assessment; Laplace transform; exponential stability. 
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V. K. Andreev. On the solution of an inverse problem simulating two-dimensional motion of a viscous fluid. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 9 (2016) no. 4, pp. 5-16. http://geodesic.mathdoc.fr/item/VYURU_2016_9_4_a0/

[1] V. K. Andreev, “Unsteady 2D Motions a Viscous Fluid Described by Partially Invariant Solutions to the Navier–Stokes Equations”, Journal of Siberian Federal University. Mathematics and Physics, 8:2 (2015), 140–147 | DOI | MR

[2] V. K. Andreev, “On an Inverse Problem for Two-Dimensional Navier–Stokes Equation”, Abstracts of the International Conference “Differential Equations and Mathematical Modeling” (Ulan-Ude, 2015), 44–45

[3] V. K. Andreev, Yu. A. Gaponenko, O. N. Goncharova et al., Mathematical Models of Convection, Walter de Gruyter GmbH Co. KG, Berlin, 2012, 417 pp. | DOI | MR

[4] A. I. Prilepko, D. G. Orlovsky, I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, N.-Y., 1999 | MR

[5] J. R. Cannot, Y. Lin, “Determination of a Parameter $p(t)$ in Some Quasi-Linear Parabolic Differential Equations”, Inverse Problems, 4 (1988), 35–45 | DOI | MR

[6] Vasin I. A., “On Some Inverse Problems of the Dynamics of a Viscous Incompressible Fluid in the Case of Integral Overdetermination”, Computational Mathematics and Mathematical Physics, 32:7 (1992), 955–963 | MR | Zbl

[7] Vasin I. A., Kamynin V. L., “On the Asymptotic Behavior of the Solutions of Inverse Problems for Parabolic Equations”, Siberian Mathematical Journal, 38:4 (1997), 647–662 | DOI | MR | Zbl

[8] Kozhanov A. I., “Parabolic Equations with an Unknown Time-Dependent Coefficient”, Computational Mathematics and Mathematical Physics, 45:12 (2005), 2085–2101 | MR | Zbl

[9] Pyatkov S. G., Safonov E. I., “On Some Classes of Linear Inverse Problems for Parabolic Equations”, Siberian Electronic Mathematical Reports, 11 (2014), 777–799 (in Russian) | Zbl

[10] Olver F., Introduction to Asymptotics and Special Functions, Nauka, M., 1978, 375 pp.

[11] Michlin S. G., Linear Partial Differential Equations, Vysshaia shkola, M., 1977, 431 pp.

[12] Ilyin V. A., “On the Solubility of Mixed Problems for Hyperbolic and Parabolic Equations”, Russian Mathematical Surveys, 15:2(92) (1960), 97–154 (in Russian) | Zbl

[13] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integrals and Series, Nauka, M., 1981, 800 pp.

[14] Lavrentyev M. A., Shabat B. V., Methods of Theory of Complex Variable Functions, Nauka, M., 1973, 736 pp.

[15] Doetsch G., Anleitung zum praktischen Gebrauch der Laplace-Transformation, Nauka, M., 1965, 288 pp.

[16] Polyanin A. D., Manzhirov A. V., Handbook of Integral Equations, Factorial Press, M., 2000, 384 pp.