On uniqueness in the problems of determining point sources in mathematical models of heat and mass transfer

On uniqueness in the problems of determining point sources in mathematical models of heat and mass transfer

Автор: Neustroeva L.V.

Журнал: Вестник Южно-Уральского государственного университета. Серия: Математика. Механика. Физика @vestnik-susu-mmph

Рубрика: Математика

Статья в выпуске: 2 т.14, 2022 года.

Бесплатный доступ

We consider the problem of determining point sources for mathematical models of heat and mass transfer. The values of a solution (concentrations) at some points lying inside the domain are taken as overdetermination conditions. A second-order parabolic equation is considered, on the right side of which there is a linear combination of the Dirac delta functions δ(x-xi) with coefficients that depend on time and characterize the intensities of sources. Several different problems are considered, including the problem of determining the intensities of sources if their locations are given. In this case, we present the theorem of uniqueness of solutions, the proof of which is based on the Phragmén-Lindelöf theorem. Next, in the model case, we consider the problem of simultaneous determining the intensities of sources and their locations. The conditions on the number of measurements (the ovedetermination conditions) are described which ensure that a solution is uniquely determined. Examples are given to show the accuracy of the results. This problem arises when solving environmental problems, first of all, the problems of determining the sources of pollution in a water basin or atmosphere. The results are important when developing numerical algorithms for solving the problem. In the literature, such problems are solved numerically by reducing the problem to an optimal control problem and minimizing the corresponding objective functional. The examples show that this method is not always correct since the objective functional can have a significant number of minima.

Еще

Eat and mass transfer, parabolic equation, uniqueness, inverseproblem, point source

Короткий адрес: https://sciup.org/147237460

IDR: 147237460

Список литературы On uniqueness in the problems of determining point sources in mathematical models of heat and mass transfer

  • Marchuk G.I. Mathematical Models in Environmental Problems. Studies in Mathematics and its Applications. Elsevier Science Publishers, Amsterdam, 1986, Vol. 16, 216 p.
  • Ozisik M.N., Orlande H.R.B. Inverse Heat Transfer: Fundamentals and Applications. Taylor & Francis, New York, 2000, 330 p.
  • Alifanov O.M., Artyukhin E.A., Nenarokomov A.V. Obratnye zadachi v issledovanii slozhnogo teploobmena (Inverse problems of complex heat transfer). Moscow, Yanus-K Publ., 2009, 299 p. (in Russ).
  • Panasenko E.A., Starchenko A.V. Numerical Solution of Some Inverse Problems with Various Types of Sources of Atmospheric Pollution. Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2008, no. 2(3), pp. 47-55. (in Russ.).
  • Penenko V.V. Variational methods of data assimilation and inverse problems for studying the atmosphere, ocean, and environment. Numerical Analysis and Applications, 2009, Vol. 2, Iss. 4, pp. 341-351.
  • Yang, C.Y. Solving the two-dimensional inverse heat source problem through the linear least-squares error method. International Journal of Heat and Mass Transfer, 1998, Vol. 41, no. 2, pp. 393398. DOI: 10.1016/S0017-9310(97)00125-7
  • Starchenko A.V., Panasenko E.A. Parallel Algorithms for the Decision of Return Problems of Carrying over of the Impurity. Vestnik USATU, 2010, Vol. 14, no. 5(40), pp. 133-139. (in Russ.).
  • Mamonov A. V., Tsai Y-H. R. Point source identification in nonlinear advection-diffusion-reaction systems. Inverse Problems, 2013, Vol. 29, no. 3, p. 26. DOI: 10.1088/0266-5611/29/3/035009
  • Deng X., Zhao Y., Zou J. On linear finite elements for simultaneously recovering source location and intensity. Int. J. Numer. Anal. Model, 2013, Vol. 10, no. 3, pp. 588-602.
  • Verdiere N., Joly-Blanchard G., Denis-Vidal L. Identifiability and Identification of a Pollution Source in a River by Using a Semi-Discretized Model. Applied Mathematics and Computation, 2013, vol. 221, pp. 1-9. DOI: 10.1016/j.amc.2013.06.022
  • Mazaheri M., Samani J.M.V., Samani H.M.V. Mathematical Model for Pollution Source Identification in Rivers. Environmental Forensics, 2015, Vol. 16, Iss. 4, pp. 310-321. DOI: 10.1080/15275922.2015.1059391
  • Su J. Heat Source Estimation with the Conjugate Gradient Method in Inverse Linear Diffusive Problems. J. Braz. Soc. Mech. Sci., 2001, Vol. 23, no. 3, pp. 321-334. DOI: 10.1590/s0100-73862001000300005
  • Neto A.J.S., Oziik M.N. Twodimensional inverse heat conduction problem of estimating the timevarying strength of a line heat source. Journal of Applied Physics, 1992, Vol. 71, Iss. 11, pp. 53-57. DOI: 10.1063/1.350554
  • Milnes E., Perrochet P. Simultaneous Identification of a Single Pollution Point-Source Location and Contamination Time under Known Flow Field Conditions. Advances in Water Resources, 2007, Vol. 30, iss. 12, pp. 2439-2446. DOI: 10.1016/j.advwatres.2007.05.013
  • Liu F.B. A modified genetic algorithm for solving the inverse heat transfer problem of estimating plan heat source. International Journal of Heat and Mass Transfer, 2008, Vol. 51, Iss. 15-16, pp. 3745-3752. DOI: 10.1016/j.ijheatmasstransfer.2008.01.002
  • Penenko A.V., Rachmetullina S. Algorithms for Atmospheric Emission Source Localization Based on the Automated Ecological Monitoring System Data. Siberian Electronic Mathematical Reports. Proceedings of conferences, 2013, Vol. 10, pp. S35-S54. (in Russ.).
  • Badia A. El, Ha-Duong T., Hamdi A. Identification of a Point Source in a Linear Advection-Dispersion-Reaction Equation: Application to a Pollution Source Problem. Inverse Problems, 2005, Vol. 21, no. 3, pp. 1121-1136. DOI: 10.1088/0266-5611/21/3/020
  • Badia A. El, Hamdi A. Inverse Source Problem in an Advection-Dispersion-Reaction System: Application to Water Pollution. Inverse Problems, 2007, Vol. 23, no. 5, pp. 2103-2120. DOI: 10.1088/0266-5611/23/5/017
  • Badia A. El, Ha-Duong T. Inverse Source Problem for the Heat Equation: Application to a Pollution Detection Problem. J. Inverse Ill-Posed Probl., 2002, Vol. 10, Iss. 6, pp. 585-599. DOI: 10.1515/jiip.2002.10.6.585
  • Badia A. El, Ha-Duong T. An inverse Source Problem in Potential Analysis. Inverse Problems, 2000, Vol. 16, Iss. 3, pp. 651-663. DOI: 10.1088/0266-5611/16/3/308
  • Ling L., Takeuchi T. Point Sources Identification Problems for Heat Equations. Commun. Comput. Phys., 2009, Vol. 5, no. 5, pp. 897-913.
  • Pyatkov S.G., Safonov E.I. Point Sources Recovering Problems for the One-Dimensional Heat Equation. Journal of Advanced Research in Dynamical and Control Systems, 2019, Vol. 11, Iss. 01, pp. 496-510. http://www.jardcs.org/abstract.php?id=100#
  • Pyatkov S.G., Neustroeva L.V. On Some Asymptotic Representations of Solutions to Elliptic Equations and Their Applications. Complex Variables and Elliptic Equations, 2021, Vol. 66, no. 6-7, pp. 964-987. DOI: 10.1080/17476933.2020.1801656
  • Triebel H. Interpolation Theory. Function Spaces. Differential Operators. Berlin: VEB Deutscher Verlag der Wissenschaften, 1978, 528 p.
  • Amann H. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems. Function Spaces, Differential Operators and Nonlinear Analysis. Teubner-Texte Math. Bd. 133. Stuttgart: Teubner, 1993, pp. 9-126. DOI: 10.1007/978-3-663-11336-2_1
  • Neustroeva L.V., Pyatkov S.G. On recovering a point source in some heat and mass transfer problems. AIP Conference Proceedings, 2021, Vol. 2328, p. 020006. DOI: 10.1063/5.0042357
  • Denk R., Hieber R.M., Prüss J. R-boundedness, Fourier multipliers and problems of elliptic and parabolic type. Mem. Amer. Math. Soc., 2003, Vol. 166, no. 788. DOI: 10.1090/memo/0788
  • Hsu Y-K., Holsen T.M., Hopke P.K. Comparison of hybrid receptor models to locate PCB sources in Chicago. Atmospheric Environment, 2003, Vol. 37, pp. 545-562.
  • Safonov E., Pyatkov S. On Some Classes of Inverse Problems on Determining the Source Function. Proc. of the 8th Scientific Conference on Information Technologies for Intelligent Decision Making Support (ITIDS 2020). pp. 242-248. DOI: 10.2991/aisr.k.201029.047
  • Vladimirov V.S., Zharinov V.V. Uravneniya matematicheskoy fiziki: ucheb. dlya studentov vuzov (Equations of Mathematical Physics: Textbook for University Students). Moscow Fizmatlit Publ., 2004, 398 p. (in Russ.).
  • Sveshnikov A.G., Bogolyubov A.N., Kravtsov V.V. Lektsii po matematicheskoy fizike (Lectures on Mathematical Physics). Moscow, MGU Publ., 1993, 351 p. (in Russ.).
  • Watson G.N. A Treatise on the Theory of Bessel Functions. Cambridge: Cambridge University Press, 1944, 804 p.
  • Kozhevnikova M.F., Levenets V.V., Rolik I.L. The Pollution Source Identification: Computational Approach. Problems of Atomic Science and Technology, 2011, no. 6, pp. 149-156.
  • Zhou L., Hopke P.K., Liu W. Comparison of Two Trajectory Based Models for Locating Particle Sources for Two Rural New York Sites. Atmospheric Environment, 2004, Vol. 38, Iss. 13, pp. 19551963. DOI: 10.1016/j.atmosenv.2003.12.034
  • Han Y.-J., Holsen T.M., Hopke P.K., Cheong J.-P., Kim H., Yi S.-M. Identification of Source Location for Atmospheric Dry Deposition of Heavy Metals During Yellow-Sand Events in Seoul, Korea in 1998 Using Hybrid Receptor Models. Atmospheric Environment, 2004, Vol. 38, pp. 5353-5361. DOI: 10.1016/j.atmosenv.2004.02.069
  • Pekney N.J., Davidson C.I., Zhow L., Hopke P.K. Application of PSCF and CPF to PMF-Modeled Sources of PM25 in Pittsburgh. Aerosol Science and Technology, 2006, Vol. 40, Iss. 10, pp. 952-961. DOI: 10.1080/02786820500543324
  • Tichmarsh E.C. Theory of functions. Oxford, Oxford University press, 1939, 454 p.
Еще
Статья научная
SciUp 2024 © 2024 ©