# About some classes of reverse tasks on determining the source function

Keywords:
second-order parabolic equation, inverse problem, initial-boundary value problem, well-posedness, unconditional solvability, existence, uniqueness

### Abstract

We examine the questions of solvability, uniqueness, and some qualitative properties of solutions of inverse problems of determining point sources (the right-hand side of a special form) in a one-dimensional parabolic equation. The values of the solution at a collection of points are taken as the overdetermination data. Examples are displayed showing that the uniqueness fails without additional conditions on the mutual location of sources and measurement points. An asymptotic formula for determining the coordinates of one source from two measurements is obtained.

### References

[1]

Marchuk G. I., Mathematical Models in Environmental Problems, Elsevier Sci. Publ., Amsterdam (1986) (Stud. Math. Appl.; V. 16).

[2]

Badia A. El, Ha-Duong T., and Hamdi A., “Identification of a point source in a linear advection-dispersion-reaction equation: application to a pollution source problem,” Inverse Probl., 21, No. 3, 1121–1136 (2005).

[3]

Badia A. El and Hamdi A., “Inverse source problem in an advection-dispersion-reaction system: application to water pollution,” Inverse Probl., 23, 2103–2120 (2007).

[4]

Boano F., Revelli R., and Ridolfi L., “Source identification in river pollution problems: a geostatistical approach,” Water Resources Res., 41, 1–13 (2005).

[5]

Fan Y. and Li D. G., “Identifying the heat source for the heat equation with convection term,” Int. J. Math. Anal., 3, No. 27, 1317–1323 (2009).

[6]

Liu C.-S. and Chang C.-W., “A global boundary integral equation method for recovering space-time dependent heat source,” Int. J. Heat Mass Transf., 92, 1034–1040 (2016).

[7]

Panasenko A. E. and Starchenko A. V. “Numerical solution of some inverse problems with different types of sources of the atmospheric pollution [in Russian],” Vestn. Tomsk. Gos. Univ., Mat., Mekh., No. 2, 47–55 (2008).

[8]

Penenko V. V., “Variational methods of data assimilation and inverse problems for studying the atmosphere, ocean, and environment,” Numer. Anal. Appl, 2, No. 4, 341–351 (2009).

[9]

Badia A. El and Ha-Duong T., “Inverse source problem for the heat equation: application to a pollution detection problem,” J. Inverse Ill-Posed Probl., 10, No. 6, 585–599 (2002).

[10]

Ling L. and Takeuchi T., “Point sources identification problems for heat equations,” Commun. Comput. Phys., 5, No. 5, 897–913 (2009).

[11]

Mamonov A. V. and Tsai Y.-H. R., “Point source identification in nonlinear advection-diffusion-reaction systems,” Inverse Probl., 29, No. 3, 035009 (2013).

[12]

Murray-Bruce J. and Dragotti P. L., “Estimating localized sources of diffusion fields using spatiotemporal sensor measurements,” IEEE Trans. Signal Process., 63, No. 12, 3018–3031 (2015).

[13]

Ozisik M. N. and Orlande H. R. B., Inverse Heat Transfer, Taylor & Francis, New York (2000).

[14]

Deng X., Zhao Y., and Zou J., “On linear finite elements for simultaneously recovering source location and intensity,” Int. J. Numer. Anal. Model., 10, No. 3, 588–602 (2013).

[15]

Pyatkov S. G. and Safonov E. I., “Point sources recovering problems for the one-dimensional heat equation,” Sib. Adv. Math., 27, No. 2, 119–132 (2017).

[16]

Verdiere N., Joly-Blanchard G., and Denis-Vidal L., “Identifiability and identification a pollution source in a river by using a semi-discretized scheme,” Appl. Math. Comput., 221, 1–9 (2013).

[17]

Kozhanov A. I., “Inverse problems of recovering the right-hand side of a special form in a parabolic equation,” Mat. Zametki SVFU, 23, No. 4, 31–45 (2016).

[18]

Pyatkov S. G. and Safonov E. I., “Point sources recovering problems for the one-dimensional heat equation,” J. Adv. Res. Dyn. Control Syst., 11, No. 01, 496–510 (2019).

[19]

Triebel H., Interpolation Theory, Function Spaces, Differential Operators, VEB Deutscher Verlag Wissenschaften, Berlin (1978).

[20]

Grisvard P., “Equations différentielles abstraites,” Ann. Sci. École Norm. Sup. (4), 2, 311–395 (1969).

[21]

Denk R., Hieber M., and Prüss J., “R-boundedness, Fourier multipliers and problems of elliptic and parabolic type,” Mem. Amer. Math. Soc., 166, No. 788 (2003).

[22]

Ladyzhenskaya O. A., Solonnikov V. A., and Ural’tseva N. N., Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence, RI (1967) (Transl. Math. Monogr.; V. 23).

[23]

Agranovich M. S. and Vishik M. I., “Elliptic problems with a parameter and parabolic problems of a general type,” Russ. Math. Surv., 19, No. 3, 53–157 (1964).

[24]

Amann H., “Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems,” in: Function Spaces, Differential Operators and Nonlinear Analysis, pp. 9–126, Teubner, Stuttgart (1993) (Teubner-Texte Math.; Bd. 133).

[25]

Amann H., “Nonautonomous parabolic equations involving measures,” J. Math. Sci., 130, No. 4, 4780–4802 (2005).

[26]

Naimark M. A., Linear Differential Operators, Frederick Ungar Publ. Co., New York (1967).

[27]

Gilbarg D. and Trudinger N. S., Elliptic Partial Differential Equations of Second order,Springer-Verl., Berlin (2001).

Received

03-09-2019

How to Cite

*Mathematical notes of NEFU*, 27(1), pp. 21-40. doi: https://doi.org/10.25587/SVFU.2020.77.95.002.

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Mathematics

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