Volume 14, no. 1Pages 5 - 25

On Evolutionary Inverse Problems for Mathematical Models of Heat and Mass Transfer

S.G. Pyatkov
This article is a survey. The results on well-posedness of inverse problems for mathematical models of heat and mass transfer are presented. The unknowns are the coefficients of a system or the right-hand side (the source function). The overdetermination conditions are values of a solution of some manifolds or integrals of a solution with weight over the spatial domain. Two classes of mathematical models are considered. The former includes the Navier-Stokes system, the parabolic equations for the temperature of a fluid, and the parabolic system for concentrations of admixtures. The right-hand side of the system for concentrations is unknown and characterizes the volumetric density of sources of admixtures in a fluid. The unknown functions depend on time and some part of spacial variables and occur in the right-hand side of the parabolic system for concentrations. The latter class is just a parabolic system of equations, where the unknowns occur in the right-hand side and the system as coefficients. The well-posedness questions for these problems are examined, in particular, existence and uniqueness theorems as well as stability estimates for solutions are exposed.
Full text
inverse problem; heat and mass transfer; filtration; diffusion; well-posedness.
1. Bejan A. Convection Heat Transfer. New York, Jon Wiley and Sons, 2004.
2. Joseph D.D. Stability of Fluid Motions. Berlin, Heidelberg, New York, Springer, 1976. DOI: 10.1007/978-3-642-80994-1
3. Polezhaev V.I., Bune A.V., Verozub N.A. Matematicheskoe modelirovanie konvektivnogo teplomassoperenosa na osnove sistemy Nav'e-Stoksa [Mathematical Modeling of Convective Heat and Mass Transfer on the Base of Navier-Stokes System]. Moscow, Nauka, 1987. (in Russian)
4. Lykov A.V., Mikhailov Yu.A. Teoriya teplomassoobmena [The Theory of Heat and Mass Transfer]. Leningrad, Gosenergoizdat, 1963. (in Russian)
5. Korotkova E.M., Pyatkov S.G. Inverse Problems of Recovering the Source Function for Heat and Mass Transfer Systems. Mathematical Notes of NEFU , 2015, vol. 22, no. 1, pp. 44-61.
6. Korotkova E.M., Pyatkov S.G. On Some Inverse Problems for a Linearized System of Heat and Mass Transfer. Siberian Advances in Mathematics, 2015, vol. 25, no. 2, pp. 110-123. DOI: 10.3103/S1055134415020029
7. Pyatkov S.G., Samkov M.L. Solvability of Some Inverse Problems for the Nonstationary Heat-And-Mass-Transfer System. Journal of Mathematical Analysis and Applications, 2017, vol. 446, no. 2, pp. 1449-1465.
8. Alekseev G.V. Optimizacija v stacionarnom Problemy teplomassoobmena i Magnitogidrodinamika [Optimization in Stationary Problems of Heat-And-Mass Transfer and Magnetohydrodynamics]. Moscow, Nauchnui Mir, 2010. (in Russian)
9. Levandowsky M., Childress W.S., Hunter S.H., Spiegel E.A. A Mathematical Model of Pattern Formation By Swimming Microorganisms. The Journal of Protozoology, 1975, vol. 22, no. 2, pp. 296-306.
10. Capatina A., Stavre R. A Control Problem in Bioconvective Flow. Kyoto Journal of Mathematics, 1998, vol. 37, pp. 585-595. DOI: 10.1215/kjm/1250518205
11. Babeshko O.M., Evdokimova O.V., Evdokimov S.M. On Taking into Account the Types of Sources and Settling Zones of Pollutants. Doklady Mathematics, 2000, vol. 61, no. 2, pp. 283-285.
12. Prilepko A.I., Orlovsky D.G., Vasin I.A. Methods for Solving Inverse Problems in Mathematical Physics. New York, Marcel Dekker, 1999.
13. Marchuk G.I. Mathematical Models in Environmental Problems. Amsterdam, Elsevier Science, 1986.
14. Ozisik M.N., Orlande H.R. Inverse Heat Transfer. New York, Taylor and Francis, 2000.
15. Belov Ya.Ya. Inverse problems for Parabolic Equations. Utrecht, VSP, 2002. DOI: 10.1515/9783110944631
16. Frolenkov I.V., Kriger E.N. An Identification Problem of the Source Function of the Special Form in Two-Dimensional Parabolic Equation. Journal of Siberian Federal University. Mathematics and Physics, 2010, vol. 3, no. 4, pp. 556-564.
17. Frolenkov I.V., Kriger E.N. Existence of a Solution to the Problem of Recovering a Coefficient for the Source Function. Siberian Journal of Pure and Applied Mathematics, 2013, vol. 13, no. 1, pp. 120-134.
18. Pyatkov, S.G. On Some Classes of Inverse Problems with Overdetermination Data on Spatial Manifolds. Siberian Mathematical Journal, 2016, vol. 57, no. 5, pp. 870-880. DOI: 10.1134/S0037446616050177
19. Pyatkov S.G., Samkov M.L. On Some Classes of Coefficient Inverse Problems for Parabolic Systems of Equations. Siberian Advances in Mathematics, 2012, vol. 22, no. 4, pp. 287-302. DOI: 10.3103/S1055134412040050
20. Pyatkov S.G., Tsybikov B.N. On Some Classes of Inverse Problems for Parabolic and Elliptic Equations. Journal of Evolution Equations, 2011, vol. 11, no. 1, pp. 155-186. DOI: 10.1007/s00028-010-0087-6
21. Pyatkov S.G. On Some Classes of Inverse Problems for Parabolic Equations. Journal of Inverse and Ill-posed Problems, 2011, vol. 18, no. 8, pp. 917-934.
22. Vabishchevich P.N., Vasil'ev V.I., Vasil'eva M.V., Nikiforov D.Ya. Numerical Solution of an Inverse Filtration Problem. Lobachevskii Journal of Mathematics, 2016, vol. 37, no. 6, pp. 777-786.
23. Prilepko A.I., Solov'ev V.V. Solvability Theorems and Rothe's Method for Inverse Problems for a Parabolic Equation. I. Differential Equations, 1987, vol. 23, no. 10, pp. 1230-1237.
24. Ivanchov M. Inverse Problems for Equation of Parabolic Type. Lviv, WNTL, 2003.
25. Prilepko A.I., Solov'ev V.V. Solvability of the Inverse Boundary-Value Problem of Finding a Coefficient of a Lower-Order Derivative in a Parabolic Equation. Differential Equations, 1987, vol. 23, no. 1, pp. 101-107.
26. Kuliev, M.A. Multi-Dimensional Inverse Problem for a Parabolic Equation in a Bounded Domain. Nonlinear Boundary Value Problem, 2004, vol. 14, pp. 138-145.
27. Yang Fan, DunGang Li. Identifying the Heat Source for the Heat Equation with Convection Term. International Journal of Mathematical Analysis, 2009, vol. 3, no. 27, pp. 1317-1323.
28. Belov Yu.Ya., Korshun K.V. An Identification Problem of Source Function in the Burgers-Type Equation. Journal of Siberian Federal University, Mathematics and Physics, 2012, vol. 5, no. 4, pp. 497-506.
29. Solov'ev V.V. Global Existence of a Solution to the Inverse Problem of Determining the Source Term in a Quasilinear Equation of Parabolic Type. Differential Equations, 1996, vol. 32, no. 4, pp. 538-547.
30. Pyatkov S.G., Rotko V.V. Inverse Problems with Pointwise Overdetermination for Some Quasilinear Parabolic Systems. Siberian Advances in Mathematics, 2020, vol. 30, no. 2, pp. 124-142.
31. Pyatkov S.G., Rotko V.V. On Some Parabolic Inverse Problems with the Pointwise Overdetermination. AIP Conference Proceedings, 2017, vol. 1907, article ID: 020008.
32. Pyatkov S.G., Rotko V.V. On Recovering the Source Function in Quasilinear Parabolic Problems With The Pointwise Overdetermination. Bulletin of the South Ural State University. Series: Mathematics. Mechanics, Physics, 2017, vol. 9, no. 4, pp. 19-26. (in Russian)
33. Rotko V.V. Inverse Problems for Mathematical Models of Convection-Diffusion with the Pointwise Overdetermination. Bulletin of the Yugra State University, 2018, no. 3 (50), pp. 57-66.
34. Pyatkov S.G. On Some Inverse Problems for First Order Operator-Differential Equations. Siberian Mathematical Journal, 2019, vol. 60, no. 1, pp. 140-147. DOI: 10.1134/S0037446619010154
35. Guidetti D. Asymptotic Expansion of Solutions to an Inverse Problem of Parabolic Type. Advances in Difference Equations, 2008, vol. 13, no. 5-6, pp. 399-426.
36. Vabishchevich P.N., Vasil'ev V.I. Computational Determination of the Lowest Order Coefficient in a Parabolic Equation. Doklady Mathematics, 2014, vol. 89, no. 2, pp.179-181. DOI: 10.1134/S1064562414020161
37. Dehghan M. Numerical Computation of a Control Function in a Partial Differential Equation. Applied Mathematics and Computation, 2004, vol. 147, pp. 397-408. DOI: 10.1016/S0096-3003(02)00733-6
38. Mamonov A.V., Yen-Hsi Richard Tsai. Point Source Identification in Nonlinear Advection-Diffusion-Reaction Systems. Inverse Problems, 2013, vol. 29, no. 3, article ID: 035009, 26 p. DOI: 10.1088/0266-5611/29/3/035009
39. Samarskii A.A., Vabishchevich P.N. Numerical Methods for Solving Inverse Problems of Mathematical Physics. Berlin; Boston, De Gruyter, 2007.
40. Kabanikhin S.I. Inverse and Ill-Posed Problems. Berlin, Boston, De Gruyter, 2012. DOI: 10.1515/9783110224016
41. Alifanov O.M. Inverse Heat Transfer Problems. Berlin, Heidelberg, Springer, 1994. DOI: 10.1007/978-3-642-76436-3
42. Alifanov O.M., Artyukhov E.A., Nenarokomov A.V. Obratnye zadachi slozhnogo teploobmena [Inverse Problems of Complex Heat Exchange]. Moscow, Yanus-K, 2009.
43. Pyatkov S.G., Safonov E.I. On Some Classes of Inverse Problems of Recovering a Source Function. Siberian Advances in Mathematics, 2017, vol. 27, no. 2, pp. 119-132. DOI: 10.3103/S1055134417020031
44. Pyatkov S.G., Uvarova M.V. On Determining the Source Function in Heat and Mass Transfer Problems under Integral Overdetermination Conditions. Journal of Applied and Industrial Mathematics, 2016, vol. 10, no. 4, pp. 93-100. DOI: 10.17104/1863-8937-2016-2-93
45. Panasenko E.A., Starchenko A.V. Numerical Solution of Some Inverse Problems with Different Types of Atmospheric Pollution. Bulletin of the Tomsk State University. Mathematics and Mechanics, 2008, vol. 2, no. 3, pp. 47-55.
46. 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, pp. 341-351.
47. Murray-Bruce J., Dragotti P.L. Estimating Localized Sources of Diffusion Fields Using Spatiotemporal Sensor Measurements. Transactions on Signal Processing, 2015, vol. 63, no. 12, pp. 3018-3031.
48. Badia A.El., Hamdi A. Inverse Source Problem in an Advection-Dispersion- Reaction System: Application to Water Pollution. Inverse Problems, 2007, vol. 23, pp. 2103-2120. DOI: 10.1088/0266-5611/23/5/017
49. Badia A.El., Tuong Ha-Duong, 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.
50. Badia A.El., Tuong Ha-Duong. Inverse Source Problem for the Heat Equation. Application to a Pollution Detection Problem. Journal of Inverse and Ill-posed Problems, 2002, vol. 10, no. 6, pp. 585-599.
51. Badia A.El., Tuong Ha-Duong. An Inverse Source Problem in Potential Analysis. Inverse Problems, 2000, vol. 16, pp. 651-663. DOI: 10.1088/0266-5611/16/3/308
52. Leevan Ling, Tomoya Takeuchi. Point Sources Identification Problems for Heat Equations. Communications in Computational Physics, 2009, vol. 5, no. 5, pp. 897-913.
53. 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, no. 1, pp. 496-510.
54. Triebel H. Interpolation Theory, Function Spaces, Differential Operators. Leipzig, Barth, 1995.
55. Amann H. Compact Embeddings of Vector-Valued Sobolev and Besov Spaces. Glasnik matematicki, 2000, vol. 35(55), pp. 16-177. DOI: 10.1016/S0026-0657(01)80042-4
56. Prilepko A.I., Ivankov A.L., Solov'ev V.V. Inverse Problems for Transport Equations and Parabolic Equations. Uniqueness, Stability, and Methods of Solving Ill-Posed Problems of Mathematical Physics. Novosibirsk, Computer Center of SB RAS, 1984, pp. 37-142.
57. Cannon J.R. A Class of Non-Linear Non-Classical Parabolic Equations. Journal of Differential Equations, 1989, vol. 79, pp. 266-288. DOI: 10.1016/0022-0396(89)90103-4
58. Cannon J.R. An Inverse Problem of Finding a Parameter in a Semi-linear Heat Equation. Journal of Mathematical Analysis and Applications, 1990, vol. 145, pp. 470-484. DOI: 10.1016/0022-247X(90)90414-B
59. Iskenderov A.D., Akhundov A.Ya. Inverse Problem for a Linear System of Parabolic Equations. Doklady Mathematics, 2009, vol. 79, no. 1, pp. 73-75. DOI: 10.1134/S1064562409010219
60. Ismailov M.I., Kanca F. Inverse Problem of Finding the Time-Dependent Coefficient of Heat Equation from Integral Overdetermination Condition Data. Inverse Problems In Science and Engineering, 2012, vol. 20, no. 24, pp. 463-476. DOI: 10.1007/s10612-012-9161-4
61. Ismailov M., Erkovan S. Inverse Problem of Finding the Coefficient of the Lowest Term in Two-Dimensional Heat Equation with Ionkin-Type Boundary Condition. Computational Mathematics and Mathematical Physics, 2012, vol. 59, no. 5, pp. 791-808.
62. Ivanchov M.I. Inverse Problem of Simulataneous Determination of Two Coefficients in a Parabolic Equation. Ukrainian Mathematical Journal, 2000, vol. 52, no. 3, pp. 379-387. DOI: 10.1007/BF02513132
63. Li Jing, Xu Youjun. An Inverse Coefficient Problem with Nonlinear Parabolic Equation. Journal of Applied Mathematics and Computing, 2010, vol. 34, pp. 195-206. DOI: 10.1007/s12190-009-0316-8
64. Kamynin V.L., Franchini E. An Inverse Problem for a Higher-Order Parabolic Equation. Mathematical Notes, 1998, vol. 64, no. 5, pp. 590-599. DOI: 10.1007/BF02316283
65. Kamynin V.L. The Inverse Problem of Determining the Lower-Order Coefficient in Parabolic Equations with Integral Observation. Mathematical Notes, 2013, vol. 94, no. 2, pp. 205-213. DOI: 10.1134/S0001434613070201
66. Kerimov N.B., Ismailov M.I. An Inverse Coefficient Problem for the Heat Equation in the Case of Nonlocal Boundary Conditions. Journal of Mathematical Analysis and Applications, 2012, no. 396, pp. 546-554.
67. Kozhanov A.I. Parabolic Equations with an Unknown Coefficient Depending on Time. Computational Mathematics and Mathematical Physics, 2005, vol. 45, no. 12, pp. 2085-2101.
68. Hussein M.S., Lesnic D. Simultaneous Determination of Time-Dependent Coefficients and Heat Source. International. Journal for Computational Methods in Engineering Science and Mechanics, 2016, vol. 17 (5-6), pp. 401-411. DOI: 10.1080/15502287.2016.1231241
69. Vasin I.A., Kamynin V.L. On the Asymptotic Behavior of Solutions to Inverse Problems for Parabolic Equations. Siberian Mathematical Journal, 1997, vol. 38, no. 4, pp. 647-662. DOI: 10.1007/BF02674572
70. Hazanee A., Lesnic D., Ismailov M.I., Kerimov N.B. Inverse Time-Dependent Source Problems for the Heat Equation with Nonlocal Boundary Conditions. Applied Mathematics and Computation, 2019, vol. 346, pp. 800-815. DOI: 10.1016/j.amc.2018.10.059
71. Prilepko A.I., Orlovskij D.G. Determination of a Parameter in an Evolution Equation and Inverse Problems of Mathematical Physics. II. Differential Equations, 1985, vol. 21, no. 4, pp. 472-477.
72. Ewing R.E., Tao Lin. A Class of Parameter Estimation Techniques for Fluid Flow in Porous Media. Advances in Water Resources, 1991, vol. 14, no. 2, pp. 89-97. DOI: 10.1016/0309-1708(91)90055-S
73. Pyatkov S.G., Safonov E.I. On Some Classes of Linear Inverse Problems for Parabolic Systems of Equations. Bulletin of Belgorod State University, 2014, vol. 35, no. 7(183), pp. 61-75.
74. Pyatkov S.G., Safonov E.I. On Some Classes of Linear Inverse Problems for Parabolic Systems of Equations.Journal of Siberian Federal University. Mathematics and Physics, 2014, vol. 11, pp. 777-799.
75. Isakov V. Inverse Problems for Partial Differential Equations. Applied Mathematical Sciences. Berlin, Springer, 2006.
76. Kozhanov A.I. Composite Type Equations and Inverse Problems. Utrecht, VSP, 1999. DOI: 10.1515/9783110943276
77. Favini A., Fragnelli G., Mininni R.M. New Prospects in Direct, Inverse and Control Problems for Evolution Equations. Cham, Heidelberg, New York, Dordrecht, London, Springer, 2014.
78. Colton D., Engl H., Louis A.K., McLaughlin J., Rundell W. Surveys on Solution Methods for Inverse Problems. Wien, Springer, 2000.
79. Sabatier P.C. Past and Future of Inverse Problems. Journal of Mathematical Physics, 2000, vol. 41, article ID: 4082. DOI: 10.1063/1.533336
80. Engl H.W., Rundell W. Inverse Problems in Diffusion Processes. Philadelphia, SIAM, 1995.
81. Danilaev P.G. Coefficient Inverse Problems for Parabolic Type Equations and Their Application. Utrecht, VSP, 2001.
82. Denk R., Hieber M., Pr''uss J. R-Boundedness, Fourier Multipliers, and Problems of Elliptic and Parabolic Type. Memoirs of the AMS, 2003, vol. 166, pp. 111-114.
83. Ladyzhenskaya O.A., Solonnikov V.A., Ural'tseva N.N. Linear and Quasi-Linear Equations of Parabolic Type. Providence, American Mathematical Society, 1968.
84. Amann H. Linear and Quasilinear Parabolic Problems. Basel, Birkhauser, 1995. DOI: 10.1007/978-3-0348-9221-6