Abstract
The inverse problem of determining the source for the heat equation in a bounded domain on the plane is studied. The trace of the solution of the direct problem on two straight line segments inside the domain is given as overdetermination (i.e., additional information on the solution of the direct problem). A Fredholm alternative theorem for this problem is proved, and sufficient conditions for its unique solvability are obtained. The inverse problem is considered in classes of smooth functions whose derivatives satisfy the Hölder condition.
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References
O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type (Nauka, Moscow, 1967; Am. Math. Soc., Providence, R.I., 1968).
A. Friedman, Partial Differential Equations of Parabolic Type (Prentice Hall, Englewood Cliffs, N.J., 1964).
S. M. Nikol’skii, A Course of Calculus (Nauka, Moscow, 1991) [in Russian].
A. M. Denisov, “Uniqueness and nonuniqueness of the solution to the problem of determining the source in the heat equation,” Comput. Math. Math. Phys. 56 (10), 1737–1742 (2016).
A. M. Denisov, Elements of the Theory of Inverse Problems (Mosk. Gos. Univ., Moscow, 1994; VSP, Utrecht, 1999).
M. M. Lavrent’ev, Conditionally Well-Posed Problems for Differential Equations (Novosibirsk. Gos. Univ., Novosibirsk, 1973) [in Russian].
A. I. Prilepko, D. G. Orlovsky, and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics (Marcel Dekker, New York, 2000).
V. Isakov, Inverse Problems for Partial Differential Equations (Springer, New York, 2006).
A. B. Kostin, “The inverse problem of recovering the source in a parabolic equation under a condition of nonlocal observation,” Sb. Math. 204 (10), 1391–1434 (2013).
V. V. Solov’ev, “Solvability of an inverse problem of determining a source, with overdetermination on the upper cap, for a parabolic equation,” Differ. Equations 25 (9), 1114–1119 (1989).
E. M. Landis, Second-Order Equations of Elliptic and Parabolic Type (Nauka, Moscow, 1971; Am. Math. Soc., Providence, R.I., 1998).
A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Nauka, Moscow, 1981; Dover, New York, 1999).
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Original Russian Text © V.V. Solov’ev, 2018, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2018, Vol. 58, No. 5, pp. 778–804.
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Solov’ev, V.V. On Determining Sources with Compact Supports in a Bounded Plane Domain for the Heat Equation. Comput. Math. and Math. Phys. 58, 750–760 (2018). https://doi.org/10.1134/S0965542518050159
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DOI: https://doi.org/10.1134/S0965542518050159