On the uniqueness of kernel determination in the integro-differential equation of parabolic type

D. K. Durdiev

Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences, Tome 19 (2015) no. 4, pp. 658-666

Résumé

We study the problem of determining the kernel of the integral term in the one-dimensional integro-differential equation of heat conduction from the known solution of the Cauchy problem for this equation. First, the original problem is replaced by the equivalent problem where an additional condition contains the unknown kernel without integral. We study the question of the uniqueness of the determining of the kernel. Next, assuming that there are two solutions $ k_1 (x, t) $ and $ k_2 (x, t), $ integro-differential equations, Cauchy and additional conditions for the difference of solutions of the Cauchy problem corresponding to the functions $ k_1 (x, t), $ $ k_2 (x, t)$ are obtained. Further research is being conducted for the difference $k_1 (x, t) - k_2 (x, t) $ of solutions of the problem and using the techniques of integral equations estimates it is shown that if the unknown kernel $ k (x, t) $ can be represented as $ k_j (x, t) = \sum_ {i = 0} ^ N a_i (x) b_i (t)$, $ j = 1, 2, $ then $ k_1 (x, t ) \equiv k_2 (x, t). $ Thus, the theorem on the uniqueness of the solution of the problem is proved.

Détail

Keywords: inverse problem, Cauchy problem, integral equation, uniqueness.
Mots-clés : parabolic equation

Bibliographie

[1] Durdiev D. K., Rashidov A. Sh., “Inverse problem of determining the kernel in an integro-differential equation of parabolic type”, Differ. Equ., 50:1 (2014), 110–116 | DOI | DOI | MR | Zbl

[2] Kasemets K., Janno J., “Inverse problems for a parabolic integro-differential equation in convolutional weak form”, Abstract and Applied Analysis, 2013 (2013), 297104, 16 pp. | DOI | Zbl

[3] von Wolfersdorf L., Janno J., “On the theory of convolution equations of the third kind, II”, Journal of Mathematical Analysis and Applications, 342:2 (2008), 838–863 | DOI | Zbl

[4] Janno J., von Wolfersdorf L., “Identification of memory kernels in one-dimensional heat flow with boundary conditions of the third kind”, Inverse Problems in Engineering, 9:2 (2001), 175–198 | DOI

[5] Janno J., von Wolfersdorf L., “An inverse problem for identification of a time- and space-dependent memory kernel of a special kind in heat conduction”, Inverse problems, 15:6 (1999), 1455–1467 | DOI | Zbl

[6] Janno J., von Wolfersdorf L., “Inverse problems for identification of memory kernels in heat flow”, Journal of Inverse and Ill-Posed Problems, 4:1 (1996), 39–66 | DOI | Zbl

[7] Romanov V. G., Obratnye zadachi matematicheskoi fiziki [Inverse problems of mathematical physics], Nauka, Moscow, 1984, 264 pp. (In Russian) | MR | Zbl

[8] Prilepko A. I., Kostin A. B., “Inverse problems of the determination of the coefficient in parabolic equations. I”, Sib. Math. J., 33:3 (1992), 489–496 | DOI | Zbl

[9] Prilepko A. I., Kostin A. B., “On inverse problems of determining a coefficient in a parabolic equation. II”, Siberian Math. J., 34:5 (1993), 923–937 | DOI | MR | Zbl

[10] Iskenderov A. D., “Multidimensional inverse problems for linear and quasi-linear parabolic equations”, Sov. Math., Dokl., 16:5 (1975), 1564–1568 | MR | Zbl

[11] Beznoshchenko N. Ya., “On determining the coefficient in a parabolic equation”, Differ. Uravn., 10:1 (1974), 24–35 (In Russian)

[12] Beznoshchenko N. Ya., “Determination of coefficients of higher terms in a parabolic equation”, Siberian Math. J., 16:3 (1975), 360–367 | DOI

[13] Romanov V. G., “An abstract inverse problem and questions of its uniqueness”, Funct. Anal. Appl., 7:3 (1973), 223–229 | DOI | MR | Zbl

[14] Romanov V. G., “On one uniqueness theorem for an integral geometry problem on a set of curves”, Matematicheskie problemy geofiziki [Mathematical Problems of Geophysics], Computing Center, Siberian Branch of the USSR Acad. Sci., Novosibirsk, 1973, 140–146 (In Russian)