Uniqueness of the kernel determination problem in an integro-differential parabolic equation with variable coefficient

D. K. Durdiev ; J. Z. Nuriddinov

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2023), pp. 3-14.

Résumé

We investigate the inverse problem of determining the time and space dependent kernel of the integral term in the $n$-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 this kernel. Next, assuming that there are two solutions $k_1(x,t)$ and $k_2(x,t)$ of the stated problem, it is formed an equation for the difference of this solution. 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 $k(x,t)=\displaystyle\sum\limits_{i=0}^Na_i(x)b_i(t)$, 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] Lorenzi A., Sinestrari E., “An inverse problem in theory of materials with memory”, Nonlinear Anal. TMA, 12:12 (1988), 1317–1335 | DOI | MR | Zbl

[2] Grasselli M., “An identification problem for a linear integro-differential equation occurring in heat flow”, Math. Meth. Appl. Sci., 15:3 (1992), 167–186 | DOI | MR | Zbl

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

[4] Colombo F., “An inverse problem for a parabolic integro-differential model in the theory of combustion”, Phys. D Nonlinear Phenom., 236:2 (2007), 81–89 | DOI | MR | Zbl

[5] Janno J., Lorenzi A., “Recovering memory kernels in parabolic transmission problems”, J. Inverse Ill-Posed Probl., 16:3 (2008), 239–265 | DOI | MR | Zbl

[6] Totieva Zh.D., Durdiev D.K., “The problem of finding the one-dimensional kernel of the thermoviscoelasticity equation”, Math. Notes, 103:1 (2018), 118–132 | DOI | MR | Zbl

[7] Durdiev D.K., Rahmonov A.A., “A $2$D kernel determination problem in a visco-elastic porous medium with a weakly horizontally inhomogeneity”, Math. Meth. Appl. Sci., 43:15 (2020), 8776–8796 | DOI | MR | Zbl

[8] Romanov V.G., “Stability estimates for the solution to the problem of determining the kernel of a viscoelastic equation”, Appl. Ind. Math., 6:3 (2012), 360–370 | DOI | MR | Zbl

[9] Romanov V.G., “Inverse problems for differential equations with memory”, Eurasian J. Math. Comput. Appl., 2:4 (2014), 51–80 | MR

[10] Durdiev U.D., Totieva Zh.D., “A problem of determining a special spatial part of 3D memory kernel in an integro-differential hyperbolic equation”, Math. Meth. Appl. Sci., 42:18 (2019), 7440–7451 | DOI | MR | Zbl

[11] Durdiev U.D., Totieva Zh.D., “The problem of determining the one-dimensional matrix kernel of the system of viscoelasticity equation”, Math. Meth. Appl. Sci., 41:17 (2018), 8019–8032 | DOI | MR | Zbl

[12] Durdiev D.K., Totieva Zh.D., “The problem of determining the one dimensional kernel of viscoelasticity equation with a source of explosive type”, J. Inverse Ill Posed Probl., 28:1 (2019), 1–10 | MR

[13] Totieva Zh.D., “The problem of determining the piezoelectric module of electro viscoelasticity equation”, Math. Meth. Appl. Sci., 41:16 (2018), 6409–6421 | DOI | MR | Zbl

[14] Durdiev D.K., “Global solvability of an inverse problem for an integro-differential equation of electrodynamics”, Diff. Equat., 44:7 (2008), 893–899 | DOI | MR | Zbl

[15] Hazanee A., Lesnic D., Ismailov M.I., Kerimov N.B., “Inverse time-dependent source problems for the heat equation with nonlocal boundary conditions”, Appl. Math. Comput., 346 (2019), 800–815 | MR | Zbl

[16] Huntul M.J., Lesnic D., Hussein M.S., “Reconstruction of time-dependent coefficients from heat moments”, Appl. Math. Comput., 301 (2017), 233–253 | MR | Zbl

[17] Hussein M.S., Lesnic D., “Simultaneous determination of time and space dependent coefficients in a parabolic equation”, Commun. Nonlinear Sci. Numer. Sim., 33 (2016), 194–217 | DOI | MR | Zbl

[18] Ivanchov M.I., Saldina N.V., “Inverse problem for a parabolic equation with strong power degeneration”, Ukr. Math. J., 58:11 (2006), 1685–1703 | DOI | MR | Zbl

[19] Durdiev D.K., Nuriddinov J.Z., “On investigation of the inverse problem for a parabolic integro-differential equation with a variable coefficient of thermal conductivity”, Vestn. Udmurtsk. un-ta. Matem. Mekhan. Kompyut. nauki, 30:4 (2020), 572–584 | MR | Zbl

[20] Durdiev D.K., Zhumaev Zh.Zh., “Problem of determining the thermal memory of a conducting medium”, Diff. Equat., 56:6 (2020), 785–796 | DOI | MR | Zbl

[21] Durdiev D.K., Zhumaev Zh.Zh., “Problem of determining a multidimensional thermal memory in a heat conductivity equation”, Meth. Funct. Anal. Topology, 25:3 (2019), 219–226 | MR

[22] Durdiev D.K., Nuriddinov J.Z., “Determination of a multidimensional kernel in some parabolic integro-differential equation”, Zhurn. SFU. Ser. Matem. fiz., 14:1 (2021), 117–127 | MR

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

[24] Romanov V.G., “An inverse problem for an equation of parabolic type”, Math. Notes, 19:4 (1976), 360–363 | DOI | MR | Zbl