Determination of a multidimensional kernel in some parabolic integro--differential equation

Durdimurod K. Durdiev ; Zhavlon Z. Nuriddinov

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 14 (2021) no. 1, pp. 117-127.

Résumé

A multidimensional parabolic integro-differential equation with the time-convolution integral on the right side is considered. The direct problem is represented by the Cauchy problem for this equation. The inverse problem is studied in this paper. The problem consists in finding the time and spatial dependent kernel of the equation from the solution of direct problem in a hyperplane $x_n=0$ for $t>0 $. This problem is reduced to the more convenient inverse problem with the use of the resolvent kernel. The last problem is replaced by the equivalent system of integral equations with respect to unknown functions. The unique solvability of the direct and inverse problems is proved with use of the principle of contraction mapping.

Détail

Keywords: integro-differential equation, inverse problem, Hölder space, resolvent.
Mots-clés : kernel

Bibliographie

[1] A. Lorenzi, E. Sinestrari, “An inverse problem in theory of materials with memory”, Nonlinear Anal. TMA, 12 (1988), 411–423

[2] D.K. Durdiev, “An inverse problem for a three-dimensional wave equation in the medium with memory”, Math. Anal. and Disc. math., NGU, Novosibirsk, 1989, 19–26 (in Russian)

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

[4] D.K. Durdiev, “To the question of correctness of one inverse problem for hyperbolic integro–differential equation”, Siberian Math. J., 33 (1992), 69–77

[5] D.K. Durdiev, “Some multidimensional inverse problems of memory determination in hyperbolic equations”, Zh. Mat. Fiz. Anal. Geom., 3:4 (2007), 411–423

[6] D.K. Durdiev, “A multidimensional inverse problem for an equation with memory”, Siberian Math. J., 35:3 (1994), 514–521

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

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

[9] P. Podio-Guidugli, “A virtual power format for thermomechanics”, Continuum Mech. Thermodyn., 20 (2009), 479–487 | DOI

[10] Zh.Sh. Safarov, D.K. Durdiev, “Inverse Problem for an Integro-Differential Equation of Acoustics”, Diff. Equ., 54:1 (2018), 134–142 | DOI

[11] Zh.D. Totieva, D.K. Durdiev, “The problem of finding the one-dimensional kernel of the thermoviscoelasticity equation”, Math. Notes, 103:1-2 (2018), 118–132 | DOI

[12] D.K. Durdiev, “An identification problem of memory function of a medium and the form of an impulse source”, J. Sib. Fed. Univ. Math. Phys., 2:2 (2009), 127–136

[13] F. Colomno, “A inverse problem for a parabolic integro-differential model in the theory of combustion”, Phys. D, 236 (2007), 81–89

[14] J. Janno, K. Kasemets, “A Positivity principle for parabolic integro-differential equations and inverse problems with final over determination”, Inverse Problems and Imaging, 3:1 (2009), 17–41

[15] D.K. Durdiev, “On the uniqueness of kernel determination in the integro-differential equation of parabolic type”, J. Samara State Tech. Univ., Ser. Phys. Math. Sci., 19:4 (2015), 658–666 (in Russian) | DOI

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

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

[18] K. Karuppiah, J.K. Kim, K. Balachandran, “Parameter identification of an integro-differential equation”, Nonlinear Functional Analysis and Applications, 20:2 (2015), 169–185

[19] A. Hazanee, D. Lesnic, M.I. Ismailov, N.B. Kerimov, “Inverse time-dependent source problems for the heat equation with nonlocal boundary conditions”, Applied Mathematics and Computation, 346 (2019), 800–815

[20] A. Hazanee, D. Lesnic, M.I. Ismailov, N.B. Kerimov, “An inverse time-dependent source problems for the heat equation with a non-classical boundary condition”, Applied Mathematics Modelling, 39:4 (2015), 6258–6276

[21] M.J. Huntul, D. Lesnic, M.S. Hussein, “Reconstruction of time-dependent coefficients from heat moments”, Communications in Nonlinear Science and Numerical Simulation, 33 (2016), 194–217

[22] M.S. Hussein, D. Lesnic, “Simultaneous determination of time and space dependent coefficients in a parabolic equation”, Applied Mathematics and Computation, 301 (2017), 233–253

[23] M.I. Ivanchov, N.V. Saldina, “Inverse problem for a parabolic equation with strong power degeneration”, Ukrainian Mathematical Journal, 58:11 (2006), 1685–1703

[24] A. Ladyzhenskaya, V.A. Solonnikov, N.N. Uraltseva, Linear and quasilinear equations of parabolic type, American Mathematical Society, Providence, Rhode Island, 1968

[25] A.A. Kilbas, Integral equations: course of lectures, BSU, Minsk, 2005 (in Russian)

[26] A.N. Kolmogorov, S.V. Fomin, Elements of function theory and functional analysis, Nauka, M., 1972 (in Russian)

[27] V.P. Mikhaylov, Differential equations with partial derivatives, Nauka, M., 1983 (in Russian)