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Inverse problems for the stationary and pseudoparabolic equations of diffusion. (English) Zbl 1447.35385
Summary: The identification of an unknown coefficient in the lower term of pseudoparabolic differential equation of diffusion \((u+\eta Mu)_t + Mu+ku=f\) and elliptic second-order differential equation \(Mu+ku=f\) with the Dirichlet boundary condition is considered. The identification of \(k\) is based on an integral boundary data. The local existence and uniqueness of generalized strong solutions for the inverse problems are proved. The stability estimates are exposed.
35R30 Inverse problems for PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35J25 Boundary value problems for second-order elliptic equations
Full Text: DOI
[1] Lyubanova, As, Identification of a constant coefficient in an elliptic equation, Appl Anal, 87, 1121-1128, (2008) · Zbl 1159.35444
[2] Lyubanova, As, On an inverse problem for quasi-Linear elliptic equation, J Siberian Federal Univ Math Phys, 8, 38-48, (2015)
[3] Lyubanova, As, The inverse problem for the nonlinear pseudoparabolic equation of filtration type, J Siberian Federal Univ Math Phys, 10, 4-15, (2017)
[4] Lyubanova, As; Tani, A., An inverse problem for pseudoparabolic equation of filtration: the existence, uniqueness and regularity, Appl Anal, 90, 1557-1571, (2011) · Zbl 1237.35101
[5] Prilepko, Ai; Orlovsky, Dg; Vasin, Ia, Methods for solving inverse problems in mathematical physics, (2000), New York (NY): Marcel Dekker, New York (NY)
[6] Alekseev, Gv; Kalinina, Ea, Identification of the lowest coefficient of a stationary convection-diffusion-reaction equation, Sibirsk Zh Industr Mat, 10, 3-16, (2007)
[7] Egger, H.; Pietschmann, J-F; Schlottbom, M., Simultaneous identification of diffusion and absorption coefficients in a quasilinear elliptic problem, Inverse Prob, 30, 035009, (2014) · Zbl 1286.35263
[8] Rundell, W., Determination of an unknown nonhomogeneous term in a linear partial differential equation from overspecified boundary data, Appl Anal, 10, 231-242, (1980) · Zbl 0454.35045
[9] Kozhanov, Ai, [On the solvability of the coefficient inverse problems for equations of Sobolev type]. Nauchniye vedomosti Belgorodskogo gosudarstvennogo universiteta, Seriya Matematika Phizika, 5, 88-98, (2010)
[10] Mamayusupov, Ms, Studies in integro-differential equations, 16, The problem of determining coefficients of a pseudoparabolic equation, 290-297, (1983), Frunze: Ilim, Frunze
[11] Pyatkov, Sg; Shergin, Sn, On some mathematical models of filtration type, Bull South Ural State Univ Ser Math Model Program Comput Softw (Bulletin SUSU MMCS), 8, 105-116, (2015)
[12] Bukhgeim, Al; Klibanov, Mv, Global uniqueness of a class of multidimensional inverse problems, Soviet Math Dokl, 24, 244-247, (1981)
[13] Klibanov, Mv, Carleman estimates for global uniqueness, stability and numerical methods for coefficient inverse problems, J Inverse Ill-Posed Probl, 21, 477-560, (2013) · Zbl 1273.35005
[14] Klibanov, Mv; Timonov, A., Carleman estimates for coefficient inverse problems and numerical applications, (2004), Utrecht: VSP, Utrecht · Zbl 1069.65106
[15] Mehraliyev, Yt; Kanca, F., An inverse boundary value problem for a second order elliptic equation in a rectangle, Math Model Anal, 19, 241-256, (2004)
[16] Solov’Ev, Vv, Coefficient inverse problem for Poisson’s equation in a cylinder, Comput Math Math Phys, 51, 1738-1745, (2011)
[17] Li, T-T; White, Lw, Total flux (nonlocal) boundary value problems for pseudoparabolic equation, Appl Anal, 16, 17-31, (1983) · Zbl 0525.35021
[18] Ladyzenskaja, Oa; Uralceva, Nn, Linear and quasilinear elliptic equations, (1973), New York (NY): Academic Press, New York (NY)
[19] Lions, J-L; Magenes, E., Problemes aux limites non homogenes et applications, 17, (1968), Paris: Dunod, Paris · Zbl 0165.10801
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