AN INVERSE PROBLEM OF RECOVERING THE RIGHT HAND SIDE OF 1D PSEUDOPARABOLIC EQUATION
DOI:
https://doi.org/10.26577/JMMCS.2021.v111.i3.03Keywords:
inverse problem, pseudoparabolic equations, overdetermination condition, existence, uniqueness, classical solutionAbstract
Inverse problems of finding the right-hand side of a partial differential equations arise when an external source is unknown or impossible to measurement, for example, the source is in a high-temperature environment or underground. The partial differential equations with mixed derivatives by time and space variables are usually called pseudo-parabolic equations or Sobolev type equations. Pseudo-parabolic equations occur in mathematical modeling of many physical phenomena such as the motion of non-Newtonian fluids, thermodynamic processes, filtration in a porous medium, unsteady flow of second-order fluids, etc. This paper is devoted to investigate the unique solvability of two inverse problems for a linear one dimensional pseudoparabolic equation. The inverse problems consist of recovering the right hand side of the equation depending on the space variable. An additional information for the first inverse problem is given by an final overdetermination condition and for the second is given by an integral overdetermination condition. Under the suitable conditions on the initial data of the problem, the existence and uniqueness of a classical solution to these inverse problems are established. By using the Fourier method, the explicit formulas of a solutions are presented in the form of a series, which make it possible to perform the necessary numerical calculations with a given accuracy.
References
[2] Al’shin A.B., Korpusov M.O., Sveshnikov A.G., "Blow-up in nonlinear Sobolev type equations," ( Ber.: De Gruyter 2011):648.
[3] Ting T.W., "Certain nonsteady flows of second-order fluids," Arch. Rational Mech. Anal. 14 (1963): 1–26.
[4] Barenblatt G.I., Entov V.M., Ryzhik V.M., "Teoriya nestasionarnoi filtratsii jidkostei i gazov"[Theory of Unsteady
Fluids]," (M.:Nedra 1972): 288.
[5] Huilgol R., "A second order fluid of the differential type," Int. J. Non-Linear Mech. 3 (1968): 471–482.
[6] Zvyagin V.G., Turbin M.V., "The study of initial-boundary value problems for mathematical models of the motion of Kelvin-Voigt fluids," J. Math. Sci. 168 (2017): 157–308.
[7] Kabanikhin S.I., "Obratnye i nekorrectnye zadachi [Inverse and some problems]," (Nobosibirsk.:Sib. nauch. izd., 2009): 458.
[8] Belov Yu.Ya., "Inverse problems for parabolic equations," (Utrecht. VSP. 2002).
[9] Prilepko A. I., Orlovsky D. G., Vasin I. A., "Methods for solving inverse problems in mathematical physics," (New York: Marcel Dekker 1999) : 744.
[10] Sattorov E.N., Ermamatova Z.E., "Recovery of Solutions to Homogeneous System of Maxwell’s Equations With Prescribed Values on a Part of the Boundary of Domain," Russ Math. 63 (2019) 35–43.
[11] Khompysh Kh., Shakir A., "The inverse problem for determining the right part of the pseudo-parabolic equation," Journal of Math. Mech. Comp. Sci. -105:1 (2020): 87–98.
[12] Kaliev I.A, Sabitova M.M., "Problems of determining the temperature and density of heat sources from the initial and final temperatures," Russian Mathematics 56:2 (2012): 60–64.
[13] Rruzhansky M., Serikbaev D., Tokmagambetov N., "An inverse problem for the pseudo-parabolic equation for Laplace operator," International Journal of Mathematics and Physics 10:1 (2019): 23–28.
[14] Ablabekov B.S., "Obratnaiye zadachi dlya pseudoparabolicheskikh urabnenii [Inverse Problems for Pseudoparabolic Equations]," (B.:Ilim, 2001): 181.
[15] Asanov A., Atamanov E.R., "Nonclassical and Inverse Problems for Pseudoparabolic Equations," (Ber.: De Gruyter, 1997): 152.
[16] Fedorov V.E., Urasaeva A.V., "An inverse problem for linear Sobolev type equation," J. of Inverse and Ill-posed Prob. 12 (2001): 387–395.
[17] Kozhanov A.I., "On the solvability of inverse coefficient problems for some Sobolev-type equations," Nauchn. Vedom. Belgorod. Univ. 18:5 (2010): 88–97.
[18] Khompysh Kh., "Inverse Problem for 1D Pseudo-parabolic Equation," Fun. Anal. in Interdisciplinary Appl. 216 (2017): 382–387.
[19] Lorenzi A., Paparoni E., "Identification problems for pseudoparabolic integrodifferential operator equations J. Inverse. Ill-Posed Probl. 5 (1997): 235–253.
[20] Lyubanova A.Sh., Tani A., "An inverse problem for pseudoparabolic equation of filtration. The existence, uniqueness and regularity," Appl. Anal. 90 (2011): 1557-1568.
[21] Lyubanova A.Sh., Velisevich A.V., "Inverse problems for the stationary and pseudoparabolic equations of diffusion," Applicable Anal. 98 (2019): 1997–2010.
[22] Antontsev S.N., Aitzhanov S.E., Ashurova G.R., "An inverse problem for the pseudo-parabolic equation with p-Laplacian," Evol. Eq. and Cont. Theo. (2021).
[23] Abylkairov U.U., Khompysh Kh., "An inverse problem of identifying the coefficient in Kelvin-Voight equations," Appl. Math. Sci. 101:9 (2015): 5079–5088.
[24] Khompysh Kh., "Inverse problem with integral overdetermination for system of equations of Kelvin-Voight fluids," Advanced Materials Research 705 (2013) 15–20.
[25] Khompysh Kh., Nugymanova N.K., "Inverse Problem For Integro-Differential Kelvin-Voigt Equation," Jour. of Inverse and Ill-Posed Prob. (Submitted 2021).
[26] Yaman M., "Blow-up solution and stability to an inverse problem for a pseudo-parabolic equation," Jour. of Ineq. and App. (2012): 1–8.
