Linear inverse problems for degenerate evolution equations with the Gerasimov–Caputo derivative in the sectorial case

  • Nagumanova Anna V., urazaeva_anna@mail.ru Chelyabinsk State University, 129 Kashirin Brothers Street, Chelyabinsk, Russia 454001
  • Fedorov Vladimir E., kar@csu.ru Chelyabinsk State University, 129 Kashirin Brothers Street, Chelyabinsk, Russia 454001; South Ural State University (National Research University), 76 Lenin Avenue, Chelyabinsk, Russia 454080
Keywords: inverse coefficient problem, fractional Gerasimov-Caputo derivative, degenerate evolution equation, analytic in a sector resolving family of operators

Abstract

We investigate the unique solvability of linear inverse problems for the evolution equation in a Banach space with a degenerate operator at the fractional Gerasimov-Caputo derivative and with a time-independent unknown coefficient. It is assumed that a pair of operators in the equation (at the unknown function and at its fractional derivative) generates a family of resolving operators of the corresponding degenerate linear homogeneous equation of the fractional order. The original problem is reduced to a system of two problems: the problem for an algebraic equation on the degeneration subspace of the original equation and the problem for the equation solved with respect to the fractional derivative, on the complement to the degeneration subspace. Two approaches are demonstrated. The first involves the study of the inverse problem for the equation solved with respect to the derivative and the direct problem for the algebraic equation. In the second approach, the inverse problem for the equation on the degeneration subspace is investigated firstly, then the direct problem for the second equation is researched. Abstract results are used to study initial-boundary value problems for a class of time-fractional order partial differential equations with an unknown coefficient depending on the spatial variables.

References


[1]
Kozhanov A. I., Composite Type Equations and Inverse Problems. VSP, Utrecht (1999).

[2]
Prilepko A. I., Orlovskii D. G., and Vasin I. A., Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker Inc., New York; Basel (2000).

[3]
Favini A. and Lorenzi A., Differential Equations. Inverse and Direct Problems, Chapman and Hall/CRC, New York (2006).

[4]
Pyatkov S. G. and Samkov M. L., “On some classes of coefficient inverse problems for parabolic systems of equations,” Sib. Adv. Math., 22, No. 4, 287–302 (2012).

[5]
Tikhonov I. V. and Eidel’man Yu. S., “Problems on correctness of ordinary and inverse problems for evolutionary equations of a special form,” Math. Notes, 56, No. 2, 830–839 (1994).

[6]
Tikhonov I. V. and Eidel’man Yu. S., “An inverse problem for a differential equation in a Banach space and the distribution of zeros of an entire Mittag-Leffler function,” Differ. Equ., 38, No. 5, 669–677 (2002).

[7]
Pyatkov S. G., “On some inverse problems for first order operator-differential equations,” Sib. Math. J., 60, No. 1, 140–147 (2019).

[8]
Glushak A. V. and Popova V. A., “Inverse problem for Euler–Poisson–Darboux abstract differential equation,” J. Math. Sci., 149, No. 4, 1453–1468 (2008).

[9]
Glushak A. V., “On an inverse problem for an abstract differential equation of fractional order,” Math. Notes, 87, No. 5, 654–662 (2010).

[10]
Orlovsky D. G., “Parameter determination in a differentia equation of fractional order with Riemann–Liouville fractional derivative in a Hilbert space,” J. Sib. Federal Univ., Math. Phys., 8, No. 1, 55–63 (2015).

[11]
Liu Y., Rundell W., and Yamamoto M., “Strong maximum principle for fractional diffusion equations and an application to an inverse source problem,” Fract. Calc. Appl. Anal., 19, No. 4, 888–906 (2016).

[12]
Fedorov V. E. and Nagumanova A. V., “Inverse problem for evolutionary equation with the Gerasimov–Caputo fractional derivative in the sectorial case [in Russian],” Vestn. Irkutsk. Gos. Univ., Ser. Mat., 28, 124–138 (2019).

[13]
Abasheeva N. L., “Some inverse problems for parabolic equations with changing time direction,” J. Inverse Ill-Posed Probl., 12, No. 4, 337–348 (2004).

[14]
Fedorov V. E. and Urazaeva A. V., “An inverse problem for linear Sobolev type equations,” J. Inverse Ill-Posed Probl., 12, No. 4, 387–395 (2004).

[15]
Urazaeva A. V. and Fedorov V. E., “Prediction-control problem for some systems of equations of fluid dynamics,” Differ. Equ., 44, No. 8, 1147–1156 (2008).

[16]
Urazaeva A. V. and Fedorov V. E., “On the well-posedness of the prediction-control problem for certain systems of equations,” Math. Notes, 85, No. 3, 426–436 (2009).

[17]
Falaleev M. V., “Degenerated abstract problem of prediction-control in Banach spaces,” Vestn. Irkutsk. Gos. Univ., Ser. Mat., 3, No. 1, 126–132 (2010).

[18]
Fedorov V. E. and Urazaeva A. V., “Linear evolution inverse problem for Sobolev type equations [in Russian],” in: Nonclassical Equations of Mathematical Physics, pp. 293–310, Sobolev Inst. Math. SB RAS, Novosibirsk (2010).

[19]
Fedorov V. E. and Ivanova N. D., “Nonlinear evolution inverse problem for some Sobolev type equations [in Russian],” Sib. Electron. Math. Rep., vol. 8, Proc. 2nd Int. Youth School-Conf. Theory and Numerical Methods of Inverse and Ill-Posed Problems Solving, Part I, 363–378 (2011).

[20]
Ivanova N. D., Fedorov V. E., and Komarova K. M., “Nonlinear inverse problem for the Oskolkov system, which is linearized in a neighborhood of a stationary solution [in Russian],” Vestn. Chelyab. Gos. Univ., No. 26, 49–70 (2012).

[21]
Al Horani M. and Favini A., “Degenerate first-order inverse problems in Banach spaces,” Nonlinear Anal., 75, No. 1, 68–77 (2012).

[22]
Plekhanova M. V. and Fedorov V. E., Optimal Control for Degenerate Distributed Systems [in Russian], South Ural State Univ. Publ., Chelyabinsk (2013).

[23]
Fedorov V. E. and Ivanova N. D., “Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel,” Discrete Contin. Dyn. Syst., Ser. S, 9, No. 3, 687–696 (2016).

[24]
Fedorov V. E. and Ivanova N. D., “Identification problem for degenerate evolution equations of fractional order,” Fract. Calc. Appl. Anal., 20, No. 3, 706–721 (2017).

[25]
Fedorov V. E. and Nazhimov R. R., “Inverse problems for a class of degenerate evolution equations with the Riemann–Liouville derivative,” Fract. Calc. Appl. Anal., 22, No. 2, 271–286 (2019).

[26]
Romanova E. A. and Fedorov V. E., “Resolving operators of the linear degenerate evolution equation with the Caputo derivative. The sectorial case,” Mat. Zametki SVFU, 23, No. 4, 58–72 (2016).

[27]
Fedorov V. E., Romanova E. A., and Debbouche A., “Analytic in a sector resolving families of operators for degenerate evolutional equations,” J. Math. Sci., 228, No. 4, 380–394 (2018).

[28]
Fedorov V. E., Gordievskikh D. M., Baleanu D., and Tas K., “Approximate controllability criterion for a class of degenerate distributed systems with the Riemann–Liouville,” Mat. Zametki SVFU, 26, No. 2, 41–59 (2019).

[29]
Bajlekova E. G., Fractional Evolution Equations in Banach Spaces: PhD thes., Eindhoven Univ. Technology, Univ. Press Fac, Eindhoven (2001).

[30]
Fedorov V. E. and Romanova E. A., “Inhomogeneous evolution fractional order equation in the sectorial case [in Russian],” Itogi Nauki i Tekhniki, Ser. Sovremen. Mat. Prilozh., 149, 103–112 (2018).

[31]
Fedorov V. E. , “A class of fractional order semilinear evolutions in Banach spaces,” in: Integral Equations and Their Applications, Proc. Univ. Network Seminar on the occasion of The Third Mongolia–Russia– Vietnam Workshop on NSIDE 2018 (Hung Yen, Viet Nam, Oct. 27–28, 2018), pp. 11–20, Hanoi Math. Soc., Hung Yen Univ. Technology and Education, Hung Yen (2018).

[32]
Triebel H., Interpolation Theory, Function Spaces, Differential Operators, VEB Deutscher Verl. Wissenschaften, Berlin (1978).

[33]
Dzektser E. S., “Generalization of the equation of motion of ground waters with a free surface,” Sov. Phys., Dokl., 17, 108–110 (1972).

[34]
Bateman H. and Erdelyi A., Higher Transcendental Functions, vol. 3, McGraw-Hill Book Co., New York; Toronto; London (1953).

[35]
Popov A. Yu. and Sedletskii A. M., “Distribution of roots of Mittag-Leffler functions,” J. Math. Sci., 2008) 190, No. 2, 209–409.
How to Cite
Nagumanova, A. and Fedorov, V. (2020) “Linear inverse problems for degenerate evolution equations with the Gerasimov–Caputo derivative in the sectorial case”, Mathematical notes of NEFU, 27(2), pp. 54-76. doi: https://doi.org/10.25587/SVFU.2020.57.76.004.
Section
Mathematics