Abstract
In the present paper, identification elliptic problem in a Hilbert space with Dirichlet and integral boundary conditions is discussed. Identification problem is reduced to auxiliary nonlocal boundary value problem with nonlocal integral condition. Operator approach is used to prove stability and coercive stability inequalities for solution of source identification elliptic problem in the case of a self-adjoint positive definite operator in a differential equation. As applications, four mixed boundary value problems for strongly elliptic multidimensional partial differential equation are investigated. Theorems on stability of solutions of these boundary value problems are established.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Ashyralyev, A., Ashyralyyev, C.: On the problem of determining the parameter of an elliptic equation in a Banach space. Nonlinear Anal. Model. Control 19, 350–366 (2014)
Ashyralyev, A., Emharab, F.: Identification hyperbolic problems with the Neumann boundary condition. Bull. Karaganda Univ.-Math. 91, 89–98 (2018)
Ashyralyev, A., Sazaklioglu, A.U.: Investigation of a time-dependent source identification inverse problem with integral overdetermination. Numer. Funct. Anal. Optim. 38, 1276–1294 (2017)
Ashyralyev, A., Sobolevskii, P.E.: New Difference Schemes for Partial Differential Equations. Operator Theory Advances and Applications. Birkhäuser Verlag, Basel, Boston, Berlin (2004)
Ashyralyyev, C.: Stability estimates for solution of Neumann type overdetermined elliptic problem. Numer. Funct. Anal. Optim. 38, 1226–1243 (2017)
Ashyralyyev, C.: Numerical solution to Bitsadze–Samarskii type elliptic overdetermined multipoint NBVP. Bound. Value Probl. 2017, 74 (2017)
Ashyralyyev, C.: A fourth order approximation of the Neumann type overdetermined elliptic problem. Filomat 31, 967–980 (2017)
Ashyralyyev, C., Akyuz, G.: Finite difference method for Bitsadze–Samarskii type overdetermined elliptic problem with Dirichlet conditions. Filomat 32, 859–872 (2018)
Ashyralyyev, C., Cay, A.: Well-posedness of Neumann-type elliptic overdetermined problem with integral condition. AIP Conf. Proc. 1997, 020026 (2018)
Ashyralyyev, C., Akyuz, G., Dedeturk, M.: Approximate solution for an inverse problem of multidimensional elliptic equation with multipoint nonlocal and Neumann boundary conditions. Electron. J. Differ. Equ. 2017, 197 (2017)
Ashyralyyeva, M.A., Ashyraliyev, M.: On the numerical solution of identification hyperbolic-parabolic problems with the Neumann boundary condition. Bull. Karaganda Univ.-Math. 91, 69–74 (2018)
Erdogan, A.S., Kusmangazinova, D., Orazov, I., Sadybekov, M.A.: On one problem for restoring the density of sources of the fractional heat conductivity process with respect to initial and final temperatures. Bull. Karaganda Univ.-Math. 91, 31–44 (2018)
Kabanikhin, S.I.: Inverse and Ill-Posed Problems: Theory and Applications. Walter de Gruyter, Berlin (2011)
Kirane, M., Malik, S.A., Al-Gwaiz, M.A.: An inverse source problem for a two dimensional time fractional diffusion equation with nonlocal boundary conditions. Math. Methods Appl. Sci. 36, 056–069 (2013)
Klibanov, M.V., Romanov, V.G.: Two reconstruction procedures for a 3D phaseless inverse scattering problem for the generalized Helmholtz equation. Inverse Probl. 32, 1 (2016)
Krein, S.G.: Linear Differential Equations in Banach Space. Nauka, Moscow, Russia (1966)
Orazov, I., Sadybekov, M.A.: On a class of problems of determining the temperature and density of heat sources given initial and final temperature. Sib. Math. J. 53, 146–151 (2012)
Sazaklioglu, A.U., Ashyralyev, A., Erdogan, A.S.: Existence and uniqueness results for an iinverse problem for semilinear parabolic equations. Filomat 31, 1057–1064 (2017)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Ashyralyyev, C. (2021). Identification Elliptic Problem with Dirichlet and Integral Conditions. In: Ashyralyev, A., Kalmenov, T.S., Ruzhansky, M.V., Sadybekov, M.A., Suragan, D. (eds) Functional Analysis in Interdisciplinary Applications—II. ICAAM 2018. Springer Proceedings in Mathematics & Statistics, vol 351. Springer, Cham. https://doi.org/10.1007/978-3-030-69292-6_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-69292-6_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-69291-9
Online ISBN: 978-3-030-69292-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)