Abstract
In this paper we use finite volume element method to solve a parameter identification problem of parabolic equations with overspecified-data. We provide the numerical scheme of the unknown function and control parameters and obtain the error estimates of approximate solution. The results of the numerical experiment are presented and are compared with the exact solution to confirm the good accuracy of the presented scheme.
Similar content being viewed by others
References
Bednar JB, MacBain JA (1986) Existence and uniqueness properties for the one-dimensional magnetotellurics inversion problem. J Math Phys 27(2):645–649
Cai ZQ (1991) On the finite volume element method. Numer Math 58(4):713–735
Dehghan M (2003) Identifying a control function in two-dimensional parabolic inverse problems. Appl Math Comput 143(2–3):375–391
Dehghan M (2005) Parameter determination in a partial differential equation from the overspecified data. Math Comput Model 41:197–213
Shakeri F, Dehghan M (2009) Method of lines solutions of the parabolic inverse problem with an overespecification at a point. Numer Algorithms 50(4):417–437
Lin Y, Cannon JR (1988) Determination of parameter p(t) in holder classes for some semilinear parabolic equations. Inverse Probl 4(3):595–606
Lin Y, Cannon JR (1990a) An inverse problem of finding a parameter in a semi-linear heat equation. Math Anal Appl 145(2):470–484
Liu YP, Yan XC, Xiong ZG, Deng K (2013) The finite volume element method for a class of parameter identification problem with overspecified-data. In: Fourth international conference on emerging intelligent data and web technologies, Xi’an, pp 100–103
MacBain JA (1987) Inversion theory for a parameterized diffusion problem. SIAM J Appl Math 47(6):1386–1391
McCormick S, Cai ZQ (1990) On the accuracy of the finite volume element method for diffusion equations on composite grids. Numer Anal 27(3):636–655
Tatarii M, Dehghan M (2007) The radial basis functions method for identifying an unknown parameter in a parabolic equation with overspecified data. Numer Methods Partial Differ Equ 23(5):984–997
Wu ZM, Ma LM (2010) Identifying the temperature distribution in a parabolic equation with over specified data using a multiquadric quasi-interpolation method. Chin Phys B 19(1):1–6
Wu W, Li RH, Chen ZY (2000) Generalized difference methods for differential equations. Numerical analysis of finite volume methods. Marcel Dekker, Inc., New York
Yin HM, Cannon JR (1989) A class of non-linear non-classical parabolic equations. J Differ Equ 79(2):266–268
Yin HM, Cannon JR (1990b) Numerical solutions of some parabolic inverse problems. Numer Methods Partial Differ Equ 6(2):177–191
Acknowledgments
A Project Supported by Scientific Research Fund of Hunan Provincial Education Department (No. 12A050) and Hunan Science and Technology Project (No. 2014FJ3045). This work was supported by the special funds of modern information service industry of Guangdong Province (No. GDEID2011IS038) and Combination of Guangdong Province, the Ministry of Education project (Nos. 2011B090400480 and 2012B091000073). The authors express their thanks to the referees for their helpful suggestions, which led to improvements of the presentation.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Xiong, Z., Deng, K., Liu, Z. et al. The finite volume element method for a parameter identification problem. J Ambient Intell Human Comput 6, 533–539 (2015). https://doi.org/10.1007/s12652-014-0238-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12652-014-0238-7