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Radiogenic Source Identification for the Helium Production-Diffusion Equation

Published online by Cambridge University Press:  03 June 2015

Gang Bao ,
Todd A. Ehlers  and
Peijun Li
Gang Bao*
Affiliation:
Department of Mathematics, Zhejiang University, Hangzhou 310027, China; Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA
Todd A. Ehlers*
Affiliation:
Department of Geosciences, Wilhelmstrasse 56, Universitä Tübingen, D-72074, Tübingen, Germany
Peijun Li*
Affiliation:
Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA
*
Email:[email protected]
Corresponding author.Email:[email protected]
Email:[email protected]
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Abstract

Knowledge of helium diffusion kinetics is critical for materials in which helium measurements are made, particulary for thermochronology. In most cases the helium ages were younger than expected, an observation attributes to diffusive loss of helium and the ejection of high energy alpha particles. Therefore it is important to accurately calculate the distribution of the source term within a sample. In this paper, the prediction of the helium concentrations as function of a spatially variable source term are considered. Both the forward and inverse solutions are presented. Under the assumption of radially symmetric geometry, an analytical solution is deduced based on the eigenfunction expansion. Two regularization methods, the Tikhonov regularization and the spectral cutoff regularization, are considered to obtain the regularized solution. Error estimates with optimal convergence order are shown between the exact solution and the regularized solution. Numerical examples are presented to illustrate the validity and effectiveness of the proposed methods

Keywords

65M3235Q80Inverse source problemproduction-diffusion equationTikhonov regularization
Type
Research Article
Information
Communications in Computational Physics , Volume 14 , Issue 1 , July 2013 , pp. 1 - 20
Copyright
Copyright © Global Science Press Limited 2013

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References

[1]Bao, G., Dou, Y., Ehlers, T., Li, P., Wang, Y., and Xu, Z., Quantifying tectonic and geomorphic interpretations of thermochronometer data: the reconstruction of mountain surface, Commun. Comput. Phys., 9 (2011), 129–146.Google Scholar
[2]Bushuyev, I., Global uniqueness for inverse parabolic problems with final observation, Inverse Problems, 11 (1995), Lll–L16.CrossRefGoogle Scholar
[3]Cannon, J., The One-Dimensional Heat Equation, Cambridge University Press, 1984.Google Scholar
[4]Cannon, J. and Peérez-Esteva, S., Uniqueness and stability of 3D heat sources, Inverse Problems, 7 (1991), 57–62.CrossRefGoogle Scholar
[5]Cheng, W., Zhao, L., Fu, C., Source term identification for an axisymmetric inverse heat conducting problem, Comput. Math. Appl., 59 (2010), 142–148.CrossRefGoogle Scholar
[6]Choulli, M. and Yamamoto, M., Conditional stability in determing a heat source, J. Inverse Ill-posed Probl., 12 (2004), 233–243.Google Scholar
[7]Engl, H., Hanke, M., and Neubauer, A., Regularization of Inverse Problems, Dordrecht: Kluwer, 1996.Google Scholar
[8]Farcas, A. and Lesnic, D., The boundary element method for the determination of a heat source dependent on one variable, Journal of Engineering Mathematics, 54 (2006), 375–388.Google Scholar
[9]Farley, K., Wolf, R., and Silver, L., The effects of long alpha-stopping distance on (U-Th)/He ages, Geochim. Cosmochim. Acta, 60 (1996), 4223–4230.CrossRefGoogle Scholar
[10]Hettlich, F. and Rundell, W., Identification of a discontinuous source in the heat equation, Inverse Problems, 17 (2001), 1465–1482.Google Scholar
[11]Isakov, V., Inverse Source Problems, American Mathematical Society, Providence, RI, 1990.Google Scholar
[12]Isakov, V., Inverse parabolic problems with the final overdetermination, Comm. Pure Appl. Math., 44 (1991), 185–209.Google Scholar
[13]Isakov, V., Inverse Problems for Partial Differential Equations, Springer-Verlag, New York, 1998.Google Scholar
[14]Johansson, T. and Lesnic, D., A variational method for identifying a spacewise-dependent heat source, IMA J. Appl. Math., 72 (2007), 748–760.CrossRefGoogle Scholar
[15]Kirsch, A., An Introduction to the Mathematical Theory of Inverse Problem, Springer-Verlag, New York, 1996.CrossRefGoogle Scholar
[16]Ladyzˇenskaja, O. A., Solonnikov, V. A., and Ural’ceva, N. N., Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, RI, 1967.Google Scholar
[17]Persson, P. and Strang, G., A simple mesh generator in Matlab, SIAM Rev., 46 (2004), 329–345.Google Scholar
[18]Press, W., Teukolsky, S., Vetterling, W., and Flannery, B., Numerical Recipes in Fortran 90: The Art of Parallel Scientific Computing, 2nd ed., Cambridge University Press, 1996.Google Scholar
[19]Sakamoto, K. and Yamamoto, M., Inverse heat source problem from time distributing overde-termination, Applicable Analysis, 88 (2009), 735–748.Google Scholar
[20]Shuster, D. and Farley, K., 4He/3He thermochronometry, Earth Planet. Sci. Lett., 217 (2003), 1–17.CrossRefGoogle Scholar
[21]Watson, G., A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1995Google Scholar
[22]Wolf, R., Farley, K., and Kass, D., Modeling of the temperature sensitivity of the apatite (U-Th)/He thermochrometer, Chem. Geol., 148 (1998), 105–114.Google Scholar
[23]Wolf, R., Farley, K., and Silver, L., Helium diffusion and low temperature thermochronometry of apatite, Geochim. Cosmochim. Acta, 60 (1996), 4231–4240.Google Scholar
[24]Xie, J. and Zou, J., Numerical reconstruction of heat fluxes, SIAM J. Numer. Anal., 43 (2005), 1504–1535.CrossRefGoogle Scholar
[25]Yamamoto, M. and Zou, J., Simultaneous reconstruction of the initial temperature and heat radiative coefficient, Inverse Problems, 17 (2001), 1181–1202.CrossRefGoogle Scholar
[26]Yan, L., Fu, C., and Dou, F., A computational method for identifying a spacewise-dependent heat source, Int. J. Numer. Meth. Biomed. Engng., 26 (2010), 597–608.Google Scholar
[27]Yi, Z. and Murio, D., Source term identification in the 1-D IHCP, Computers Math. Applic., 47 (2004), 1921–1933.Google Scholar
[28]Yi, Z. and Murio, D., Source term identification in the 2-D IHCP, Computers Math. Applic., 47 (2004), 1517–1533.Google Scholar