Commun. Comput. Phys., 14 (2013), pp. 1-20. |
Radiogenic Source Identification for the Helium Production-Diffusion Equation Gang Bao ^{1}, Todd A. Ehlers ^{2}, Peijun Li ^{3*} 1 Department of Mathematics, Zhejiang University, Hangzhou 310027, China; Department of Mathematics, Michigan State University, East Lansing, MI 48824, USA.2 Department of Geosciences, Wilhelmstrasse 56, Universitat Tubingen, D-72074, Tubingen, Germany. 3 Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA. Received 3 January 2012; Accepted (in revised version) 25 May 2012 Available online 18 October 2012 doi:10.4208/cicp.030112.250512a 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. AMS subject classifications: 65M32, 35Q80Notice: Undefined variable: pac in /var/www/html/issue/abstract/readabs.php on line 164 Key words: Inverse source problem, production-diffusion equation, Tikhonov regularization. *Corresponding author. Email: bao@math.msu.edu (G. Bao), todd.ehlers@uni-tuebingen.de (T. A. Ehlers), lipeijun@math.purdue.edu (P. Li) |