Radiogenic Source Identification for the Helium Production-Diffusion Equation

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.