Inverse random source problem for the Helium production-diffusion equation

Abstract

In this paper, we consider the prediction of the helium concentrations as function of a spatially variable source term perturbed by fractional Brownian motion. For the direct problem, we show that it is well-posed and has a unique mild solution under some conditions. For the inverse problem, the uniqueness and the instability are given. In the meanwhile, we determine the statistical properties of the source from the expectation and covariance of the final-time data $u(r,T)$. Finally, numerical implements are given to verify the effectiveness of the proposed reconstruction.

Keywords:

• Helium production-diffusion equation
• inverse source problem
• fractional Brownian motion
• ill-posedness

AMS 2010 Mathematics Subject Classifications:

• 35R30
• 35R60
• 65M32

Acknowledgments

The authors would like to thank Professor Xiaoli Feng (School of Mathematics and Statistics, Xidian University)'s help. In addition, the authors are greatly indebted to the reviewers' of this paper for their most helpful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).