Two fractional regularization methods for identifying the radiogenic source of the Helium production-diffusion equation

Xuemin Xue; Xiangtuan Xiong; Yuanxiang Zhang; Xuemin Xue; Xiangtuan Xiong; Yuanxiang Zhang

doi:10.3934/math.2021662

AIMS Mathematics

2021, Volume 6, Issue 10: 11425-11448. doi: 10.3934/math.2021662

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Research article

Two fractional regularization methods for identifying the radiogenic source of the Helium production-diffusion equation

1.
Department of Mathematics, Northwest Normal University, Lanzhou, Gansu 730070, China
2.
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730070, China

Received: 27 April 2021 Accepted: 22 July 2021 Published: 09 August 2021
MSC : 65M32, 35Q80

The predication of the helium diffusion concentration as a function of a source term in diffusion equation is an ill-posed problem. This is called inverse radiogenic source problem. Although some classical regularization methods have been considered for this problem, we propose two new fractional regularization methods for the purpose of reducing the over-smoothing of the classical regularized solution. The corresponding error estimates are proved under the a-priori and the a-posteriori regularization parameter choice rules. Some numerical examples are shown to display the necessarity of the methods.
- inverse radiogenic source problem,
- fractional regularization methods,
- regularized solution,
- regularization parameter choice rules,
- error estimates
Citation: Xuemin Xue, Xiangtuan Xiong, Yuanxiang Zhang. Two fractional regularization methods for identifying the radiogenic source of the Helium production-diffusion equation[J]. AIMS Mathematics, 2021, 6(10): 11425-11448. doi: 10.3934/math.2021662

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Abstract

The predication of the helium diffusion concentration as a function of a source term in diffusion equation is an ill-posed problem. This is called inverse radiogenic source problem. Although some classical regularization methods have been considered for this problem, we propose two new fractional regularization methods for the purpose of reducing the over-smoothing of the classical regularized solution. The corresponding error estimates are proved under the a-priori and the a-posteriori regularization parameter choice rules. Some numerical examples are shown to display the necessarity of the methods.

References

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