Direct and inverse problem for the parabolic equation with initial value and time-dependent boundaries

Abstract

In this paper, we deal with dual problem of a class of non-classical parabolic equations in which the boundaries are time-dependent function instead of fixed values, which arise from Ductal carcinoma in situ (DCIS). In the direct problem part, on using the several transformation and heat potential theory, we established the integral form of solution and proved the existence and uniqueness of solution. Then we consider the inverse problem of finding the control parameter of known moving boundaries, which means determining the potential function of model from incisional biopsy information in the view of DCIS. Algorithm and numerical simulation for both problems are included to demonstrate the validity and applicability of solutions.

Keywords:

• parabolic equation
• time-dependent boundary
• inverse problem
• numerical approximation
• DCIS

AMS Subject Classifications:

• 35K05
• 35K15
• 35K20
• 65M32

Acknowledgements

The authors would like to gratefully and sincerely thank Dr Yongzhi Xu for his guidance and valuable suggestions. Also, we would like to thank Dr David Swanson for reading this paper.

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The present investigation was supported in part by the National Natural Science Foundation of the People’s Republic of China [grant number 11201070]; the Science Research Fund of Department of Guangdong Province of the People’s Republic of China [grant number Yq2013161].