Date on Master's Thesis/Doctoral Dissertation

8-2017

Document Type

Doctoral Dissertation

Degree Name

Ph. D.

Department

Mathematics

Degree Program

Applied and Industrial Mathematics, PhD

Committee Chair

Xu, Yongzhi

Committee Co-Chair (if applicable)

Gill, Ryan

Committee Member

Gill, Ryan

Committee Member

Hu, Changbing

Committee Member

Gie, Gung-Min

Committee Member

Wu, Dongfeng

Author's Keywords

parabolic equation; nonlocal problem; free boundary problem; inverse problem; numerical solution; ductal carcinoma in situ (DCIS) model

Abstract

Ductal carcinoma in situ (DCIS) is the earliest form of breast cancer. Three mathematical models in the one dimensional case arising from DCIS are proposed. The first two models are in the form of parabolic equation with initial and known moving boundaries. Direct and inverse problems are considered in model 1, existence and uniqueness are proved by using tool from heat potential theory and Volterra integral equations. Also, we discuss the direct problem and nonlocal problem of model 2, existence and uniqueness are proved. And approximation solution of these problems are implemented by Ritz-Galerkin method, which is the first attempt to deal with such problems. Based on the finding of the previous two models, the more general free boundary problem model - nonlinear parabolic partial differential equation with initial, boundary and free boundary condition is presented. Well-posedness theorems are proved by applying knowledge of semigroup solution operators. Illustrative examples are included to demonstrate the validity and applicability of the technique for all three models.

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