Date on Master's Thesis/Doctoral Dissertation


Document Type

Doctoral Dissertation

Degree Name

Ph. D.



Degree Program

Applied and Industrial Mathematics, PhD

Committee Chair

Xu, Yongzhi

Committee Member

Swanson, David

Committee Member

Hu, Changbing

Committee Member

Gie, Gung-Min

Committee Member

Sumanasekera, Gamini

Author's Keywords

partial differential equation; sobolev space; regularization


The heat equation with a terminal condition problem is not well-posed in the sense of Hadamard so regularization is needed. In general, partial differential equations (PDE) with terminal conditions are those in which the solution depends uniquely but not continuously on the given condition. In this dissertation, we explore how to find an approximation problem for a nonlinear heat equation which is well-posed. By using a small parameter, we construct an approximation problem and use a modified quasi-boundary value method to regularize a time dependent thermal conductivity heat equation and a quasi-boundary value method to regularize a space dependent thermal conductivity heat equation. Finally we prove, in both cases, the approximation solution converges to the original solution whenever the parameter goes to zero.