Date on Master's Thesis/Doctoral Dissertation
Applied and Industrial Mathematics, PhD
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.
Hapuarachchi, Sujeewa Indika, "Regularized solutions for terminal problems of parabolic equations." (2017). Electronic Theses and Dissertations. Paper 2776.