Regularized solutions for terminal problems of parabolic equations.

partial differential equation;sobolev space;regularization

Sujeewa Indika Hapuarachchi

University:University of Louisville

Department:Mathematics

关键词: partial differential equation;    sobolev space;    regularization;   

Others  :  https://ir.library.louisville.edu/cgi/viewcontent.cgi?article=3893&context=etd

美国|英语

来源: The Universite of Louisville's Institutional Repository

【 摘 要 】

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.