The Study on Some Basic Properties of the Heat Equation

Abstract

In this thesis we study the heat equation and its properties based on Evans' ``Partial Differential Equation' First we derive the heat equation and learn to solve the initial-value problem and the nonhomogeneous problem Next we talk about some basic properties of the heat equation including the mean-value formula maximum principle regularity and uniqueness Last but not least with full knowledge of the heat equation on the whole space when it comes to half-line problems we first introduce the standard reflection method extending the problems to the whole line and solving the Dirichlet boundary value problems and Neumann boundary value problems on the half line However the general initial-boundary value problem cannot be solved by the reflection method Therefore we introduce Professor Liu Tai-Ping and Professor Yu Shih-Hsien's method They derive a relation between the Dirichlet and the Neumann boundary value through Laplace transform and furthermore construct a solution formula using Dirichlet-Neumann relation and the fundamental solution of the heat equation

Date of Award	2019
Original language	English
Supervisor	Yu-Chu Lin (Supervisor)