One-Dimensional Heat Conduction

Abstract

In this paper we will introduce the one-dimensional heat conduction problem and present methods for searching a solution, based on different boundary conditions which are imposed on the corresponding eqaution. First, we will get acquainted with partial differential equations, specifically: second order linear equations. After we do the proper classification, the canonical form for parabolic type of equation will be derived, since this is the type of equation that a heat equation belongs to. In the following we will specify and explain in details the initial and boundary conditions which are unavoidable parts of the heat conduction problem. We will define the Fourier series, list their main properties and state several basic theorems regarding their convergence. This part of mathematical theory is essential for understanding the solving process for the heat conduction problem. In the main part of this paper we will derive the heat equation by using some basic physical laws. Then, we will thoroughly analyse the homogeneous and nonhomogeneous equation, show the way in which we can complexify some types of homogeneous problems and give some examples along with their illustrations.