Tomas Nilsson: Värmeledningsekvationen

Abstract:

This thesis discusses and presents the heat equation, a famous classical linear partial differential equation. This equation was initially developed by the French mathematician and phycisist J. Fourier (1768-1830) in order to model how heat diffuses through a region of a certain medium or material.We perform a detailed study of the solution method separation of variables in one and several dimensions through a deduction of analytical solutions to the heat equation subject to different boundary conditions. Included in this is to deduce at generalization of the solution method based on the fact that the Laplace operator can be expressed in terms of its eigenvalues and eigenfunctions. Furthermore, a discussion and presentation follows about heat kernel, a fundamental solution of the heat equation that can be used to express solutions over certain domains. We then study and present the maximum principle for the heat equation in a theorem with proof. This principle implies that the maximum value of the temperature in a region M, where M contains both the space and the time variable, will not surpass the maximum value that earlier existed in M, unless it is on the boundary of M. Finally we motivate and deduce a basic numerical method with the purpose to solve a simpler case of the heat equation.