Chapter 12 - Fourier Series and Applications to Partial Differential Equations

Abstract

This chapter provides an overview of Fourier series and its applications to partial differential equations. When the eigenfunctions are sines and cosines, the expansion is known as a Fourier series. To explain the convergence of the Fourier series, it defined as: a function is piecewise continuous on a finite interval if it is continuous at each point in the interval except possibly at a finite number of points. The chapter explains the Convergence theorem. It provides an overview of Bessel functions and their properties and examines the eigenfunction expansion called a generalized Fourier series involving Bessel functions. The chapter reviews one of the more important differential equations, the one-dimensional heat equation. In one spatial dimension, the solution to the heat equation translates into the temperature in a thin rod or wire of length a. As the rate at which heat flows through the rod depends on the material that makes up the rod, the constant k, which is related to the thermal diffusivity of the material, is included in the heat equation. The chapter further discusses one-dimensional wave equation, Laplace's equation, often called the potential equation, and two-dimensional wave equation in a circular region.