ABSTRACT
This chapter looks at some applications using Fourier series and the tools used for proving various types of convergence. It begins by looking briefly at a problem in ordinary differential equations. Some kinds of ordinary differential equation models are amenable to solution using Fourier series techniques. The chapter looks at a simple linear partial differential equation called the cable equation which arises in information transmission such as in a model of a neuron. One can use the ideas of the linear independence of functions in the solution of nonhomogeneous differential equations. The linear independence of solutions of the homogeneous linear second order model is utilized to build a solution to the nonhomogeneous model. The chapter describes boundary value problems, the Kernel function, linear partial differential equations, convergence analysis for Fourier series, power series for ordinary differential equations and sample Horner’s loop method.
