A Fourier Series Framework for Homogeneous Fredholm Integral Equations of the Second Kind

Homogeneous Fredholm integral equations of the second kind commonly occur in a wide range of areas within applied mathematics, particularly within those contexts in which the evolution of an unknown function is described by an underlying integral operator equation. For this study, a Fourier series approach will be used for a systematic and transparent investigation of these equations. This will be done by first rewriting both the kernel and the solution series in terms of a trigonometric series so that an algebraic equation set can be derived directly from a given Fredholm equation of the second kind involving relations among Fourier coefficients. Such a procedure will enable one to determine the existence of non-trivial solutions based on specific algebraic constraints among coefficients, along with an investigation of the eigenvalues related to the given Fredholm equation's underlying operator. This approach is best used for those Fredholm equations involving periodic and smooth kernels, for which the Fourier series converges rapidly and captures most features of a given problem accurately. A few examples will be used throughout this study to better understand the effectiveness of these Fourier series tools within an investigation of the solution of the Fredholm equation space in a rather efficient alternative manner, rather than the traditionally used methods within this area of mathematics.