Analytic study of Fourier Series Convergence with applications to the One Dimensional Heat Equation

Abstract

This paper explores the mathematical properties and practical application of the Fourier Series, focusing on the transition from periodic function approximation to solving Partial Differential Equations (PDEs). Through the analysis of a square wave function, we demonstrate the efficiency of using half-range expansions and odd extensions in simplifying the calculation of Fourier coefficients. Furthermore, the study extends these mathematical results to solve the one-dimensional heat equation using the method of separation of variables. Ultimately, this study demonstrates  the effectiveness of Fourier series methods in solving boundary value problems and modeling heat diffusion in finite domains.