Applications of series in physics (heat conduction, electromagnetic field)

Introduction

Mathematical series are foundational tools in physics, enabling the approximation of complex phenomena through iterative, convergent representations. In the study of heat conduction and electromagnetic fields, series methods provide rigorous frameworks for solving partial differential equations (PDEs) that govern physical systems. This article explores how Fourier series and Taylor series are applied in these domains, emphasizing their convergence properties and practical implications. The focus is on the interplay between theoretical mathematics and physical reality, illustrating how series solutions enable the modeling of thermal diffusion and wave propagation.

Core Content

Heat Conduction

The heat equation, a PDE describing the distribution of thermal energy in a medium, is a quintessential example of the application of series in physics. Given by
$$ \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{