L3 Chimie et Physique

This document presents provisional notes for a mathematics course targeted at chemistry and physics undergraduate students. The objective is to illustrate the application of Fourier analysis in solving the heat equation and the wave equation, focusing on problems that lead to linear differential equations with non-constant coefficients, where solutions cannot be expressed in terms of elementary functions. The study delves into special functions, specifically Bessel functions, and concludes with an introduction to probability theory on a finite set. The methodology involves separation of variables to address the heat equation, modeling the propagation of heat in a finite rod, and establishing compatibility conditions for initial conditions. Results show that the Sturm-Liouville problem yields distinct eigenvalues and corresponding eigenfunctions representing the solutions for both the heat equation and the wave equation through Fourier series in sine form. The linearity of the problem leads to the principle of superposition, allowing combinations of solutions. This foundational course aids in understanding essential mathematical tools for further studies in chemistry and physics.