Diffusion Into a Catalyst: Regular Perturbation Analysis

Applied Mathematics in Chemical and Biological Systems Understanding Complex Phenomena Through Perturbation, Diffusion, and Nonlinear Dynamics Nature thrives on complexity — nonlinearities, instabilities, and competing physical timescales. This graduate-level lecture series explores how applied mathematics reveals structure within that complexity. From the stability of diffusion to the onset of nonlinear behavior, we’ll develop mathematical intuition through concrete, physically grounded examples drawn from chemistry, biology, and heat/mass transfer. What You’ll Learn Across these lectures, we’ll move from linear diffusion to nonlinear dynamics, and from simple perturbations to multi-scale asymptotics—building a conceptual toolkit that every scientist and engineer can use to reason about complex systems: L1: Building intuition for nonlinear dynamics using the heat equation, negative diffusivity, dispersion relations, and the emergence of the Kuramoto–Sivashinsky equation. L2: Regular perturbation analysis of a diffusion–reaction system where the reaction rate is small compared to diffusion. L3: Singular perturbation analysis of the same system when the reaction rate dominates diffusion, revealing boundary layers and sharp gradients. L4: Diffusion into a sphere with oscillating boundary conditions, exploring how external frequency competes with internal diffusion through both regular and singular perturbations. Intended for graduate students, researchers, and educators in science and engineering, this series blends physical intuition, historical context, and mathematical rigor to illuminate the subtle beauty of applied mathematics in chemistry and biology.