A Unified Dynamical System for Nonlinear Diffusion in the Human Respiratory System

Abstract

In this paper, we develop a unified mathematical framework for nonlinear diffusion in the human respiratory system, coupling gas exchange with fluid dynamics in both healthy and diseased lungs. We generalize Fick’s second law by making diffusivity concentration-dependent D(C) = D0(1 + αC), modeling effects such as inflammation or tissue damage. Three finite difference schemes (explicit, implicit, Crank-Nicolson) are used to solve the nonlinear PDE, with the Crank-Nicolson method being most accurate (second-order convergence) and stable. Analytical solutions to the linear problem confirm the numerical methodology. Through a traveling wave transformation C(x, t) = U(z), z = x − vt, the PDE is reduced to an ODE system, allowing phase-plane analysis of wave propagation and steady states. Theoretical and computational results are bridged by this framework, which provides a flexible tool to investigate oxygen transport and fluid buildup in diseases such as emphysema or pleural effusion. Predicting spatial-temporal disease progression, the results demonstrate its potential, with applications in clinical modeling and therapeutic studies.