Numerical Sensitivity Analysis of Diffusion Coefficients under Optimized Lipschitz Perturbations in Functional Spaces

Diffusion processes play a fundamental role in chemical, biological, and physical systems, with thermal diffusivity governing heat propagation under temperature variations. The Arrhenius relation links activation energy to reaction rates, providing a quantitative framework for studying temperature-dependent diffusion. This work investigates the Lipschitz stability of heat diffusion models with respect to both the diffusion coefficient and the activation energy. Using a one-dimensional heat equation, an analytical solution via separation of variables is derived, with the diffusion coefficient expressed in Arrhenius form. By applying the Mean Value Theorem, we establish bounds on the solution differences corresponding to variations in diffusion parameters, proving that the solution is Lipschitz continuous with respect to the diffusion coefficient. Extending this analysis to the activation energy, we demonstrate that small perturbations in the activation energy, Q lead to proportionally small changes in concentration, confirming the model’s robustness under parameter uncertainties. Numerical simulations illustrate that diffusion coefficients and concentrations remain bounded for realistic ranges of temperature and activation energy, supporting the theoretical findings. The results highlight the reliability of Arrhenius-based diffusion models for practical and industrial applications, including thermal transport, biomedical modelling, and corrosion studies. This work provides a rigorous mathematical foundation for the sensitivity analysis and stable numerical simulation of diffusion-driven processes under physically relevant parameter variations.