Of MatUrban Heat Island Modeling via Fractional Calculus: A Comparative Study Relative to Classical Models

Liu W, Ihedioha A, Oso BO and Ani S

Published on: 2026-02-17

Abstract

The Urban Heat Island (UHI) effect, characterized by elevated temperatures in urban areas relative to their rural surroundings, poses growing challenges to urban sustainability, public health, and energy management. Traditional mathematical descriptions of UHI dynamics are commonly based on classical heat diffusion equations, which assume local interactions and short-memory processes. However, urban thermal behavior is strongly influenced by heat storage in built materials, heterogeneous land surfaces, and delayed nocturnal cooling, all of which indicate the presence of long-memory and nonlocal effects. This study develops a fractional calculus–based framework for modeling the UHI effect and presents a systematic comparison with classical integer-order heat models. By replacing the standard time derivative with a fractional-order operator, the proposed model explicitly incorporates memory-dependent heat transfer and anomalous diffusion. Analytical formulation and numerical simulations are carried out under representative boundary and initial conditions, and model performance is evaluated using comparative error metrics. The results demonstrate that fractional-order models provide a closer fit to observed urban temperature dynamics, particularly in capturing persistent warming and delayed cooling patterns. Sensitivity analysis further reveals that sub-diffusive fractional orders offer the most realistic representation of UHI behavior. Overall, the study highlights the advantages of fractional calculus in improving both the descriptive and predictive capability of urban heat models, and it underscores the potential of this approach as a valuable tool for urban climate analysis and heat mitigation planning.