Modeling Urban Heat Island Dynamics Using Fractional Calculus: A Comparative Study with Classical Heat Diffusion Models

Abstract

The Urban Heat Island (UHI) effect, characterized by higher temperatures in urban areas compared with their surrounding rural environments, presents increasing challenges to urban sustainability, public health, and energy management. Conventional mathematical models describing UHI dynamics are typically based on classical heat diffusion equations that assume local interactions and short-memory processes. However, urban thermal behavior is significantly affected by heat storage within built materials, heterogeneous land surfaces, and delayed nocturnal cooling, all of which suggest the presence of long-memory and nonlocal effects. This study develops a fractional calculus–based framework for modeling the UHI effect and provides a systematic comparison with classical integer-order heat models. By replacing the standard time derivative with a fractional-order operator, the proposed model explicitly accounts for memory-dependent heat transfer and anomalous diffusion processes. Analytical formulations and numerical simulations are performed under representative initial and boundary conditions, and model performance is evaluated using comparative error metrics. The results indicate that fractional-order models achieve a closer agreement with observed urban temperature dynamics, particularly in capturing persistent warming and delayed cooling patterns. Sensitivity analysis further shows that sub-diffusive fractional orders provide the most realistic representation of UHI behavior. Overall, the study demonstrates the advantages of fractional calculus in enhancing both the descriptive and predictive capabilities of urban heat models and highlights its potential as an effective tool for urban climate analysis and heat-mitigation planning.