A Fractional Calculus Approach to Energy Balance Modeling: Incorporating Memory for Responsible Forecasting

Global climate change demands modeling approaches that are both computationally efficient and physically faithful to the system’s long-term dynamics. Classical Energy Balance Models (EBMs), while valuable, are fundamentally limited by their memoryless exponential response, which fails to represent the prolonged thermal inertia of the climate system—particularly that associated with deep-ocean heat uptake. In this study, we introduce a fractional Energy Balance Model (fEBM) by replacing the classical integer-order time derivative with a Caputo fractional derivative of order α(0<α≤1), thereby embedding long-range memory directly into the model structure. We establish a rigorous mathematical foundation for the…

• KF
  King Faisal University