Modeling Extreme-Event Precursors with the Fractional Diffusion Equation

Abstract

Abstract Extreme catastrophic events such as earthquakes, terrorism and economic collapses are difficult to predict. We propose a tentative mathematical model for the precursors of these events based on a memory formalism and apply it to earthquakes suggesting a physical interpretation. In this case, a precursor can be the anomalous increasing rate of events (aftershocks) following a moderate earthquake, contrary to Omori's law. This trend constitute foreshocks of the main event and can be modelled with fractional time derivatives. A fractional derivative of order 0 < v < 2 replaces the first-order time derivative in the classical diffusion equation. We obtain the frequency-domain Green's function and the corresponding time-domain solution by performing an inverse Fourier transform. Alternatively, we propose a numerical algorithm, where the time derivative is computed with the Grünwald-Letnikov expansion, which is a finitedifference generalization of the standard finite-difference operator to derivatives of fractional order. The results match the analytical solution obtained from the Green function. The calculation requires to store the whole field in the computer memory since anomalous diffusion “remembers the past”.