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# Fractional Calculus and Applied Analysis

Editor-in-Chief: Kiryakova, Virginia

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Volume 18, Issue 1

# Modeling Extreme-Event Precursors with the Fractional Diffusion Equation

Michele Caputo
/ José M. Carcione
• Istituto Nazionale di Oceanografia e di Geofisica Sperimentale (OGS) Borgo Grotta Gigante 42c, 34010 Sgonico, Trieste, ITALY
• Email
• Other articles by this author:
/ Marco A. B. Botelho
Published Online: 2015-02-10 | DOI: https://doi.org/10.1515/fca-2015-0014

## 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”.

MSC 2010: Primary 35R11; Secondary 86A15; 86A17; 86-08

Key Words and Phrases: extreme events; precursors; forecast; earthquakes; Omori's law memory; fractional derivatives

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Published Online: 2015-02-10

Citation Information: Fractional Calculus and Applied Analysis, Volume 18, Issue 1, Pages 208–222, ISSN (Online) 1314-2224,

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