Tracking fronts and interfaces is a fundamental task in several real world applications of mathematical modelling. The most widely used and successful technique for tracking front propagation is the so-called Level-Set Method (LSM). However, many applications require to track fronts embedded into a random environment. The LSM is generalized to describe such random situations according to the probability density function (PDF) of the displacement of interface particles around the average frontline determined by the ordinary LSM. The correct determination of the PDF, which describes the physics of the specific underlying process, is therefore of paramount importance for any applications.

The research follows three general themes:

- Mathematical investigation and theoretical development
- Development of models for anomalous diffusion in the framework of Fractional Calculus
- Applications in turbulent premixed combustion, wildland fire propagation and groundwater infiltration.

*Effective 1D front propagation in a subdiffusive system driven by the time-fractional diffusion equation*

Key Publications:

Pagnini G., Short note on the emergence of fractional kinetics. Physica A 409, 29-34 (2014)

Scalas E., Viles N., A Functional Limit Theorem for stochastic integrals driven by a time-changed symmetric α-stable Lévy process. Stochastic Processes and Their Applications 124, 385-410 (2014)

Pagnini G., Mentrelli A., Modelling wildland fire propagation by tracking random fronts. Nat. Hazards Earth Syst. Sci. Discuss. 1, 6521-6557 (2013)

Baleanu D., Diethelm K., Scalas E., Trujillo J.J., Fractional Calculus. Models and Numerical Methods. CNC Series on Complexity, Nonlinearity and Chaos, Vol. 3, World Scientific, (2012)

Pagnini G., Bonomi E., Lagrangian formulation of turbulent premixed combustion. Phys. Rev. Lett. 107, 044503 (2011)

Mura A., Pagnini G., Characterizations and simulations of a class of stochastic processes to model anomalous diffusion. J. Phys. A: Math. Theor. 41, 285003 (2008)

Mainardi F., Luchko Yu., Pagnini G., The fundamental solution of the space- time fractional diffusion equation. Fract. Calc. Appl. Anal. 4, 153-192 (2001)

Mathematical Aspects and Applications of Fractional Differential Equations Special Session organized by Carlota Cuesta (UPV/EHU) and Gianni Pagnini First Joint International Meeting RSME-SCM-SEMA-SIMAI-UMI Bilbao, June 30-July 4, 2014

Fractional Calculus, Probability and Non-local Operators: Applications and Recent Developments.

A workshop on the occasion of the retirement of Francesco Mainardi.

Bilbao - Basque Country - Spain, 6-8 November 2013