Analytic solution of nonlinear fractional Burgers-type equation by invariant subspace method

In this paper we study the analytic solutions of Burgers-type nonlinear fractional equations by means of the Invariant Subspace Method. We first study a class of nonlinear equations directly related to the time-fractional Burgers equation. Some generalizations linked to the forced time-fractional Burgers equations and variable-coefficient diffusion are also considered. Finally we study a Burgers-type equation involving both space and time-fractional derivatives

Nonlinear time-fractional dispersive equations

In this paper we study some cases of time-fractional nonlinear dispersive equations (NDEs) involving Caputo derivatives, by means of the invariant subspace method. This method allows to find exact solutions to nonlinear time-fractional partial differential equations by separating variables. We first consider a third order time-fractional NDE that admits a four-dimensional invariant subspace and we find a similarity solution. We also study a fifth order NDE. In this last case we find a solution involving Mittag-Leffler functions. We finally observe that the invariant subspace method permits to find explicit solutions for a wide class of nonlinear dispersive time-fractional equations.Comment: 14 pages; in press in Communications in Applied and Industrial Mathematics (2014

Weak solutions to Allen-Cahn-like equations modelling consolidation of porous media

We study the weak solvability of a system of coupled Allen–Cahn–like equations resembling cross–diffusion which is arising as a model for the consolidation of saturated porous media. Besides using energy like estimates, we cast the special structure of the system in the framework of the Leray–Schauder fixed point principle and ensure this way the local existence of strong solutions to a regularised version of our system. Furthermore, weak convergence techniques ensure the existence of weak solutions to the original consolidation problem. The uniqueness of global-in-time solutions is guaranteed in a particular case. Moreover, we use a finite difference scheme to show the negativity of the vector of solutions. Keywords: Weak solutions; cross–diffusion system; energy method; Leray–Schauder fixed point theorem; finite differences; consolidation of porous media

Centroid moment tensor catalog with 3D lithospheric wavespeed model. The 2016–2017 Central Apennines sequence

Moment tensor inversions of broadband velocity data are usually managed by adopting Green's functions for 1D layered seismic wave speed models. This assumption can impact on source parameter estimates in regions with complex 3D heterogeneous structures and discontinuities in rock properties. In this work, we present a new centroid moment tensor (CMT) catalog for the Amatrice-Visso-Norcia (AVN) seismic sequence based on a recently generated 3D wave speed model for the Italian lithosphere. Forward synthetic seismograms and Fréchet derivatives for CMT-3D inversions of 159 earthquakes with Mw ≥ 3.0 are simulated using a spectral-element method (SEM) code. By comparing the retrieved solutions with those from time domain moment tensor (TDMT) catalog, obtained with a 1D wave speed model calibrated for Central Apennines (Italy), we observe a remarkable degree of consistency in terms of source geometry, kinematics, and magnitude. Significant differences are found in centroid depths, which are more accurately estimated using the 3D model. Finally, we present a newly designed parameter, τ, to better quantify and compare a-posteriori the reliability of the obtained MT solutions. τ measures the goodness of fit between observed and synthetic seismograms accounting for differences in amplitude, arrival time, percentage of fitted seconds, and the usual L2-norm estimate. The CMT-3D solutions represent the first Italian CMT catalog based on a full-waveform 3D wave speed model. They provide reliable source parameters with potential implications for the structures activated during the sequence. The developed approach can be readily applied to more complex Italian regions where 1D models are underperforming and not representative of the area

What Can We Learn from a Simple Physics-Based Earthquake Simulator?

Physics-based earthquake simulators are becoming a popular tool to investigate on the earthquake occurrence process. So far, the development of earthquake simulators is commonly led by the approach “the more physics, the better”. However, this approach may hamper the comprehension of the outcomes of the simulator; in fact, within complex models, it may be difficult to understand which physical parameters are the most relevant to the features of the seismic catalog at which we are interested. For this reason, here, we take an opposite approach and analyze the behavior of a purposely simple earthquake simulator applied to a set of California faults. The idea is that a simple simulator may be more informative than a complex one for some specific scientific objectives, because it is more understandable. Our earthquake simulator has three main components: the first one is a realistic tectonic setting, i.e., a fault data set of California; the second is the application of quantitative laws for earthquake generation on each single fault, and the last is the fault interaction modeling through the Coulomb Failure Function. The analysis of this simple simulator shows that: (1) the short-term clustering can be reproduced by a set of faults with an almost periodic behavior, which interact according to a Coulomb failure function model; (2) a long-term behavior showing supercycles of the seismic activity exists only in a markedly deterministic framework, and quickly disappears introducing a small degree of stochasticity on the recurrence of earthquakes on a fault; (3) faults that are strongly coupled in terms of Coulomb failure function model are synchronized in time only in a marked deterministic framework, and as before, such a synchronization disappears introducing a small degree of stochasticity on the recurrence of earthquakes on a fault. Overall, the results show that even in a simple and perfectly known earthquake occurrence world, introducing a small degree of stochasticity may blur most of the deterministic time features, such as long-term trend and synchronization among nearby coupled faults. © 2018, Springer International Publishing AG, part of Springer Nature

Heterogeneous perturbation of fluid density and solid elastic strain in consolidating porous media

International audienc