Innovations in Fractional Calculus and Applications to Functional and Biological Materials
CECAM-HQ-EPFL, Lausanne, Switzerland
Fractional partial differential equations (FPDEs) are emerging as a powerful tool for modeling challenging multiscale phenomena including microstructure in materials, overlapping microscopic and macroscopic scales, anomalous transport, and long-range time memory or spatial interactions. Furthermore, fractional calculus is an excellent framework for modelling nonconventional fractal and non-local media, opening valuable prospects on future engineered materials. Compared to integer-order PDEs, the fractional order of the derivatives in FPDEs may be a function of space and time or even a distribution, opening up great opportunities for modeling and simulation of multi-physics phenomena, e.g. seamless transition from wave propagation to diﬀusion, or from local to non-local dynamics. In addition, data-driven fractional diﬀerential operators may be constructed to ﬁt data from a particular experiment or speciﬁc phenomenon, including the eﬀect of uncertainties, in which the fractional orders are determined directly from the data, and introducing nonlinearities leading to more complex operators, with one or more fractional orders, capable to model less typical phenomena (such as, for instance, wave propagation in heterogeneous systems). Similarly, in imaging applications the variable and even distributed fractional order that may change in space offer great flexibility that can be used to suppress noise while preserving edge sharpness.
In short, FPDEs lead to a paradigm shift, according to which data-driven fractional operators may be constructed to model a speciﬁc phenomenon instead of the current practice of tweaking free parameters that multiply pre-set integer-order differential operators. Also important is that the misspeciﬁcation of physical models using integer order derivatives leads to a variable coeﬃcient ﬁt (struggling to ﬁt the data at each location, for example) whereas it was shown in the literature that the “correct” fractional order model can ﬁt all the data with a constant coeﬃcient model.
The main reasons that FPDE modeling has not been used extensively so far is that FPDEs are non-unique and that they are quite expensive to solve numerically as they typically generate dense linear algebraic systems due to the nonlocality of fractional differential operators. Furthermore, FPDEs present additional mathematical and numerical difficulties, which are not encountered in the context of integer-order PDEs.
This workshop will focus on the use of fractional calculus in different areas of materials, addressing multiscale structure, porous media, crack propagation, visco-elasto-plasticity, wave propagation, non-local continua, dynamic fracture in brittle and quasi-brittle solids.etc.. We will invite researchers who work on multiscale modeling of materials as well as applied mathematicians who have made significant progress in advancing the fundamentals of FPDEs.
Igor Podlubny (Technical University of Kosice) - Organiser
Ignacio Pagonabarraga (University of Barcelona) - Organiser
Jan Hesthaven (EPFL) - Organiser
George Karniadakis (Brown University, Providence, RI) - Organiser