Special Issue on "Fractional PDEs: Theory, Numerics, and Applications"

Fractional Partial Differential Equations (FPDEs) are emerging as a powerful tool for modeling challenging multiscale phenomena including overlapping microscopic and macroscopic scales, anomalous transport, and long-range temporal or spatial interactions. The fractional order may be a function of space–time or even be a distribution, opening up tremendous opportunities for modeling and simulation of multiphysics phenomena, e.g. seamless transition from wave propagation to diffusion, or from local to non-local dynamics. It is even possible to construct data-driven fractional differential operators that 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. In other words, a new (and simple) data assimilation paradigm can be formulated to determine the fractional order (or possibly a distribution) by taking into account a diverse set of sources of information, including available experimental data albeit of variable fidelity. Researchers who are new to the field of FPDEs wonder as to when “to think fractionally?” and in which areas factional modeling may be useful. Fractional modeling is effective in systems with self-similarity and scale-invariance, systems characterized by non-Markovian behavior and long-range interactions or power laws as well as for lossy or disordered media. In applications such as anomalous diffusion or propagation through random media, in turbulence, in porous and granular media and sediment transport, in biophysics and cell migration, in crack propagation, in viscoplastic materials, etc., we encounter processes that are perhaps best described by timeor space-fractional derivatives. In the context of computational physics, even classical long-standing issues of monotonicity, boundary conditions, anisotropy, and limited regularity, typical of problems involving interfaces, e.g. multiphase flows and fluid–structureinteractions, can be re-formulated and re-interpreted in terms of fractional derivatives as well as probabilistically. Examples include the classical second Stokes flow problem of vorticity diffusion, which can be reformulated as “half-order” advection equation with a transport velocity proportional to the square-root of viscosity. Another example involves Lévy flights that can explain super-diffusion and acceleration of steep fronts in reaction–diffusion systems. Fractional calculus (FC) is of course not new, initiated in 1665 in a letter of L’Hospital to Leibnitz requesting clarification on the meaning of the half derivative of the function f(x) = x. Later, Leibnitz, Liouville (1834) and Riemann (1892) developed the basic mathematical ideas, and subsequently, FC was brought to the attention of the engineering community by Heaviside in the 1890s. However, until the end of the previous century only isolated efforts were undertaken on the theoretical or application side. Hence, it is reasonable to wonder why fractional calculus was not adopted much earlier by the computational science community. This is a difficult question to answer but two reasons that fractional modeling has not been used extensively so far is that FPDEs are non-unique, and that they are quite expensive to solve numerically. We believe, however, that both issues can be effectively addressed in the near-future, especially at this juncture, as the first is closely related to availability of (big) data, while the second to advanced discretizations and solvers. In both areas, there has been impressive progress during the last three decades. The number of papers on FC that have been published since 1985 is increasing exponentially, and so is the number of scientific fields affected by FC and FPDEs. For example, according to the Web of Science, in 1985 there were only 14 fields affected by the FC and FPDEs while at the present time, there are ten times more, see Fig. 1. Hence, the current number – 145 fields in the Web of Science – means that now FC and FPDEs affect all fields of science and engineering as the Web of Science lists currently exactly 145 fields/categories. With this ever-expanding range of applications and models based on fractional calculus comes a need for the development of rigorous theory and corresponding robust, accurate and efficient computational methods for approximating the solutions of FPDEs. On the theoretical side, the well-posedness of ad hoc fractional reformulations of conservation laws is in doubt. Even basic questions, e.g. what are the proper boundary conditions for non-local problems, is a complex and generally open problem. Moreover, non-local features and the singular kernels involved in fractional operators give rise to significant computational challenges to the extent that current numerical methods for fractional modeling in three-dimensions and for long-time integration become prohibitively expensive. Accordingly, new research is needed on two main fronts: fast linear