Fractional modelling is a new meta-discipline that naturally addresses the complexity-induced research barriers in divers’ areas of mathematics, science, and engineering. Fractional partial differential equations (FPDEs) rigorously extend the standard integer-order calculus and differential equations to those of fractional orders. The area of FPDEs is a rapidly growing field of research at the interface between computational mathematics, probability, and mathematical physics, where FPDEs emerge as powerful tools for modelling challenging multi-physics multiscale phenomena that exhibit anomalous (sub- or super-) diffusion, nonlocal interactions, long-memory dependence, power-law effects, and self-similar structures. In such complex processes, FPDEs naturally appear as physically correct governing equations leading to high-fidelity modelling and predictive simulations, which otherwise cannot be achieved using classical PDEs. Fractional modelling also plays a crucial role in modelling transport phenomena, where rates of change in the quantity of interest depend on space and/or time. In this context, FPDEs with ‘variable-to-distributed’ orders can be exploited in diverse physical and biological applications.
The present IJCM special issue on FPDEs Advances on Computational Fractional Partial Differential Equations is a timely effort to highlight and promote the recent international research activities on this fast-growing field of study. The 22 papers of the present special issue can be categorized into three distinguished categories: (I) Theory: including papers on the theoretical aspects of FPDEs, e.g. variational forms, delay forms, distributed-order wave-phenomenon, total fractional derivatives, and nonlinear time-fractional Schrödinger equation, (II) Numerical Methods: including a wide range of numerical methods such as finite-difference methods, finite-element method, spectral methods (Petrov–Galerkin, pseudo-spectral, etc.), iterative reproducing kernel method, in addition to time-integration methods, and (III) Applications: including, anomalous diffusion, dilation, and erosion in image processing, image denoizing, fractional viscoelastic modelling of human ear, in addition to non-Fickian reaction–diffusion with nonlocal effects.
The editorial team would like to specially thank the IJCM team at Taylor & Francis for their kind support on the publication of this special issue and for their editorial help. We also acknowledge all authors and referees from many countries for their constant interest and help and for their contribution to the quality of the issue.