In this chapter, we selectively present global methods for efficiently solving FPDEs, employing the basis functions introduced in Chapters 2 and 3. Here, we adopt the term global often in the context of space-time, considering time as another (space-like) spectral direction. We examine a number of typical FPDEs, which we introduced and probabilistically interpreted in Chapter 1, including: the subdiffusion equation, tempered fractional diffusion on the half/whole line, in addition to the generalized and unified (1+d)-dimensional sub-to-superdiffusion FPDE model for d≥1, where a single FPDE form can model a range of physical processes by just varying the corresponding temporal/spatial fractional derivatives in the model, hence, rendering the FPDE elliptic, parabolic, and/or hyperbolic on the (1+d)-dimensional space-time hypercube. In this chapter, we employ one-sided, two-sided, constant/variable-order, and fully distributed order fractional operators, introduced in Chapters 1 and 2.

As highlighted in Chapter 1, anomalous transport phenomena can be observed in a wide variety of complex, multi-scale, and multi-physics systems such as: sub-/super-diffusion in subsurface transport, kinetic plasma turbulence, aging polymers, glassy materials, in addition to amorphous semiconductors, biological cells, heterogeneous tissues, and fractal disordered media. In this chapter, we focus on some selective applications of FPDEs and the methods presented in earlier chapters, reporting the scientific evidence of how and why fractional modeling naturally emerges in each case, along with a review of selected nonlocal mathematical models that have been proposed. The applications of interest are: (i) concentration transport in surface/subsurface dynamics, (ii) complex rheology and material damage, and (iii) fluid turbulence and geostrophic transport.

We initially introduce the standard diffusion model solving the PDF of the Brownian motion/process, satisfying the normal scaling property. This happens through a new definition of the process increments, where they are no longer drawn from a normal distribution, leading to α-stable Lévy flights at the microscopic level and correspondingly an anomalous diffusion model with a fractional Laplacian at the macroscopic scale. Next, we show how the Riemann–Liouville fractional derivatives emerge in another anomalous diffusion model corresponding to the asymmetric α-stable Lévy flights at small scales. Subsequently, we introduce the notion of subdiffusion stochastic processes, in which the Caputo time-fractional derivative appears in the anomalous subdiffusion fractional model. We combine the previous two cases, and construct continuous-time random walks, where a space-time fractional diffusion model will solve the evolution of the probability density function of the stochastic process. Next, we motivate and introduce many other types of fractional derivatives that will code more complexity and variability at micro-to-macroscopic scales, including fractional material derivatives, time-variable diffusivity for the fractional Brownian motion, tempered/variable-order/distributed-order/vector fractional calculus, etc.

This chapter provides a comprehensive presentation of global numerical methods for solving FODEs employing the polynomial and non-polynomial bases, introduced in Chapter 2. The FODEs of interest will be initial-/boundary-value problems, posed using a variety of fractional derivatives (e.g., Caputo, Riemann–Liouville, Riesz, one-sided, two-sided, variable-order, distributed order, etc.), introduced in Chapters 1 and 2. We devote Sections 3.1 and 3.2 to introducing a series of variational and non-variational spectral methods in single domains, where the solution singularities can occur at the initial or boundary points. In a variational formulation of an FODE, one first obtains the weak (variational) form of the given equation, where the highest derivative order is reduced using integration-by-parts, and then solves the variational formulation by constructing the corresponding (finite-dimensional) solution and test subspaces. In non-variational problems, one rather directly solves the strong (original) FODE, hence assuming a higher regularity in the solution. Moreover, we introduce spectral element methods (SEM) for FODEs in multiple domains for the main purpose of capturing possible interior/boundary singularities.

We present the need for new fractional spectral theories, explicitly yielding rather non-polynomial, yet orthogonal, eigensolutions to effectively represent the singularities in solutions to FODEs/FPDEs. To this end, we present the regular/singular theories of fractional Sturm–Liouville eigen-problems. We call the corresponding explicit eigenfunctions of these problems Jacobi poly-fractonomials. We demonstrate their attractive properties including their analytic fractional derivatives/integrals, three-term recursions, special values, function approximability, etc. Subsequently, we introduce the notion of generalized Jacobi poly-fractonomials (GJPFs), expanding the range of admissible parameters also allowing function singularities of negative indices at both ends. Next, we present a rigorous approximation theory for GJPFs with numerical examples. We further generalize our fractional Sturm–Liouville theories to regular/singular tempered fractional Sturm–Liouville eigen-problems, where a new exponentially tempered family of fractional orthogonal basis functions emerges. We finally introduce a variant of orthogonal basis functions suitable for anomalous transport that occurs over significantly longer time-periods.

Fractional diffusion equations are naturally derived on unbounded domains, and their solutions usually decay very slowly at infinity. A usual approach to dealing with unbounded domains is to use a domain truncation with exact or approximate transparent boundary conditions. But since accurate transparent boundary conditions at truncated boundaries are not easily available, we develop in this chapter efficient spectral methods for FPDEs on unbounded domains so as to avoid errors introduced by domain truncation. Formulation of Laplacians in bounded domains will be presented in Chapter 6.

The fractional Laplacian has multiple equivalent characterizations. Moreover, in bounded domains, boundary conditions must be incorporated in these characterizations in mathematically distinct ways, and there is currently no consensus in the literature as to which definition of the fractional Laplacian in bounded domains is most appropriate for a given application. Although the study of the fractional Laplacian is far from complete, this chapter can serve as a proper educational/research starting point for students/researchers in order to employ these operators to model complex anomalous systems. The Riesz (or integral) definition, for example, admits a nonlocal boundary condition, where the value of a function must be prescribed on the entire exterior of the domain in order to compute its fractional Laplacian. By contrast, the spectral definition requires only the standard local boundary condition. We compare several commonly used definitions of the fractional Laplacian theoretically, through their stochastic interpretations as well as their analytical properties. Then, we present quantitative comparisons using a sample of state-of-the-art methods. We finally discuss recent advances on nonzero boundary conditions and present new methods to discretize such boundary value problems.

We present efficient time-stepping schemes for accurate and long-time integration of time-fractional models. We direct our attention to introducing local multi-step finite-difference methods for time-fractional models. We introduce the fractional Adams family of schemes, which seamlessly generalize the classical explicit Adams–Bashforth and implicit Adams–Moulton schemes. Next, we combine the fractional Adams implicit-explicit (IMEX) schemes for stable and long-time integrations along with employing new correction terms, which enrich the underlying approximation space, especially in the context of nonlinear FODEs. We also investigate the linear stability of the fractional IMEX methods along their fast implementations. To this end, we present a fast approximate inversion scheme and fast computation of hypergeometric functions, which makes the IMEX algorithms amenable for accurate long-time integration of FODEs. To reduce the number of correction terms, we formulate a self-singularity-capturing scheme, which automatically captures the singular structure of the unknown solution (with even several random singularities without any prior knowledge), employing a two-stage time-integration algorithm. We will test the ease and efficiency of the method in the context of challenging cases, e.g., long-integration of singular-oscillatory solutions and nonlinear FODEs.