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Fractional PDEs, which involve a fractional-order derivative in time or/and space and more generally nonlocal derivatives (e.g., fractional Laplacian), have been established as extremely powerful modeling tools. Fractional PDEs can adequately describe transport processes collectively known as anomalous diffusion, where the mean squared displacement grows sublinearly/superlinearly with the time, as opposed to the linear growth for normal diffusion. The spectrum of their practical applications is very broad and spans many diverse disciplines, e.g., subsurface flow, thermal diffusion in fractal domains, and dynamics of protein molecules. Compared with mathematical models describing normal diffusion, nonlocal models pose substantial challenges to both their rigorous analysis and their numerical treatment. Hence, the development of reliable and efficient numerical schemes and their analysis is of significant theoretical and practical interest. In the last two decades, intensive research has been carried out in this area. This special issue consists of fourteen invited contributions (four in the current issue, and the remaining ten in issue 18(1)) on novel numerical methods for nonlocal models and their mathematical analysis. These papers cover a broad range of topics including construction of computational schemes, stability and error analysis, and testing and numerical simulations. Below we briefly describe the content of the special issue. The papers [2, 14, 17, 18, 27, 35] deal with the time fractional diffusion problem, which involves a fractional-order derivative of either Riemann–Liouville or Caputo type in time of order α ∈ (0, 1). It is frequently employed to describe subdiffusion processes, and hence also known as the subdiffusion equation in the literature. This model can be viewed as the macroscopic counterpart of continuous time random walk. The solution to the subdiffusion problem usually contains a weak singularity at time t = 0, even for smooth problem data. Thus spectral methods using polynomials generally do not work well. One promising idea is to use nonpolynomial functions [9]. The paper [17] proposes two spectral methods for the subdiffusion equation by approximating the solution by Müntz polynomials, based on Galerkin formulation and Petrov–Galerkin formulation, respectively, and shows that both methods are exponentially convergent for general source terms, for which the exact solution has very limited regularity. Due to the limited smoothing property of the solution operator, many time stepping schemes are only first-order accurate in time, if implemented straightforwardly. The paper [35] studies a time stepping scheme, which can be derived via either Diethelm’s method [12] or piecewise constant discontinuous Galerkin method [30]. Just as expected, a direct implementation of the method has only an O(τ) rate, with τ being the time step size. The authors propose an interesting corrected variant of the scheme, which modifies only the first step of the scheme while improves the convergence rate to O(τ1+α), in a manner similar to the works [10, 20, 21].

Preface: Numerical Analysis of Fractional Differential Equations