Superconvergence Points For The Spectral Interpolation Of Riesz Fractional Derivatives

In this paper, superconvergence points are located for the approximation of the Riesz derivative of order $\alpha$ using classical Lobatto-type polynomials when $\alpha \in (0,1)$ and generalized Jacobi functions (GJF) for arbitrary $\alpha > 0$, respectively. For the former, superconvergence points are zeros of the Riesz fractional derivative of the leading term in the truncated Legendre-Lobatto expansion. It is observed that the convergence rate for different $\alpha$ at the superconvergence points is at least $O(N^{-2})$ better than the optimal global convergence rate. Furthermore, the interpolation is generalized to the Riesz derivative of order $\alpha > 1$ with the help of GJF, which deal well with the singularities. The well-posedness, convergence and superconvergence properties are theoretically analyzed. The gain of the convergence rate at the superconvergence points is analyzed to be $O(N^{-(\alpha+3)/2})$ for $\alpha \in (0,1)$ and $O(N^{-2})$ for $\alpha > 1$. Finally, we apply our findings in solving model FDEs and observe that the convergence rates are indeed much better at the predicted superconvergence points

Two families of novel second-order fractional numerical formulas and their applications to fractional differential equations

In this article, we introduce two families of novel fractional $\theta$-methods by constructing some new generating functions to discretize the Riemann-Liouville fractional calculus operator $\mathit{I}^{\alpha}$ with a second order convergence rate. A new fractional BT-$\theta$ method connects the fractional BDF2 (when $\theta=0$) with fractional trapezoidal rule (when $\theta=1/2$), and another novel fractional BN-$\theta$ method joins the fractional BDF2 (when $\theta=0$) with the second order fractional Newton-Gregory formula (when $\theta=1/2$). To deal with the initial singularity, correction terms are added to achieve an optimal convergence order. In addition, stability regions of different $\theta$-methods when applied to the Abel equations of the second kind are depicted, which demonstrate the fact that the fractional $\theta$-methods are A($\vartheta$)-stable. Finally, numerical experiments are implemented to verify our theoretical result on the convergence analysis.Comment: 16 pages, 10 figure

Numerical methods for nonlocal and fractional models

Partial differential equations (PDEs) are used, with huge success, to model phenomena arising across all scientific and engineering disciplines. However, across an equally wide swath, there exist situations in which PDE models fail to adequately model observed phenomena or are not the best available model for that purpose. On the other hand, in many situations, nonlocal models that account for interaction occurring at a distance have been shown to more faithfully and effectively model observed phenomena that involve possible singularities and other anomalies. In this article, we consider a generic nonlocal model, beginning with a short review of its definition, the properties of its solution, its mathematical analysis, and specific concrete examples. We then provide extensive discussions about numerical methods, including finite element, finite difference, and spectral methods, for determining approximate solutions of the nonlocal models considered. In that discussion, we pay particular attention to a special class of nonlocal models that are the most widely studied in the literature, namely those involving fractional derivatives. The article ends with brief considerations of several modeling and algorithmic extensions which serve to show the wide applicability of nonlocal modeling.Comment: Revised/Improved version. 126 pages, 18 figures, review pape