A q-analog of the Seidel generation of Genocchi numbers

A new $q$-analog of Genocchi numbers is introduced through a q-analog of Seidel's triangle associated to Genocchi numbers. It is then shown that these $q$-Genocchi numbers have interesting combinatorial interpretations in the classical models for Genocchi numbers such as alternating pistols, alternating permutations, non intersecting lattice paths and skew Young tableaux.Comment: 17 page

A Finite Element Method With Singularity Reconstruction for Fractional Boundary Value Problems

We consider a two-point boundary value problem involving a Riemann-Liouville fractional derivative of order \al\in (1,2) in the leading term on the unit interval $(0,1)$. Generally the standard Galerkin finite element method can only give a low-order convergence even if the source term is very smooth due to the presence of the singularity term x^{\al-1} in the solution representation. In order to enhance the convergence, we develop a simple singularity reconstruction strategy by splitting the solution into a singular part and a regular part, where the former captures explicitly the singularity. We derive a new variational formulation for the regular part, and establish that the Galerkin approximation of the regular part can achieve a better convergence order in the $L^2(0,1)$, H^{\al/2}(0,1) and $L^\infty(0,1)$-norms than the standard Galerkin approach, with a convergence rate for the recovered singularity strength identical with the $L^2(0,1)$ error estimate. The reconstruction approach is very flexible in handling explicit singularity, and it is further extended to the case of a Neumann type boundary condition on the left end point, which involves a strong singularity x^{\al-2}. Extensive numerical results confirm the theoretical study and efficiency of the proposed approach.Comment: 23 pp. ESAIM: Math. Model. Numer. Anal., to appea

An Analysis of Galerkin Proper Orthogonal Decomposition for Subdiffusion

In this work, we develop a novel Galerkin-L1-POD scheme for the subdiffusion model with a Caputo fractional derivative of order $\alpha\in (0,1)$ in time, which is often used to describe anomalous diffusion processes in heterogeneous media. The nonlocality of the fractional derivative requires storing all the solutions from time zero. The proposed scheme is based on continuous piecewise linear finite elements, L1 time stepping, and proper orthogonal decomposition (POD). By constructing an effective reduced-order scheme using problem-adapted basis functions, it can significantly reduce the computational complexity and storage requirement. We shall provide a complete error analysis of the scheme under realistic regularity assumptions by means of a novel energy argument. Extensive numerical experiments are presented to verify the convergence analysis and the efficiency of the proposed scheme.Comment: 25 pp, 5 figure