Can Huang

Spectral Methods for Substantial Fractional Differential Equations

In this paper, a non-polynomial spectral Petrov–Galerkin method and its associated collocation method for substantial fractional differential equations are proposed, analyzed, and tested. We modify a class of generalized Laguerre polynomials to form our trial basis and test basis. After a proper scaling of these bases, our Petrov–Galerkin method results in diagonal and well-conditioned linear systems for certain types of fractional differential equations. In the meantime, we provide superconvergence points of the Petrov–Galerkin approximation for associated fractional derivative and function value of true solution. Additionally, we present explicit fractional differential collocation matrices based upon Laguerre–Gauss–Radau points. It is noteworthy that the proposed methods allow us to adjust a parameter in the basis according to different given data to maximize the convergence rate. All these findings have been proved rigorously in our convergence analysis and confirmed in our numerical experiments.

Optimal Fractional Integration Preconditioning and Error Analysis of Fractional Collocation Method Using Nodal Generalized Jacobi Functions

In this paper, a nonpolynomial-based spectral collocation method and its well-conditioned variant are proposed and analyzed. First, we develop fractional differentiation matrices of nodal Jacobi polyfractonomials [M. Zayernouri and G. E. Karniadakis, J. Comput. Phys., 252 (2013), pp. 495--517] and generalized Jacobi functions [S. Chen, J. Shen, and L. L. Wang, Math. Comp., 85 (2016), pp. 1603--1638] on Jacobi--Gauss--Lobatto (JGL) points. We show that it suffices to compute the matrix of order

Convergence of a p-version/hp-version method for fractional differential equations

\mu\in (0,1)

A spectral collocation method for eigenvalue problems of compact integral operators

to compute that of any order

On the Spectrum Computation of Non-oscillatory and Highly Oscillatory Kernel with Weak Singularity

k +\mu

Spectral Collocation Methods for Differential-Algebraic Equations with Arbitrary Index

with integer

Well-conditioned fractional collocation methods using fractional Birkhoff interpolation basis

k \geq 0

Supergeometric convergence of spectral collocation methods for weakly singular Volterra and fredholm integral equations with smooth solutions

. With a different definition of the nodal basis, our approach also fixes a deficiency of the polyfractonomial fractional collocation method in [M. Zayernouri and G. E. Karniadakis, SIAM J. Sci. Comput., 38 (2014), pp. A40--A62]. Second, we provide explicit and compact formulas for computing the inverse of direct fractional differential collocation matrices at “interior” points by virtue of fractional JGL Birkhoff interpolation. This leads t...

Polynomial preserving recovery for quadratic elements on anisotropic meshes

Recently, M. Zayernouri and G.E. Karniadakis (2014) 10] proposed a new spectral method for fractional differential equations and observed an exponential rate of convergence. In this paper, we will prove a convergence rate of their spectral method and thus, provide an explanation for their observations.

The spectral collocation method for stochastic differential equations

We propose and analyze a new spectral collocation method to solve eigenvalue problems of compact integral operators, particularly, piecewise smooth operator kernels and weakly singular operator kernels of the form 1 |t−s|μ , 0 < μ < 1. We prove that the convergence rate of eigenvalue approximation depends upon the smoothness of the corresponding eigenfunctions for piecewise smooth kernels. On the other hand, we can numerically obtain a higher rate of convergence for the above weakly singular kernel for some μ’s even if the eigenfunction is not smooth. Numerical experiments confirm our theoretical results.