A multi-domain spectral method for time-fractional differential equations

Abstract

This paper proposes an approach for high-order time integration within a multi-domain setting for timefractional differential equations. Since the kernel is singular or nearly singular, two main difficulties arise after the domain decomposition: how to properly account for the history/memory part and how to perform the integration accurately. To address these issues, we propose a novel hybrid approach for the numerical integration based on the combination of three-term-recurrence relations of Jacobi polynomials and high-order Gauss quadrature. The different approximations used in the hybrid approach are justified theoretically and through numerical examples. Based on this, we propose a new multi-domain spectral method for high-order accurate time integrations and study its stability properties by identifying the method as a generalized linear method. Numerical experiments confirm hp-convergence for both time-fractional differential equations and time-fractional partial differential equations.

An Analysis of the Modified L1 Scheme for Time-Fractional Partial Differential Equations with Nonsmooth Data

“…There has been much recent interest in developing numerical methods for (1.1), especially spectral methods, [4], [5], [43], [45], and the discontinuous Galerkin method [8], [32], [33], [34]. In this paper, we will consider some time discretization schemes for (1.1) using the direct approximation of the time fractional derivative.…”

A parareal method for time-fractional differential equations

“…Next, define the approximating space: 4) where P N is the polynomial space of degree N defined on Ω i , i.e., we represent the global solution as piecewise polynomials. For simplicity, we consider a uniform decomposition (h i ≡ h) and the same number of degrees of freedom on each element.…”

“…The expression for {M i } is provided in [4]. In practice, it is preferred to work directly in physical space.…”

Shifted fractional-order Jacobi orthogonal functions: Application to a system of fractional differential equations

“…Due to all types of spectral methods are global, they very convenient for approximating the solution of linear and nonlinear FDEs [32,33,34,35]. Doha et al [36] presented and developed spectral tau and collocations techniques to solve the multi-term FDEs including linear and nonlinear terms, using Jacobi polynomials, in which the authors generalized the quadrature Legendre tau method [37] and the Chebyshev spectral methods [38].…”