Optimal Error Estimates of Spectral Petrov--Galerkin and Collocation Methods for Initial Value Problems of Fractional Differential Equations

Abstract

We present optimal error estimates for spectral Petrov-Galerkin methods and spectral collocation methods for linear fractional ordinary differential equations with initial value on a finite interval. We also develop Laguerre spectral Petrov-Galerkin methods and collocation methods for fractional equations on the half line. Numerical results confirm the error estimates. Introduction.Numerical methods for fractional differential equations have been investigated for decades; see, e.g., [8,12,33]. However, spectral methods for these equations have a rather short history since the solutions usually have some singular lower-order derivatives. The recent investigation of spectral methods for these singular problems is motivated by the following facts. First, all numerical methods for these problems are nonlocal as spectral methods are. For example, in finite difference methods, see, e.g., [33], and in finite element methods, see, e.g., [13,18], data at almost all grids or all elements are exploited to approximate the integral operators at one grid or element. These methods are still nonlocal, even when a "fixed memory principle" is applied to reduce the use of all data; see, e.g., [15,19]. Second, spectral methods lead to high-order accuracy when the solutions are smooth; see, e.g., [7,24] for integerorder differential equations. For fractional differential equations with weakly singular kernels, solutions can be smooth even when some inputs of the equations are singular.Consider the following fractional ordinary differential equation (FODE) over the interval I = (−1, 1):

Implicit-Explicit Difference Schemes for Nonlinear Fractional Differential Equations with Nonsmooth Solutions

“…When solving these FDEs, the singularity requires special attention to obtain the expected high accuracy. Several approaches have been proposed to deal with the weak singularity, such as using adaptive grids (nonuniform grids) to keep errors small near the singularity [18,44,51], or employing non-polynomial basis functions to include the correct singularity index [2,14,52], or using the correction terms to remedy the loss of accuracy and recover high-order schemes [18,30,46,49,50].…”

A unified spectral method for FPDEs with two-sided derivatives; Part II: Stability, and error analysis

“…For anomalous transport, it has been shown that fractional ordinary/partial differential equations FODEs/FPDEs are the most tractable models that rigorously code memory effects, self-similar structures, and power-law distributions [26,34,15,24,27]. In addition to finite difference and higher-order compact methods [20,25,33,9,5,39,3,16,38,40], a great progress has been made on developing finite-element methods [23,10,28] and spectral/spectral-element methods [31,37,36,35,4,6,42,22,43,14,41,32,19,12] to obtain higher accuracy for FODEs/FPDEs.…”

An approach to construct higher order time discretisation schemes for time fractional partial differential equations with nonsmooth data

“…(2) Use some nonpolynomial (or singular) basis functions or collocation spectral methods to capture the singularity of the solutions of (1.1), see [1], [5], [13], [14], [34], [27], [59], [63], [67]. (3) Separate the solution into two parts: smooth and nonsmooth parts.…”