Preconditioning fractional spectral collocation
Abstract
Fractional spectral collocation (FSC) method based on fractional Lagrange interpolation has recently been proposed to solve fractional differential equations. Numerical experiments show that the linear systems in FSC become extremely illconditioned as the number of collocation points increases. By introducing suitable fractional Birkhoff interpolation problems, we present fractional integration preconditioning matrices for the illconditioned linear systems in FSC. The condition numbers of the resulting linear systems are independent of the number of collocation points. Numerical examples are given.
ractional Lagrange interpolation, fractional Birkhoff interpolation, fractional spectral collocation, preconditioning
65L60, 41A05, 41A10
1 Introduction
Fractional spectral collocation (FSC) methods [7, 8, 2] based on fractional Lagrange interpolation have recently been proposed to solve fractional differential equations. By a spectral theory developed in [6] for fractional SturmLiouville eigenproblems, the corresponding fractional differential matrices can be obtained with ease. However, numerical experiments show that the involved linear systems become extremely illconditioned as the number of collocation points increases. Typically, the condition number behaves like , where is the number of collocation points and is the order of the leading fractional term. Efficient preconditioners are highly required when solving the linear systems by an iterative method.
Recently, Wang, Samson, and Zhao [5] proposed a wellconditioned collocation method to solve linear differential equations with various types of boundary conditions. By introducing a suitable Birkhoff interpolation problem, they constructed a pseudospectral integration preconditioning matrix, which is the exact inverse of the pseudospectral discretization matrix of the thorder derivative operator together with boundary conditions. Essentially, the linear system in the wellconditioned collocation method [5] is the one obtained by right preconditioning the original linear system; see [1]. By introducing suitable fractional Birkhoff interpolation problems and employing the same techniques in [5], Jiao, Wang, and Huang [3] proposed fractional integration preconditioning matrices for linear systems in fractional collocation methods base on Lagrange interpolation. In the RiemannLiouville case, it is necessary to modify the fractional derivative operator in order to absorb singular fractional factors (see [3, §3]). In this paper, we extend the Birkhoff interpolation preconditioning techniques in [5, 3] to the fractional spectral collocation methods [7, 8, 2] based on fractional Lagrange interpolation. Unlike that in [3], there are no singular fractional factors in the RiemannLiouville case. Numerical experiments show that the condition number of the resulting linear system is independent of the number of collocation points.
The rest of the paper is organized as follows. In §2, we review several topics required in the following sections. In §3, we introduce fractional Birkhoff interpolation problems and the corresponding fractional integration matrices. In §4, we present the preconditioning fractional spectral collocation method. Numerical examples are also reported. We present brief concluding remarks in §5.
2 Preliminaries
2.1 Fractional derivatives
The definitions of fractional derivatives of order , on the interval are as follows [4]:

Leftsided RiemannLiouville fractional derivative:

Rightsided RiemannLiouville fractional derivative:

Leftsided Caputo fractional derivative:

Rightsided Caputo fractional derivative:
By the definitions of fractional derivatives, we have
(2.1) 
and
(2.2) 
Therefore,
and
2.2 Fractional Lagrange interpolation
Throughout the paper, let be a set of distinct points satisfying
(2.3) 
Given , the fractional Lagrange interpolation basis associated with the points is defined as
(2.4) 
For a function with , the fractional Lagrange interpolant of takes the form
2.3 Computations of and with
Note that , can be represented exactly as
(2.5) 
where denote the standard Jacobi polynomials. The coefficients can be obtained by solving the linear system
Remark \thetheorem
Let and be the GaussJacobi quadrature nodes and weights with the Jacobi polynomial . Then,
3 RiemannLiouville fractional Birkhoff interpolation
Let be the set of all algebraic polynomials of degree at most . Define the space
In the following, we consider two special cases.
3.1 The case
For a function with , given distinct points satisfying
consider the RiemannLiouville fractional Birkhoff interpolation problem:
(3.1) 
The interpolant for the RiemannLiouville fractional Birkhoff problem (3.1) of a function with takes the form
where
with satisfying
By , the proof of Theorem 3.1 is straightforward.
Remark \thetheorem
Let and be the GaussLegendre quadrature nodes and weights with the Legendre polynomial . Then,
Define the matrices
It is easy to show that
(3.2) 
3.2 The case
For a function with , given distinct points satisfying
consider the RiemannLiouville fractional Birkhoff interpolation problem:
(3.3) 
The interpolant for the RiemannLiouville fractional Birkhoff problem (3.3) of a function with takes the form
where
with and satisfying
By , the proof of Theorem 3.2 is straightforward.
Remark \thetheorem
Let and be the GaussJacobi quadrature nodes and weights with the Jacobi polynomial . Then,
In this subsection, let . Define the matrices
It is easy to show that
(3.4) 
4 Preconditioning fractional spectral collocation (PFSC)
In this section, we use two examples to introduce the preconditioning scheme.
4.1 An initial value problem
Consider the fractional differential equation of the form
(4.1) 
The fractional spectral collocation scheme leads to the following linear system
(4.2) 
where
and
The unknown vector is an approximation of the vector of the exact solution at the points , i.e.,
Consider the matrix as a right preconditioner for the linear system (4.2). By (3.2), we have the right preconditioned linear system
(4.3) 
It is easy to show that
where
Then, the equation (4.3) reduces to
(4.4) 
After solving (4.4), we obtain by
Example 1
We consider the fractional differential equation (4.1) with
The function is chosen such that the exact solution of (4.1) is
Let be the GaussJacobi points as in Remark 2.3 and be the GaussLegendre points as in Remark 3.1. We compare condition numbers, number of iterations (using BiCGSTAB in Matlab with TOL) and maximum pointwise errors of FSC and PFSC (see Figure 1). Observe from Figure 1 (left) that the condition number of FSC behaves like , while that of PFSC scheme remains a constant even for up to . As a result, PFSC scheme only requires about 7 iterations to converge (see Figure 1 (middle)), while the usual FSC scheme requires much more iterations with a degradation of accuracy as depicted in Figure 1 (right).
4.2 A boundary value problem
Consider the fractional differential equation of the form
(4.5) 
The fractional spectral collocation method leads to the following linear system
(4.6) 
where
and
The unknown vector is an approximation of the vector of the exact solution at the points , i.e.,
Consider the matrix as a right preconditioner for the linear system (4.6). By (3.4), we have the right preconditioned linear system
(4.7) 
It is easy to show that
where
Then, the equation (4.7) reduces to
(4.8) 
After solving (4.8), we obtain by
Example 2
We consider the fractional differential equation (4.5) with
The function is chosen such that the exact solution of (4.5) is
Let be the Chebyshev points of the second kind (also known as GaussChebyshevLobatto points) defined as
and be the GaussJacobi points as in Remark 3.2. We compare condition numbers, number of iterations (using BiCGSTAB in Matlab with TOL) and maximum pointwise errors of FSC and PFSC (see Figure 2). Observe from Figure 2 (left) that the condition number of FSC behaves like , while that of PFSC scheme remains a constant even for up to . As a result, PFSC scheme only requires about 13 iterations to converge (see Figure 2 (middle)), while the FSC scheme fails to converge (when ) within iterations as depicted in Figure 2 (right).
5 Concluding remarks
We numerically show that the Birkhoff interpolation preconditioning techniques in [5, 3] are still effective for fractional spectral collocation schemes [7, 8, 2] based on fractional Lagrange interpolation. The preconditioned coefficient matrix is a perturbation of the identity matrix. The condition number is independent of the number of collocation points. The preconditioned linear system can be solved by an iterative solver within a few iterations. The application of the preconditioning FSC scheme to multiterm fractional differential equations is straightforward.
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