Preconditioning fractional spectral collocation

# 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 ill-conditioned as the number of collocation points increases. By introducing suitable fractional Birkhoff interpolation problems, we present fractional integration preconditioning matrices for the ill-conditioned linear systems in FSC. The condition numbers of the resulting linear systems are independent of the number of collocation points. Numerical examples are given.

F

ractional Lagrange interpolation, fractional Birkhoff interpolation, fractional spectral collocation, preconditioning

{AMS}

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  for fractional Sturm-Liouville eigenproblems, the corresponding fractional differential matrices can be obtained with ease. However, numerical experiments show that the involved linear systems become extremely ill-conditioned 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  proposed a well-conditioned 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 th-order derivative operator together with boundary conditions. Essentially, the linear system in the well-conditioned collocation method  is the one obtained by right preconditioning the original linear system; see . By introducing suitable fractional Birkhoff interpolation problems and employing the same techniques in , Jiao, Wang, and Huang  proposed fractional integration preconditioning matrices for linear systems in fractional collocation methods base on Lagrange interpolation. In the Riemann-Liouville 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 , there are no singular fractional factors in the Riemann-Liouville 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 :

• Left-sided Riemann-Liouville fractional derivative:

 RL−1Dνxu(x)=1Γ(n−ν)dndxn∫x−1u(ξ)(x−ξ)ν−n+1dξ,
• Right-sided Riemann-Liouville fractional derivative:

 RL  xDν1u(x)=(−1)nΓ(n−ν)dndxn∫1xu(ξ)(ξ−x)ν−n+1dξ,
• Left-sided Caputo fractional derivative:

 C−1Dνxu(x)=1Γ(n−ν)∫x−1u(n)(ξ)(x−ξ)ν−n+1dξ,
• Right-sided Caputo fractional derivative:

 CxDν1u(x)=(−1)nΓ(n−ν)∫1xu(n)(ξ)(ξ−x)ν−n+1dξ.

By the definitions of fractional derivatives, we have

 (2.1) RL−1Dνxu(x)=n−1∑i=0u(i)(−1)Γ(1+i−ν)(x+1)i−ν+ C−1Dνxu(x),

and

 (2.2) RL  xDν1u(x)=n−1∑i=0(−1)iu(i)(1)Γ(1+i−ν)(1−x)i−ν+CxDν1u(x).

Therefore,

 RL−1Dνxu(x)= C−1Dνxu(x),ifu(i)(−1)=0,i=0,1,…,n−1,

and

 RL  xDν1u(x)=CxDν1u(x),ifu(i)(1)=0,i=0,1,…,n−1.

In this paper, we mainly deal with the left-sided Riemann-Liouville fractional problems with homogeneous boundary/initial conditions. By a simple change of variables, (2.1) and (2.2), the extension to other fractional problems is easy.

### 2.2 Fractional Lagrange interpolation

Throughout the paper, let be a set of distinct points satisfying

 (2.3) −1

Given , the -fractional Lagrange interpolation basis associated with the points is defined as

 (2.4) ℓμj(x)=(x+1xj+1)μN∏n=1,n≠jx−xnxj−xn,j=1,…,N.

For a function with , the -fractional Lagrange interpolant of takes the form

 uN(x)=N∑j=1u(xj)ℓμj(x).

### 2.3 Computations of RL−1Dμxℓμj(x) and RL−1D1+μxℓμj(x) with μ∈(0,1)

Note that , can be represented exactly as

 (2.5) ℓμj(x)=(x+1xj+1)μN∏n=1,n≠jx−xnxj−xn=(x+1)μN∑n=1αnjP(−μ,μ)n−1(x),

where denote the standard Jacobi polynomials. The coefficients can be obtained by solving the linear system

 N∑n=1(xi+1)μP(−μ,μ)n−1(xi)αnj=δij,i=1,…,N.
###### Remark \thetheorem

Let and be the Gauss-Jacobi quadrature nodes and weights with the Jacobi polynomial . Then,

 αnj=1(xj+1)μ(2n−1)(n−1)!(n−1)!2Γ(n−μ)Γ(n+μ)ωjP(−μ,μ)n−1(xj).

We now compute and . Let denote the Legendre polynomial of order . By (see )

 RL−1Dμx((x+1)μP(−μ,μ)n−1(x))=Γ(n+μ)Γ(n)Pn−1(x)

and

 RL−1D1+μxℓμj(x)=ddx(RL−1Dμxℓμj(x)),

we have

 RL−1Dμxℓμj(x)=N∑n=1αnjΓ(n+μ)Γ(n)Pn−1(x)

and

 RL−1D1+μxℓμj(x) = N∑n=2αnjΓ(n+μ)Γ(n)P′n−1(x) = N∑n=2αnjΓ(n+μ)Γ(n)n2P(1,1)n−2(x).

## 3 Riemann-Liouville fractional Birkhoff interpolation

Let be the set of all algebraic polynomials of degree at most . Define the space

 SμN=(x+1)μPN−1.

In the following, we consider two special cases.

### 3.1 The case ν=μ∈(0,1)

For a function with , given distinct points satisfying

 −1

consider the Riemann-Liouville -fractional Birkhoff interpolation problem:

 (3.1) Find  p(x)∈SμN  such that  RL−1Dνxp(yj)=RL−1Dνxu(yj),j=1,…,N.
{theorem}

The interpolant for the Riemann-Liouville -fractional Birkhoff problem (3.1) of a function with takes the form

 uνN(x)=N∑j=1RL−1Dνxu(yj)Bνj(x),

where

 Bνj(x)=(x+1)νN∑n=1˜αnjP(−ν,ν)n−1(x),j=1,…,N,

with satisfying

 N∑n=1˜αnjΓ(n+ν)Γ(n)Pn−1(yi)=δij,i=1,…,N.

By , the proof of Theorem 3.1 is straightforward.

###### Remark \thetheorem

Let and be the Gauss-Legendre quadrature nodes and weights with the Legendre polynomial . Then,

 ˜αnj=2n−12Γ(n)Γ(n+ν)ωjPn−1(yj).

Define the matrices

 D(ν)x↦y=[RL−1Dνxℓμj(yi)]Ni,j=1,B(−ν)y↦x=[Bνj(xi)]Ni,j=1.

It is easy to show that

 (3.2) D(ν)x↦yB(−ν)y↦x=IN.

### 3.2 The case ν=1+μ∈(1,2)

For a function with , given distinct points satisfying

 −1

consider the Riemann-Liouville -fractional Birkhoff interpolation problem:

 (3.3) Misplaced &
{theorem}

The interpolant for the Riemann-Liouville -fractional Birkhoff problem (3.3) of a function with takes the form

 uνN(x)=N−1∑j=1RL−1Dνxu(yj)Bνj(x),

where

 Bνj(x)=(x+1)μN−1∑n=1˜βnj(P(−μ,μ)n(x)−P(−μ,μ)n(1)),j=1,…,N−1,

with and satisfying

 N−1∑n=1˜βnjΓ(n+1+μ)Γ(n+1)n+12P(1,1)n−1(yi)=δij,i=1,…,N−1.

By , the proof of Theorem 3.2 is straightforward.

###### Remark \thetheorem

Let and be the Gauss-Jacobi quadrature nodes and weights with the Jacobi polynomial . Then,

 ˜βnj=2n+14nΓ(n+1)Γ(n+1+μ)ωjP(1,1)n−1(yj).

In this subsection, let . Define the matrices

 ˜D(ν)x↦y=[RL−1Dνxℓνj(yi)]N−1i,j=1,˜B(−ν)y↦x=[Bνj(xi)]N−1i,j=1.

It is easy to show that

 (3.4) ˜D(ν)x↦y˜B(−ν)y↦x=IN−1.

## 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) RL−1Dνxu(x)+a(x)u(x)=f(x),ν∈(0,1);u(−1)=0.

The fractional spectral collocation scheme leads to the following linear system

 (4.2) (D(ν)x↦y+diag{a}D(0)x↦y)u=f,

where

 D(0)x↦y=[ℓνj(yi)]Ni,j=1,

and

 a=[a(y1)a(y2)⋯a(yN)]T, f=[f(y1)f(y2)⋯f(yN)]T.

The unknown vector is an approximation of the vector of the exact solution at the points , i.e.,

 [u(x1)u(x2)⋯u(xN)]T.

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) (IN+diag{a}D(0)x↦yB(−ν)y↦x)v=f.

It is easy to show that

 D(0)x↦yB(−ν)y↦x=B(0−ν)y↦y,

where

 B(0−ν)y↦y=[Bνj(yi)]Ni,j=1.

Then, the equation (4.3) reduces to

 (4.4) (IN+diag{a}B(0−ν)y↦y)v=f.

After solving (4.4), we obtain by

#### Example 1

We consider the fractional differential equation (4.1) with

 ν=0.8,a(x)=2+sin(25x).

The function is chosen such that the exact solution of (4.1) is

 u(x)=ex+1−1+(x+1)46/7.

Let be the Gauss-Jacobi points as in Remark 2.3 and be the Gauss-Legendre points as in Remark 3.1. We compare condition numbers, number of iterations (using BiCGSTAB in Matlab with TOL) and maximum point-wise 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). Figure 1: Comparison of condition numbers (left), number of iterations (middle), and maximum point-wise errors (right) for Example 1.

### 4.2 A boundary value problem

Consider the fractional differential equation of the form

 (4.5) RL−1Dνxu(x)+a(x)u′(x)+b(x)u(x)=f(x),ν=1+μ∈(1,2);u(±1)=0.

The fractional spectral collocation method leads to the following linear system

 (4.6) (˜D(ν)x↦y+diag{a}˜D(1)x↦y+diag{b}˜D(0)x↦y)u=f,

where

 ˜D(1)x↦y=[ddx(ℓμj(x))∣∣x=yi]N−1i,j=1,˜D(0)x↦y=[ℓμj(yi)]N−1i,j=1,

and

 a=[a(y1)a(y2)⋯a(yN−1)]T, b=[b(y1)b(y2)⋯b(yN−1)]T, f=[f(y1)f(y2)⋯f(yN−1)]T.

The unknown vector is an approximation of the vector of the exact solution at the points , i.e.,

 [u(x1)u(x2)⋯u(xN−1)]T.

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) (IN−1+diag{a}˜D(1)x↦y˜B(−ν)y↦x+diag{b}˜D(0)x↦y˜B(−ν)y↦x)v=f.

It is easy to show that

 ˜D(1)x↦y˜B(−ν)y↦x=˜B(1−ν)y↦y,˜D(0)x↦y˜B(−ν)y↦x=˜B(0−ν)y↦y,

where

 ˜B(1−ν)y↦y=[ddx(Bνj(x))∣∣x=yi]N−1i,j=1,˜B(0−ν)y↦y=[Bνj(yi)]N−1i,j=1.

Then, the equation (4.7) reduces to

 (4.8) (IN−1+diag{a}˜B(1−ν)y↦y+diag{b}˜B(0−ν)y↦y)v=f.

After solving (4.8), we obtain by

#### Example 2

We consider the fractional differential equation (4.5) with

 ν=1.9,a(x)=2+sin(4πx),b(x)=2+cosx.

The function is chosen such that the exact solution of (4.5) is

 u(x)=ex+1−x−2−e2−34(x+1)2+(x+1)46/7−2(x+1)39/7.

Let be the Chebyshev points of the second kind (also known as Gauss-Chebyshev-Lobatto points) defined as

 xj=−cosjπN,j=0,1,…,N,

and be the Gauss-Jacobi points as in Remark 3.2. We compare condition numbers, number of iterations (using BiCGSTAB in Matlab with TOL) and maximum point-wise 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). Figure 2: Comparison of condition numbers (left), number of iterations (middle), and maximum point-wise errors (right) for Example 2.

## 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 multi-term fractional differential equations is straightforward.

### References

1. Kui Du, Preconditioning rectangular spectral collocation, arXiv preprint arXiv:1510.00195, (2015).
2. Lorella Fatone and Daniele Funaro, Optimal collocation nodes for fractional derivative operators, SIAM J. Sci. Comput., 37 (2015), pp. A1504–A1524.
3. yujian Jiao, Li-Lian Wang, and can Huang, Well-conditioned fractional collocation methods using fractional birkhoff interpolation basis, arXiv preprint arXiv:1503.07632, (2015).
4. Anatoly A. Kilbas, Hari M. Srivastava, and Juan J. Trujillo, Theory and applications of fractional differential equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, 2006.
5. Li-Lian Wang, Michael Daniel Samson, and Xiaodan Zhao, A well-conditioned collocation method using a pseudospectral integration matrix, SIAM J. Sci. Comput., 36 (2014), pp. A907–A929.
6. Mohsen Zayernouri and George Em Karniadakis, Fractional Sturm-Liouville eigen-problems: theory and numerical approximation, J. Comput. Phys., 252 (2013), pp. 495–517.
7.  , Fractional spectral collocation method, SIAM J. Sci. Comput., 36 (2014), pp. A40–A62.
8.  , Fractional spectral collocation methods for linear and nonlinear variable order FPDEs, J. Comput. Phys., 293 (2015), pp. 312–338.
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