High-Order Accurate Local Schemes for Fractional Differential Equations

Abstract

High-order methods inspired by the multi-step Adams methods are proposed for systems of fractional differential equations. The schemes are based on an expansion in a weighted $L^2$ space. To obtain the schemes this expansion is terminated after $P+1$ terms. We study the local truncation error and its behavior with respect to the step-size h and P. Building on this analysis, we develop an error indicator based on the Milne device. Methods with fixed and variable step-size are tested numerically on a number of problems, including problems with known solutions, and a fractional version on the Van der Pol equation.

Appendices

Appendix 1: Polynomial Approximation

1.1 Jacobi Polynomials

In this section we suppose $\alpha >-1$. Let $P_n^{(\alpha ,0)}$ be the Jacobi polynomial of degree n corresponding the weight $w_{\alpha }\left( \xi \right) =(1-\xi )^{\alpha }$, normalized such that $\Vert P_n^{(\alpha ,0)} \Vert _{\alpha }^2=1$, where

$$\begin{aligned} \Vert f\Vert _{\alpha }^2=\int _{-1}^1 |f|^2\, {w_{\alpha }}. \end{aligned}$$

(9.1)

Let $I=\left( -1,1\right)$,

$$\begin{aligned} \left\langle f,g \right\rangle _{\alpha }=\int _{-1}^1 fg\, {w_{\alpha }}, \end{aligned}$$

(9.2)

for $f:I\rightarrow \mathbb {R}$, $g:I\rightarrow \mathbb {R}^d$, and $L^2_{\alpha }(I,\mathbb {R}^d)$ the space of measurable functions $f:I\rightarrow \mathbb {R}^d$ such that $\Vert f\Vert _{\alpha }<\infty$. The following can be found in [17], for example. The Jacobi polynomials $P_n^{(\alpha ,0)}$ are given by Rodrigues’ formula

$$\begin{aligned} P_n^{(\alpha ,0)}\left( \xi \right) =\frac{\sqrt{2n+\alpha +1}}{2^{(\alpha +1)/2}}\ \frac{(-1)^n}{2^n n!}\, w_{\alpha }^{-1}\, \frac{\mathrm {d}^n}{\mathrm {d}\xi ^n}\Big (\left( 1-\xi ^2\right) ^n w_{\alpha }\left( \xi \right) \Big ), \end{aligned}$$

(9.3)

and are the eigenfunctions of the Sturm–Liouville problem

$$\begin{aligned} A\, v=\nu _n v \end{aligned}$$

(9.4)

where

$$\begin{aligned} A\, v=-w_{\alpha }^{-1}\Big (\left( 1-\xi ^2\right) w_{\alpha } v'\Big )' \qquad \qquad \nu _n=n\left( n+\alpha +1\right) . \end{aligned}$$

(9.5)

The operator $A:D\left( A\right) \rightarrow L^2_{\alpha }\left( I,\mathbb {R}\right)$ is self adjoint. Let $f\in L^2_{\alpha }(I,\mathbb {R}^d)$, and

$$\begin{aligned} f_n=\left\langle P_n^{(\alpha ,0)},f \right\rangle _{\alpha }. \end{aligned}$$

(9.6)

It can be shown that

$$\begin{aligned} f=\sum _{n=0}^\infty f_n P_n^{(\alpha ,0)} \end{aligned}$$

(9.7)

in the $L^2_{\alpha }(I,\mathbb {R}^d)$ norm. Equivalently, there holds

$$\begin{aligned} \lim _{N\rightarrow \infty }\Vert f-\pi _N f\Vert _{\alpha }=0, \end{aligned}$$

(9.8)

where $\pi _N$ is given by

$$\begin{aligned} \pi _N f=\sum _{n=0}^N f_n P_n^{(\alpha ,0)}. \end{aligned}$$

(9.9)

Parseval’s identity holds:

$$\begin{aligned} \Vert f\Vert _{\alpha }^2=\sum _{n=0}^\infty |f_n|^2. \end{aligned}$$

(9.10)

1.2 Approximation of $D\left( A^{\sigma /2}\right)$ Functions

In this section some results regarding polynomial approximation of functions in $L^2_{\alpha }(I,\mathbb {R}^d)$ are presented. In particular, the results of this section concern the approximation of functions which have singularities at the interval’s boundaries. For such a function f the approach taken here provides improved estimates compared to the estimates obtained by finding $\sigma$ such that $f\in H_{\alpha }^\sigma$. This approach can also be found in [18, 19].

For $0<r\in \mathbb {R}$, define

$$\begin{aligned} A^{r}f=\sum _{n=1}^\infty \nu _n^{r} f_n P_n^{(\alpha ,0)}. \end{aligned}$$

(9.11)

The domain $D\left( A^{r}\right)$ of $A^{r}$ is the space of functions $f\in L^2_{\alpha }(I,\mathbb {R}^d)$, such that

$$\begin{aligned} \Vert A^{r} f\Vert _{\alpha }^2=\sum _{n=1}^\infty \nu _n^{2r} |f_n|^2 <\infty . \end{aligned}$$

(9.12)

Lemma 1

Suppose $0<\sigma \in \mathbb {R}$, and $f\in D\left( A^{\sigma /2}\right)$. Then,

$$\begin{aligned} \Vert \left( 1-\pi _N\right) f\Vert _{\alpha }\le (N+1)^{-\sigma }\Vert A^{\sigma /2} f\Vert _{\alpha } \qquad \qquad N\ge 0. \end{aligned}$$

(9.13)

Proof

Suppose $f\in D\left( A^{\sigma /2}\right)$, and $N\ge 0$. Then

$$\begin{aligned} \Vert \left( 1-\pi _N\right) f\Vert _{\alpha }^2 =\sum _{n=N+1}^\infty |f_n|^2. \end{aligned}$$

(9.14)

Owing to

$$\begin{aligned} 1=\nu _n^{-\sigma }\nu _n^{\sigma }\le (N+1)^{-2\sigma }\nu _n^{\sigma } \qquad \qquad n\ge N+1, \end{aligned}$$

(9.15)

we get

$$\begin{aligned} \Vert \left( 1-\pi _N\right) f\Vert _{\alpha }^2\le & {} (N+1)^{-2\sigma } \sum _{n=N+1}^\infty \nu _n^{\sigma }|f_n|^2 \nonumber \\\le & {} (N+1)^{-2\sigma } \Vert A^{\sigma /2}f\Vert _{\alpha }^2 \end{aligned}$$

(9.16)

and thus the conclusion. $\square$

Proposition 4

Suppose $\gamma >0$, $f\left( \xi \right) =\left( 1+\xi \right) ^{\gamma }$, and $g\left( \xi \right) =\left( 1-\xi \right) ^{\gamma }$. Then,

$$\begin{aligned} f \in D\left( A^{\sigma /2}\right) \qquad \qquad 0<\sigma <1+2\gamma , \end{aligned}$$

(9.17)

and

$$\begin{aligned} g \in D\left( A^{\sigma /2}\right) \qquad \qquad 0<\sigma <1+\alpha +2\gamma . \end{aligned}$$

(9.18)

Proof

Here we only prove (9.17). The proof of (9.18) is similar and can be also found in [19]. We have

$$\begin{aligned} A^{\sigma /2} f =\sum _{n=1}^\infty \nu _n^{\sigma /2} f_n P_n^{(\alpha ,0)} \end{aligned}$$

(9.19)

where

$$\begin{aligned} f_n = \int _{-1}^1 \left( 1+\xi \right) ^\gamma P_n^{(\alpha ,0)}\left( \xi \right) w_{\alpha }\left( \xi \right) \,\mathrm {d}\xi . \end{aligned}$$

(9.20)

It follows that (9.17) is valid if and only if

$$\begin{aligned} \Vert A^{\sigma /2} f\Vert _{\alpha }^2=\sum _{n=1}^\infty |\nu _n|^{\sigma }\, |f_n|^2<\infty . \end{aligned}$$

(9.21)

Thus we require an estimate on $f_n$. By Rodrigues’ formula,

$$\begin{aligned} f_n = \frac{(-1)^n \sqrt{2n+\alpha +1}}{2^{n+\alpha /2+1/2} n!} \int _{-1}^1 \left( 1+\xi \right) ^\gamma \, \Big (\left( 1-\xi ^2\right) ^n w_{\alpha }\Big )^{(n)}\,\mathrm {d}\xi . \end{aligned}$$

(9.22)

We integrate by parts to get

$$\begin{aligned} f_n = \frac{(-1)^n \sqrt{2n+\alpha +1}}{2^{n+\alpha /2+1/2}\, n!}\, \frac{{\varGamma }\left( n-\gamma \right) }{{\varGamma }\left( -\gamma \right) } \int _{-1}^1 \left( 1+\xi \right) ^\gamma \, \left( 1-\xi \right) ^{n+\alpha } \,\mathrm {d}\xi \end{aligned}$$

(9.23)

which yields

$$\begin{aligned} f_n=2^{\alpha /2+\gamma +1/2} (-1)^n\frac{{\varGamma }\left( 1+\gamma \right) }{{\varGamma }\left( -\gamma \right) }\, \frac{{\varGamma }\left( n-\gamma \right) {\varGamma }\left( n+\alpha +1\right) }{n! {\varGamma }\left( n+\alpha +\gamma +2\right) }\sqrt{2n+\alpha +1}.\qquad \end{aligned}$$

(9.24)

We use Stirling’s approximation [17],

$$\begin{aligned} {\varGamma }\left( x\right) \sim \sqrt{\frac{2\pi }{x}}\left( \frac{x}{e}\right) ^x \qquad \qquad x\rightarrow \infty \end{aligned}$$

(9.25)

to get

$$\begin{aligned} f_n\sim c_{\alpha \gamma } n^{-3/2-2\gamma } \qquad \qquad n\rightarrow \infty . \end{aligned}$$

(9.26)

So, $A^{\sigma /2} f\in L^2_{\alpha }\left( I,\mathbb {R}\right)$ if and only if $\sigma <1+2\gamma$, and thus the conclusion. $\square$

Appendix 2: Computing $R_{km}$ and ${\mathcal J}_k$

Here, $0<\alpha <1$, $\beta =-1+\alpha$, $w_\beta \left( s\right) =\left( 1-s\right) ^\beta$, and $P_j^{(\beta ,0)}$ are the Jacobi polynomials associated with the weight $w_\beta$, normalized such that their norm is one. We have

$$\begin{aligned} \psi _j\left( s\right) =2^{\alpha /2}\, P_j^{(\beta ,0)}\left( 2s-1\right) , \end{aligned}$$

(10.1)

and conversely

$$\begin{aligned} 2^{-\alpha /2}\, \psi _j\left( \frac{1+\xi }{2}\right) = P_j^{(\beta ,0)}\left( \xi \right) . \end{aligned}$$

(10.2)

We transform the expressions for $R_{km}$ and ${\mathcal J}_k$ into integrals over $(-1,1)$: we get

$$\begin{aligned} R_{km}\left( \theta \right)= & {} \frac{\theta ^{1-\alpha }}{2} \int _{-1}^1 \psi _m\left( \frac{1+\xi }{2}\right) \, \psi _k\left( \theta \frac{1+\xi }{2}\right) \, \left( 1-\theta \frac{1+\xi }{2}\right) ^{\beta }\,\mathrm {d}\xi \nonumber \\= & {} \theta ^{1-\alpha } \int _{-1}^1 P_m^{(\beta ,0)}\left( \xi \right) \, P_k^{(\beta ,0)}\left( \theta \xi -\varphi \right) \, \Big (2-\theta \left( 1+\xi \right) \Big )^{\beta }\,\mathrm {d}\xi \end{aligned}$$

(10.3)

and

$$\begin{aligned} {\mathcal J}_k\left( f;t,h\right)= & {} \int _0^1 f\left( t+hs\right) \psi _k\left( \theta +\varphi s\right) \, w_{\beta }\left( s\right) \,\mathrm {d}s \nonumber \\= & {} \frac{1}{2^{\alpha /2}} \int _{-1}^1 f\left( t+h\frac{\xi +1}{2}\right) P_k^{(\beta ,0)}\left( \theta +\varphi \xi \right) \, w_{\beta }\left( \xi \right) \,\mathrm {d}\xi . \end{aligned}$$

(10.4)

In our implementation, the integrals above are approximated by a Gauss quadrature. Precisely, $R_{km}$ is computed with the Gauss–Jacobi quadrature associated with the weight $w_\beta$, and ${\mathcal J}_k$ is computed with the Gauss–Legendre quadrature.

The approximation of the matrix ${\mathcal R}\left( \theta \right) =\left( R_{km}\left( \theta \right) \right)$ requires the computation of some values every time $\theta$ changes. To make the computation more efficient, the part of ${\mathcal R}$ that does not require adaptation can be stored. The Gauss quadrature provides

$$\begin{aligned} R_{km}\left( \theta \right) \approx \theta ^{1-\alpha }\sum _{j=1}^{N_q} P_m^{(\beta ,0)}\left( \xi _j\right) P_k^{(\beta ,0)}\left( \theta \xi _j-\varphi \right) \Big (2-\theta \left( 1+\xi _j\right) \Big )^{\beta } \omega _j \end{aligned}$$

(10.5)

where $\omega _j$, and $\xi _j$ are the quadrature weights and nodes, respectively. The last equation can be written as a matrix product

$$\begin{aligned} {\mathcal R}\left( \theta \right) ={\mathcal R}_2^T\left( \theta \right) {\mathcal R}_1. \end{aligned}$$

(10.6)

Notice that $R_1$ does not change during the time-stepping, and can be stored and reused, while ${\mathcal R}_2$ must be computed whenever $\theta$ changes. We have

$$\begin{aligned} \left( {\mathcal R}_1\right) _{j,m+1}&=P_m^{(\beta ,0)}\left( \xi _j\right) w_j \qquad \qquad j=1,\ldots ,N_q \quad m=0,\ldots ,P \end{aligned}$$

(10.7)

$$\begin{aligned} \left( {\mathcal R}_2\right) _{j,k+1}\left( \theta \right)&=P_k^{(\beta ,0)}\left( \theta \xi _j-\varphi \right) \Big (2-\theta \left( 1+\xi _j\right) \Big )^{\beta } \qquad \qquad j=1,\ldots ,N_q \quad k=0,\ldots ,P. \end{aligned}$$

(10.8)