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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.

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Acknowledgments

This work was partially supported by the NSF DMS-1115416 and by OSD/AFOSRFA9550-09-1-0613.

Author information

Correspondence to Daniel Baffet.

Appendices

Appendix 1: Polynomial Approximation

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)

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)

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Baffet, D., Hesthaven, J.S. High-Order Accurate Local Schemes for Fractional Differential Equations. J Sci Comput 70, 355–385 (2017). https://doi.org/10.1007/s10915-015-0089-1

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Keywords

  • Fractional differential equations
  • Volterra equations
  • High-order methods