Computation of Fractional Derivatives of Analytic Functions

Abstract

It has recently been demonstrated that both regular derivatives and contour integrals of analytic functions can be numerically evaluated to very high orders of accuracy utilizing only grid-based function values in the complex plane. Using closely related techniques, we show here the same to be true for the task of evaluating fractional order derivatives of analytic functions across the complex plane. Several cases are illustrated.

Appendices

Appendix A: Examples of End Correction Stencils at a Singular End Point

We give here some examples of correction stencils for TR evaluation of $\int _{h}^{\infty }\frac{f(z)}{z^{\alpha +1}}dz$, to be applied to f(z)-values around $z=0$ for some $\alpha$ in the range $0<\alpha <1$, and having omitted the factor $h^{-\alpha }$ in the right hand side of (16). Below are first some examples of $n=1$ ($3\times 3$) stencils.

$$\begin{aligned} \begin{array}{rrr} \underline{\alpha =0.01}\quad \\ 0.000127+0.000107i &{} 0.000191+0.001018i &{} -0.000128+0.000148i\\ 0.000866+0.000000i &{} 1.005706+0.000000i &{} -0.001172-0.000000i\\ 0.000127-0.000107i &{} 0.000191-0.001018i &{} -0.000128-0.000148i \end{array} \end{aligned}$$

This value of $\alpha =0.01$ is close to $\alpha =0$ in which case the fractional derivative reduces to the function value at the evaluation point. Therefore, this stencil is close to one at its center point, and to zero at all other entries. In this and all following cases, the values above and below the real axis are the complex conjugates of each other, in particular being real along the real axis.

$$\begin{aligned} \begin{array}{rrr} \underline{\alpha =0.25}\quad \\ 0.0051+0.0043i &{} 0.0072+0.0401i &{} -0.0051+0.0059i\\ 0.0349+0.0000i &{} 1.1468+0.0000i &{} -0.0474+0.0000i\\ 0.0051-0.0043i &{} 0.0072-0.0401i &{} -0.0051-0.0059i \end{array} \\ \begin{array}{rrr} \underline{\alpha =0.5}\quad \\ 0.0181+0.0159i &{} 0.0218+0.1433i &{} -0.0182+0.0210i\\ 0.1286+0.0000i &{} 1.3027+0.0000i &{} -0.1685+0.0000i\\ 0.0181-0.0159i &{} 0.0218-0.1433i &{} -0.0182-0.0210i \end{array} \\ \begin{array}{rrr} \underline{\alpha =0.75}\quad \\ 0.0642+0.0591i &{} 0.0491+0.5085i &{} -0.0643+0.0710i\\ 0.4762+0.0000i &{} 1.4682-0.0000i &{} -0.5706-0.0000i\\ 0.0642-0.0591i &{} 0.0491-0.5085i &{} -0.0643-0.0710i \end{array} \\ \begin{array}{rrr} \underline{\alpha =0.99}\quad \\ 2.4600+2.4498i &{} 0.0973+19.6691i &{} -2.4600+2.4740i\\ 19.6047+0.0000i &{} 1.6376+\,\;0.0000i &{} -19.7990+0.0000i\\ 2.4600-2.4498i &{} 0.0973-19.6691i &{} -2.4600-2.4740i \end{array} \end{aligned}$$

As $\alpha$ approaches one, the weights diverge towards infinity (since the second term in the right hand side of (12) diverges).¹⁶

In the (much more accurate) $n=2$ case, the central weights differ very little from the $n=1$ case, and the outer ones are numerically close to zero. For example for $\alpha =0.5$:

$$\begin{aligned} \begin{array}{rrrrr} \underline{\alpha =0.5}\quad \\ -0.0000-0.0000i &{} 0.0001+0.0001i &{} -0.0000-0.0006i &{} -0.0001+0.0001i &{} 0.0000-0.0000i\\ 0.0001+0.0001i &{} 0.0165+0.0145i &{} 0.0222+0.1470i &{} -0.0166+0.0192i &{} -0.0001+0.0001i\\ -0.0005+0.0000i &{} 0.1318+0.0000i &{} 1.3030+0.0000i &{} -0.1729+0.0000i &{} 0.0006+0.0000i\\ 0.0001-0.0001i &{} 0.0165-0.0145i &{} 0.0222-0.1470i &{} -0.0166-0.0192i &{} -0.0001-0.0001i\\ -0.0000+0.0000i &{} 0.0001-0.0001i &{} -0.0000+0.0006i &{} -0.0001-0.0001i &{} 0.0000+0.0000i \end{array} \end{aligned}$$

If n is increased further, the stencil weights grow rapidly, with the central weight $-19.04$ for $n=3$ and around $10^{12}$ for $n=4$. If extremely high accuracy is desired, these larger stencils are nevertheless computationally cost-efficient, although their use requires extended precision arithmetic.

Appendix B: Illustrations of Fractional Derivatives and Numerical Convergence Rates

1.1 Illustrations of Some Computed Fractional Derivatives

The examples are all cases for which the analytic fractional derivatives are known, in order to allow straightforward verification that the numerical accuracy is consistently close to machine precision across the entire displayed regions. For more illustrations using this same numerical approach as developed here, see [13]. Examples of codes for the present algorithm can be found on GitHub [18].

For each function considered, we display the real and imaginary parts, the magnitude with phase angle, and the relative error of the approximation. Regarding the errors, the numbers by the colorbar correspond to $\log _{10}$ of the error, i.e., -16 matches roughly the machine precision. The plots were produced using $n=2$ (i.e., size $5\times 5$ correction stencils) and only function values at the nodes of the displayed computational domain (padded with $n=2$ layers of nodes). The end correction scheme is used in the entire domain except within the red circles shown in the error plots, where instead the Taylor expansion approaches described in Sect. 5 is used.

Fig. 13

Plots of the real and imaginary parts for both Riemann sheets of $D_{}^{\alpha }\frac{1}{1+z^{2}}$ with $\alpha =0.5$ and $h=1/20$

Fig. 14

Plots of the magnitudes and phase angles for both Riemann sheets in the same case as for Fig. 13

Fig. 15

Plots of relative error vs. h for the test functions used for Figs. 5 and 7, respectively. The different curves are explained in Sect. 10.2

Fractional derivatives of analytic functions are, except on rare occasions (such as the one illustrated in Fig. 9) multi-valued functions. The method presented here can just as well compute values on any of its sheets, according to the formula $D^{\alpha }f(z)=\frac{1}{e^{2\pi \,k\,\alpha \,i}}\left( \frac{1}{\Gamma (1-\alpha )}\int _{0}^{z}\frac{f'(\tau )}{(z-\tau )^{\alpha }}d\tau \right)$, where $k\in \mathbb {{\mathbb {Z}}}$ and f(z) is assumed to be analytic at $z=0$. Figures 13 and 14 illustrate both sheets in the case of of $D^{1/2}\frac{1}{1+z^{2}}$.

1.2 Convergence Rates

Theoretically, with end correction stencils of size $5\times 5$, we expect convergence to occur in the outer region at a rate better than $O(h^{22})$. The computations in the inner regions (as described in Sect. 5) were implemented to give matching levels of accuracy. To verify these rates in a log-log plot of error vs. h, it is necessary to have a wide range of h-values that give results that are mostly free from the influence of rounding errors (which arise at the level $O(10^{-15})$). That can be achieved by using a much larger physical domain $[-20,20]\times [-20,20]$ than used in our previous displays. As seen in Fig. 15, this allows the high convergence rate to be confirmed for both outer and inner regions, although with the misleading impression that the error levels for the two regions are strongly different (rather than comparable, as seen in the previous Figs. 5, 6, 7, 8, 9, 10, 11 and 12). In Figs. 15a, b, the thin dashed and solid lines show the errors in the inner and outer regions, respectively, for the five choices of $\alpha =0.1,$0,3, 0.5, 0.7, 0.9 and the heavy solid lines indicates the slopes for $O(h^{26})$ and for $O(10^{-15}/h^{\alpha })$ in case of $\alpha =1/2$, corresponding to theoretically expected error rates due to truncation and rounding errors, respectively.