Abstract
A Hilbert space framework for fractional calculus is presented. The utility of the approach is exemplified by applications to abstract ordinary fractional differential equations with or without delay.
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Notes
- 1.
It should be noted that \(\partial_{0}^{-\alpha}\) is largely independent of the particular choice of ρ∈ ]0,∞[. Indeed, since
$$\mathcal{L}_{\rho}\chi_{_{ ]0,\infty [}} (m_{0} )m_{0}^{\alpha-1}=\frac{1}{\sqrt{2\pi}}\frac{\varGamma (\alpha )}{ ({i}m+\rho )^{\alpha}} $$we have for \(\varphi\in\mathring{C}_{\infty} (\mathbb {R},H )\)
$$\frac{1}{\varGamma (\alpha )}\chi_{_{ ]0,\infty [}} (m_{0} )m_{0}^{\alpha-1}* \varphi=\partial _{0}^{-\alpha}\varphi $$and
$$\biggl(\frac{1}{\varGamma (\alpha )}\chi_{_{ ]0,\infty [}} (m_{0} )m_{0}^{\alpha-1}*\varphi \biggr) (t )=\frac{1}{\varGamma (\alpha )}\int _{-\infty}^{t}\frac{1}{ (t-s )^{1-\alpha}}\varphi (s ) ds. $$From this convolution integral representation we can also read off that \(\partial_{0}^{-\alpha}\) is causal.
- 2.
In the limit a→−∞ the spectral fractional derivative is formally recovered:
$$\partial_{0}^{\gamma}={_{-\infty}D_{t}}^{\gamma}={_{-\infty }^{\quad C}D_{t}}^{\gamma}. $$There is, however, a domain issue here. Whereas \(\partial_{0}^{\gamma}\) is a well-defined closed operator, the operators \({_{-\infty }D_{t}}^{\gamma}, {_{-\infty}^{\quad C}D_{t}}^{\gamma}\) are usually considered in terms of their integral representation leading to slightly different constraints and different choices of underlying spaces.
- 3.
Here χ M denotes the characteristic function or indicator function of the set M.
- 4.
That is
$$\begin{aligned} \vert f\vert _{\rho,\mathrm{Lip}}:=\inf \bigl\{ & L\in \,]0,\infty[ \bigm| \bigl\vert f (u )-f (v )\bigr\vert _{\rho,-\alpha+\gamma,0}\leq L\vert u-v\vert _{\rho,\gamma,0}\\ &\textit{for all }u,v\in H_{\rho,\gamma}(\mathbb{R},H) \bigr\} . \end{aligned}$$
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Picard, R. (2015). A Note on Real Powers of Time Differentiation. In: Mityushev, V., Ruzhansky, M. (eds) Current Trends in Analysis and Its Applications. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-12577-0_29
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