# Tagged Questions

Questions regarding derivatives and integrals of non-integer order.

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### What is the actual meaning of a fractional derivative?

We're all use to seeing differential operators of the form $\frac{d}{dx}^n$ where $n\in\mathbb{Z}$. But it has come to my attention that this generalises to all complex numbers, forming a field called ...
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### On nonlinear fractional field equations

In a paper by Vazquez I read that an interesting variant of fractional diffusion (where the fractional operator is usually $(-\Delta)^s$) could be $(I-\Delta)^s$. Therefore I am wondering if there are ...
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### Well-definedness for a singular integral

Let $T_\alpha$ be a singular integral operator defined by $$T_{\alpha}[f](t):=\int_{0}^{t}\frac{f(t)-f(s)}{(t-s)^{\alpha+1}}ds$$ for continuous functions $f$ on $[0,\infty)$ and $0<\alpha<1$. ...
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### Can I study a global stability of equilibrium point in a fractional order system?

I have a system of fractional-order differential equations in the sense of  Caputo’s derivatives. For this model, I obtained the equilibrium points and proved that some equilibrium points are locally ...
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### Can infinitely many orbifolds be “added up” to form a fractal space? [on hold]

Disclaimer: this question is rather vague and thus might not be suitable for this site. If so, feel free to tell me and I'll delete it. Intuitively, an orbifold, from what I understand, is a "...
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### Rearrangement in Bessel function spaces

I consider, for $0<s<1$, the Bessel function space $$L^{s,2}(\mathbb{R}^d) = \left\{f\in L^2(\mathbb{R}^n) : (1+|\cdot|)^{s/2}\hat{f}(\cdot)\in L^2(\mathbb{R}^n)\right\}.$$ The question I ...
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### Resolvent operator of fractional Laplacian

For $0<\alpha<2$, we define the fractional Laplacian with Fourier transform \begin{align} \widehat{(-\Delta)^{\frac{\alpha}{2}} u}(\xi) = |\xi|^\alpha \widehat u(\xi). \end{align} Consider the ...
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### What is the fractional derivative smoothness of functions from the Zygmund class?

Let $\Lambda([0,1])$ be the Zygmund class of continuous on $[0,1]$ functions for which $$\sup h^{-1}|f(x+2h)-2f(x+h)+f(x)|<\infty.$$ What would be the exact smoothness class for the fractional ...
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### positively invariant set respec to fractional system

In my research I need to show that the set $$M := \{X \in \mathbb{R}^4,X≥0\}$$ where $$X(t)=(x_1(t),x_2(t),x_3(t),x_4(t))^T$$ is positively invariant with respect the following system of ...
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### Fractional Laplacian and stereographic projection

The ordinary Laplacian on $\mathbb{R}^N$ behaves nicely under a stereographic projection onto $\mathbb{S}^N\setminus\{P\}$. (Here $P$ is either the north or south pole of the unit sphere $\mathbb{S}^N$...
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I wonder if there any established formula for fractional derivative of a function $\pi \cot (\pi x)$. I derived the following expression: $(\pi \cot (\pi q))^{(p)}=-\frac{\zeta'(p+1,q)+(\psi(-p)+\... 1answer 409 views ### Fractional integration lemma Hello everyone. I am trying to establish a fractional integration lemma of this form. For$\alpha\geq 0$, and$1\leq p,q<\infty$and$0\leq \frac{1}{q}-\frac{1}{p}=\frac{\alpha}{d}$or$1\leq p,...
I would like to know if the the following exist or are defined The Fourier transform $\mathcal{F}\left(\frac{d^{\frac{1}{2}}y}{dx^\frac{1}{2}}\right)$ of a fractional differential operator such as \$\...