Tagged Questions

Questions regarding derivatives and integrals of non-integer order.

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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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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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Green's function for fractional Laplacian

Consider the fractional differential equation \begin{align} D_{|x|}^\alpha u(x) +bu(x)=f(x) \end{align} with $0<\alpha<2$ on an unbounded domain. Instead of $D_{|x|}^\alpha$ one also often sees ...
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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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Prove that these two definitions of “natural” integration constant coincide when both converge

These are two possible definitions of antiderivative (integral) incorporating a supposedly natural choice of an integration constant (see this question for further details). The first one is based on ...
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Why calculus textbooks do not include the natural integration constants in the tables of integrals? [closed]

The formulas for integrals in the textbooks usually define indefinite integral up to a constant term. Yet the natural integration constant for antiderivative can be fixed from the following formula ...
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Is the fractional integral of order 1/2 of an L_2 function continuous

Let $R_\alpha f(t) = \int_0^t (t-s)^{-\alpha} f(s)\,ds$ the fractional integration operator. If $f \in L_q(0,1)$ for some $q>2$ then $R_{1/2} f$ is (even Hölder) continuous on $[0,1]$. My ...
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Fractional Laplacian on compact hypersurface/manifold via harmonic extension?

Let $M$ be a sufficiently smooth compact hypersurface of dimension $n-1$ in $\mathbb{R}^n$. In pages 10-11 of this paper, the authors define $\mathcal{M} = M \times (0,\infty)$ and consider the ...
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Finding the Fractional Derivative of This Function [duplicate]

I've been trying to find an answer to this question, and it seems as though the question has gone unanswered. The question regards the derivative of $f(x)=1+n^{-x}$ where $n$ is a natural number. Is ...
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Fractional laplacian of radially symmetric functions

For a "good" function $u$, I consider its (Gagliardo) fractional laplacian ($0<s<1$) $$(-\Delta)^s u(x) = \int_{\mathbb{R}^N} \frac{u(x)-u(y)}{|x-y|^{N+2s}}dx\, dy,$$ at least as a principal ...
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Fractional Derivatives Of Sums

I have a question regarding the definition of a fractional derivative. I've searched, but I can't find a definition of fractional derivatives that explain the concept in terms of an operator on some ...
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Regularized fractional derivative of distributions.

A fractional derivative of distributions is usually introduced using definition of fractional integral as a convolution of two distributions. However there is another approach based on fractional ...
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Local fractional derivative that doesn't vanish on differentiable functions

Riemann-Liouville fractional derivative is a nonlocal fractional derivative that doesn't vanish in general on differentiable functions. Kolwankar-Gangal fractional derivative is local but vanishes on ...
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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 ...