Questions tagged [fractional-calculus]

Questions regarding derivatives and integrals of non-integer order.

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48 votes
2 answers
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Geometric interpretation of the half-derivative?

For $f(x)=x$, the half-derivative of $f$ is $$\frac{d^{\frac{1}{2}}}{dx^{\frac{1}{2}}} x = 2 \sqrt{\frac{x}{\pi}} \;.$$ Is there some geometric interpretation of (Q1) this specific derivative, and, (...
Joseph O'Rourke's user avatar
6 votes
3 answers
2k views

A question on fractional derivatives

I know practically nothing about fractional calculus so I apologize in advance if the following is a silly question. I already tried on math.stackexchange. I just wanted to ask if there is a notion of ...
user avatar
6 votes
1 answer
226 views

Fractional integrals and $\sum f(n) n^x$

Preamble The following is a rather unrigorous way to obtain the Euler-Maclaurin formula. Consider some $\sum_{n=1}^\infty f(n)$. We may rewrite this as $$\sum_{n=1}^\infty f(n)=\sum_{n=1}^\infty \sum_{...
Caleb Briggs's user avatar
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0 votes
1 answer
153 views

Symmetry of fractional laplacian

Let $\Omega\subset\mathbb{R}^n$, let $s\in [1/2,1)$, let $u\in C^{1,2s-1+\epsilon}(\Omega)$ such that: $u=0$ on $\mathbb{R}^n\setminus\Omega$, and: $u\in C^{0,s}(\mathbb{R}^n)$, is true that: $$\int_{\...
inoc's user avatar
  • 339
43 votes
5 answers
9k views

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 ...
Christopher Olah's user avatar
6 votes
1 answer
832 views

How important is the Atangana-Baleanu fractional derivative, the main recent development in fractional calculus?

In 2016 a new definition of a fractional derivative was announced in this paper, which has since had more than 100 citations. This derivative, the Atangana-Baleanu derivative, is the main recent ...
willhart's user avatar
  • 113
3 votes
1 answer
902 views

A question on the use of fractional derivatives in Riemann Hypothesis

We already know that Riemann-zeta function on the critical band is defined as follows: $$(1-2^{1-\alpha})\zeta(\alpha) = \sum_{k=1}^{\infty} (-1)^{k+1}k^{-\alpha},\quad \Re(\alpha) \in ]0, 1[ $$ Is ...
Tahar Nguira's user avatar
3 votes
1 answer
69 views

prove spectral equivalence bounds for fractional power of matrices

Let $A, D \in \mathbb{R}^{n \times n}$ be two symmetric,positive definite and tri-diagonal matrices for that we know that they are spectrally equivalent, thus ist holds $$ c^- x^\top D x \le x^\top A ...
Luna947's user avatar
  • 75
3 votes
0 answers
199 views

Infinite partial fraction expansions to compute fractional iterations and recurrences

Let say a function $f$ is defined iteratively over the set of positive integers, for instance $f(t+1)=f(f(t))$ or $f(t+1)=f(t)+f(t-1)$. Based on the recurrence relationship and initial conditions, how ...
Vincent Granville's user avatar
3 votes
1 answer
489 views

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 ...
Anixx's user avatar
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2 votes
3 answers
644 views

Reference on spectral fractional Laplacian

Are there Harnack type inequalities and Schauder type estimates for the spectral fractional Laplacian. References are welcome.
sadiaz's user avatar
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2 votes
1 answer
251 views

Uniform estimation of an integral involving a Hölder-continuous function

Let $\Omega\subset\mathbb{R}^n$ be open and bounded, let $s\in(0,1)$, let $u\in C^{0,2s+\epsilon}(\Omega)$ bounded with $u\in C^{0,s}(\mathbb{R}^n)$ and such that: $u=0$, on $\mathbb{R}^n\setminus\...
inoc's user avatar
  • 339
1 vote
0 answers
830 views

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 ...
Thomas's user avatar
  • 111
1 vote
0 answers
45 views

Another uniform estimation of an integral involving an Hölder function with derivative that is Hölder

Let $\Omega\subset\mathbb{R}^n$, let $s\in [1/2,1)$, let $u\in C^{1,2s-1+\epsilon}(\Omega)$ such that: $u=0$ on $\mathbb{R}^n\setminus\Omega$, and: $u\in C^{0,s}(\mathbb{R}^n)$, is true that there ...
inoc's user avatar
  • 339
0 votes
1 answer
111 views

Uniform estimation of an integral

Let $\Omega\subset\mathbb{R}^n$ be open and bounded, let $s\in(0,1)$, let $u\in C^{0,2s+\epsilon}(\Omega)$ bounded and such that: $u=0$, on $\mathbb{R}^n\setminus\Omega$, is true that there exist a ...
inoc's user avatar
  • 339