All Questions
Tagged with fractional-calculus ca.classical-analysis-and-odes
21 questions
1
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1
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127
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Given a radial symmetric function $f$, the estimate of |$\Delta ^ {m/2}f$| in $R^{2m}$ by induction
This question might be a little strange; my order of Laplacian is related to the dimension of the space.
Actually, I’m reading a result which is obtained by induction; it is the absolute value of ...
- 809
1
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0
answers
70
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A possible upper bound for a function that satisfies a singular integral inequality
I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality:
$$
|v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left(
|...
1
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0
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64
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Variation of the fractional derivatives
$\DeclareMathOperator\AC{AC}\DeclareMathOperator\Lip{Lip}$Suppose we have $f\in L^1(\mathbb{R})\cap \AC(\mathbb{R})\cap \Lip(\mathbb{R})$ and $f$ piecewise linear function, bounded and $|f|\leqslant \...
- 89
1
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0
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47
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Functional inequality for fractional Laplacian
Let $f$ be a nonnegative function on the $d$-dimensional torus $\mathbb{T}^d$, which you can take to be smooth. Let $\bar{f}:=\int_{\mathbb{T}^d}fdx$. I am interested in whether the following ...
- 873
-1
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1
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87
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Is integration by parts allowed for the $J^s$ derivative, where $s \in \mathbb R$
I am having the following integral:
$$I = \int u\, J^s(\partial_x \overline{u})- \overline{u}\, J^s(\partial_x u))dxdy$$
where $J^S= (I-\Delta)^\frac{2}{2}$, $\mathbb{R} \ni s \geq 1$ and $u=u(x,y)$,
$...
- 159
1
vote
1
answer
506
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Fourier transform of the fractional Poisson kernel
Recall that the extension of function from $u:\mathbb{R}^n\to \mathbb{R}$ can be defined using the Poisson Kernel as follows:
$$u^{\mathrm{e}}(\mathbf{x}):=\gamma_{n} \int_{\mathbb{R}^{n}} \frac{x_{n+...
- 537
1
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0
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210
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Questions about iterating the Euler-Maclaurin summation formula
Introduction
The Euler–Maclaurin summation formula is as follows for a positive integer $p$ and a continuous function $f(\cdot)$ that is $p$ times continuously differentiable on the interval $[m,n]$ : ...
- 4,829
0
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1
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257
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Explicit expression for the fractional Laplacian of $1/(1+|x|^2)^s$
For $s\in (0,1)$, is there are an explicit expression for
$$(-\Delta)^s \left(\frac{1}{\left(1+|x|^2\right)^s}\right)?$$
Edit: My goal is to show that for the function $u(x)=\frac{1}{\left(1+|x|^2\...
- 537
2
votes
0
answers
140
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Computing the fractional laplacian of a logarithm function
Are there explicit formulas to compute the fractional laplacian $(-\Delta)^{s/2}\log |x|$ in $\mathbb{R}^n$ for $s\in (0,2)$?
- 537
2
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0
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175
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Boundary terms in integration by parts for the fractional Laplacian
Let $u,v \in C^\infty(\Omega)$ and assume that $v$ is compactly supported inside a domain $\Omega$.
Is it true that
$$
\int_\Omega v (-\Delta)^su \, d x = \int_\Omega (-\Delta)^{s/2}v(-\Delta)^{s/2}u \...
- 99
0
votes
0
answers
153
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Spectral fractional Laplacian of power-function $(-\Delta)^s x^{\alpha}$ in $(0,1) \subset \mathbb R$
How can one compute the Neumann spectral fractional Laplacian of power function, $(-\Delta)^s x^{\alpha}$, with $\alpha >0$, in an interval $(0,1)$. I'm only aware of the formula in the whole space....
- 99
1
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1
answer
342
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A mysterious expression from a discriminant
I recently investigated $\textrm{Discriminant}_u(\omega P(u) +(z-u)P'(u))$, where $P(u) := u^3 + au + b$ and $\omega$ is a real parameter (with $\omega\in(0,1)\cup(1,3)$) associated with the order of ...
- 41
7
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3
answers
2k
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A question on fractional derivatives
I know practically nothing about fractional calculus so I apologize in advance if the following is a silly question. I already tried on math.stackexchange.
I just wanted to ask if there is a notion of ...
user168611
2
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1
answer
291
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An inequality involving fractional Laplacian
I have to prove that for $s\in(0,1)$, $u\in\mathcal{S}(\mathbb{R}^n)$, (i.e. $u$ is a Schwartz function):
$$ |(-\Delta)^su(x)|\leq c_{n,s}|x|^{-n-2s},\quad\forall x\in\mathbb{R}^n\setminus B_1(0), $$
...
- 339
0
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1
answer
106
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A question on nontrivial solution of ODE
It is well known that there exists no non-trivial bounded solution of $-u''+u=0$ in $\mathbb R.$ Is this result even true, the problem
$$ \bigg(-\frac{d^2}{dx^2}\bigg)^{s} u+u=0 $$
has no bounded ...
- 99
6
votes
1
answer
1k
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How important is the Atangana-Baleanu fractional derivative, the main recent development in fractional calculus?
In 2016 a new definition of a fractional derivative was announced in this paper, which has since had more than 100 citations. This derivative, the Atangana-Baleanu derivative, is the main recent ...
- 113
2
votes
1
answer
98
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periodic solutions for a fractional ODE
Can any one give a reference on what are the periodic solutions of the linear fractional ODE $(-\frac{d^2}{dx^2} )^s u= u$ on $x\in (0, T)$ with $u(0)= u(T)$ and $s\in (0, 1)$.
An example of a ...
- 402
4
votes
1
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222
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On nonlinear fractional field equations
In a paper by Vazquez I read that an interesting variant of fractional diffusion (where the fractional operator is usually $(-\Delta)^s$) could be $(I-\Delta)^s$. Therefore I am wondering if there are ...
- 459
48
votes
2
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7k
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Geometric interpretation of the half-derivative?
For $f(x)=x$, the half-derivative of $f$ is
$$\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, (...
- 151k
1
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3
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960
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Higher order fractional laplacian
when consider the fractional laplacian $(-\triangle)^\alpha$,is there an essential difference between $0<\alpha<1$ and $\alpha>1$ ? As far as I'm concern,the higer order laplacian ($\alpha&...
- 1,644
8
votes
2
answers
1k
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Is there a known formula for fractional derivative of cot x?
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)+\...
- 10.1k