Questions tagged [fractional-calculus]
Questions regarding derivatives and integrals of non-integer order.
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Generator of an analytic semigroup
Perhaps I have a naive question. My question is as follows:
When we consider a Cauchy proposition of the following form:
$$
\begin{cases}
x'(t)= -Ax(t)+ F(t,x(t)) &\text{for}\ t> 0 \\
x(0)=...
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Extension operator maps which fractional Sobolev space to $W^{p,s}(R)$
Let us assume we have the following extension operator:
$$
\operatorname{ext}_R^\sigma u=
\begin{cases}
u(x) & \text{if }x \in (0,T)\\
u(0) & \text{if }x \in(0,T)^c.
\end{cases}
$$ We ...
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Fractional powers of Dirichlet-to-Neumann map to derive estimate for PDE
Assume $\Omega$ is an open, bounded subset of $\mathbb R^3$ with smooth boundary $\partial \Omega= \Gamma$. For $u \in
H^{1/2}(\Gamma)$, let $U \in H^1(\Omega)$ denote the weak solution of
the ...
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2
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Interpretation of the singular integral for the definition of fractional Laplacian in classical case $s=1$
Fractional Laplacian is often defined via next principal value integral (assume dimension one for the simplicity)
$$(-\Delta)^su(x):=c_sP.V.\int_\mathbb{R} \frac{u(x)-u(y)}{|x-y|^{1+2s}}\, dy.$$
This ...
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1
answer
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Prove if the fractional Laplacian of a function is bounded
Take $s\in (0, 1)$. I am trying to understand if $(-\Delta)^s (\log(1+x^2))$ is bounded, that is if there exists $R>0$ such that $|(-\Delta)^s (\log(1+x^2))|\le R$.
Here $(-\Delta)^s$ is the ...
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Matrix-order derivatives (differentiating a function a matrix number of times)
I have been exploring methods of generalizing the order of derivatives to a broader range of inputs (such as real numbers, complex, and now matrices). We are very well familiar with integer-order ...
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Given a radial symmetric function $f$, the estimate of |$\Delta ^ {m/2}f$| in $R^{2m}$ by induction
This question might be a little strange; my order of Laplacian is related to the dimension of the space.
Actually, I’m reading a result which is obtained by induction; it is the absolute value of ...
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2
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$C^{k,\alpha}$ dependence of ODE solutions on initial data
I faced such a question. Consider the Cauchy problem for an ODE:
$$
\begin{cases}
y'=F(t,y)\\ y(0)=y_0.
\end{cases}
$$ Assume $F$ has the regularity $C^{k,\alpha}$ (i.e. it has partial derivatives of ...
- 5,632
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A possible upper bound for a function that satisfies a singular integral inequality
I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality:
$$
|v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left(
|...
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Characterization of locality in Fourier multiplier
Obviously Laplacian $-\Delta$ is a local operator, and it can be written as a Fourier multiplier $|\xi|^2$. Similarly fractional Laplacian $(-\Delta)^s$ is with Fourier multiplier $|\xi|^{2s}$ and is ...
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What is the relationship between Hölder spaces and differentiability?
I'm porting this question over from MSE as it did not get any responses other than one comment on there.
Let $C^{k,\alpha}$ be a Hölder space where $0 \leq \alpha \leq 1$. I have seen various sources ...
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Could the patterns in the roots of the Riemann-Liouville differintegral of $(s-1)\,L(s, \chi_{q,j})$ be explained?
This question aims to extend this question to (automorphic) Dirichlet L-functions.
Take: $$f(s,q,j) = (s-1)\,L(s, \chi_{q,j})$$
with $q$ the modulus and $j$ the index of a character $\chi$. A fast way ...
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How to obtain the Lagrangian of the fractional classical mechanics
The Lagrangian of fractional classical mechanics $L_\alpha(\dot{q},q)$ is defined as
$$
L_\alpha(\dot{q}, q)=p\dot{q}-H_\alpha(p,q)
$$
where
$$
\begin{split}
p & =\frac{\partial L_\alpha(\dot{q}, ...
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A Riemann-Liouville differintegral for all entire Dirichlet L-series. Could it be simplified further?
It appears that the well-known relation between entire Dirichlet L-series and the Hurwitz zeta function $\zeta(s,a)$, with $k$ = modulus, $j$ = index of the Dirichlet character $\chi$:
$$(s-1)\,L\left(...
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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
$$\sum_{n=1}^\infty f(n)=\sum_{n=1}^\infty \sum_{...
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Question on possibility of uniquely defining the FRFT via certain properties
I was working around with the fractional Fourier transform (FRFT) when the mathematics undergrad found out, by brute-force computations, that the derivative of the FRFT with respect to the parameter ...
CommunityBot
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Could the patterns in the roots of the Riemann-Liouville differintegral for $(s-1)\,\zeta(s)$ be explained?
The well-known integral expression for the entire function:
$$(s-1)\,\zeta(s) = \frac{-i\,\pi}{2}\int_{1/2-i\infty}^{1/2+i\infty} \frac{\csc(\pi\,u)^2}{u^{s-1}} \, du \qquad s \in \mathbb{C} \tag{0}$$
...
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Is there a fractional derivative that preserves the composition of the one-parameter Mittag-Leffler function with $x\mapsto x^{\alpha}$?
Let $\alpha\in (0,1)$. The Riemann-Liouville fractional derivative of order $\alpha$ is defined by
$$ \sideset{_0^R}{}{D^{\alpha}f(t)}
=\frac{1}{\Gamma{(1-\alpha)}}
\frac{d}{dt}\left(\int_{0}^{t}
\...
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What fractional equation does this convolution solve
Assume that $f:\mathbb{R}\to \mathbb{R}$ is an infinitely differentiable with compact support and let $E_{\alpha}$ be the one-parameter Mittag-Leffler function with $0<\alpha<1$. Find the ...
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Maximum principle for nonlocal equation
I was reading this paper:
Luis Silvestre - On the differentiability of the solution to the
Hamilton-Jacobi equation with critical fractional diffusion (https://arxiv.org/pdf/0911.5147.pdf)
In Lemma ...
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Fractional Laplacian in higher order case
Let $n\geq 2$ and $\sigma \in (0,\frac{n}{2})$. Denote the critical Sobolev exponent $2_{\sigma}^*:=\frac{2n}{n-2\sigma}$, consider Sobolev space $E$ which is the space of real-valued functions $u\...
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views
Solution to the fractional differential equation
What is the solution of the fractional differential equation
$$
f^{(\alpha-1)}(t) = tf(t)
$$
where $(\alpha)$ denotes the fractional derivative of order $\alpha$
EDIT: Background behind this ...
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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 \...
- 60.2k
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Comparison of two versions of fractional Sobolev spaces: do we have $W^{s,p}(\mathbb{R}^{n})=H^{s,p}(\mathbb{R}^{n})$?
There are two versions of fractional Sobolev spaces.
Definition 1: (Via Gagliardo semi-norm) Let $1\leq p\leq +\infty$, $0<s<1$ and let $\Omega\subseteq \mathbb{R}^n$ be an open set. The ...
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Can we interpret fractional Sobolev spaces in terms of fractional derivatives?
Let $1 \leq p < \infty$, $0<s<1$, and $\Omega \subseteq R^n$ be a domain. The Banach space $W^{s,p}(\Omega)$ is defined as
$$W^{s,p}(\Omega) := \left\{ f \in L^p(\Omega) \colon \int_{\Omega \...
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If the Riemann-Liouville fractional integral of $f$ is zero then $f=0$ a.e
I have to prove this:
Let $\alpha\in(0,1)$ and $f\in L^q(a,b)$, $1\leq q<\frac 1\alpha$, and $\mathcal{I}_{a+}^\alpha f=0$. Then $f(x)=0$ for almost all $x\in (a,b)$.
Where $(\mathcal{I}_{a+}^\...
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General fractional integrals and derivatives
Does anyone know how i can extend the 1st level general fractional derivatives to 2nd level so that the 1st level GFD is a left inverse operator to the GFI?
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Domain where the fractional Laplacian operator is a closed operator
Consider the fractional Laplacian defined by
$$(-\Delta)^s u(x) = P.V. \int_{\mathbb{R}^N} \frac{u(x) - u(y)}{|x - y|^{N + 2s}}dy, \ s \in (0,1).$$
Also consider that
$$D((-\Delta)^s) = \{u \in H^s(\...
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When is a product of two two-parameter Mittag-Leffler functions a Mittag-Leffler function?
I am studying properties of the two-parameter Mittag-Leffler function.
$$ E_{\alpha,\beta}(z)=\sum_{k=0}^\infty \dfrac{z^k}{\Gamma(\alpha k+\beta)}.$$
I am particularly interested in recurrences and ...
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A mapping property for fractional Laplace--Beltrami operator
Suppose that $g$ is a smooth Riemannian metric on $\mathbb R^3$ that is equal to the Euclidean metric outside the unit ball. Let us define the fractional Laplace--Beltarmi operator on $\mathbb R^3$ of ...
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On the Fractional Laplace-Beltrami operator
I would appreciate it if a reference could be given for the following claim.
Let $g$ be a Riemannian metric on $\mathbb R^n$, $n\geq 2$ that is equal to the Euclidean metric outside some compact set. ...
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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, (...
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Given $a>0$, find $b>0$ for which $\|\langle x\rangle^{-b}|\partial_x|^{1/2}f\|_{L^2}\lesssim\|\partial_x f\|_{L^2}+\|\langle x\rangle^{-a}f\|_{L^2}$
I have asked the same question on MathSE. I was thinking about the following problem.
Problem. Given $\alpha>0$, find all values of $\beta\geq 0$ such that the following estimate is true for all $\...
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A question of interpolation space on homogeneous Carnot group
Let $\mathbb{G}$ be the homogeneous Carnot group on $\mathbb{R}^{n}$ defined as follows:
A homogeneous Lie group $\mathbb{G}=(\mathbb{R}^{n},\circ)$ is called a homogeneous Carnot group (or a ...
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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 ...
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Evaluating the series $\sum_{n=0}^{\infty} n! x^n$ and inverse variable-fractional-derivatives
So I was interested in formally assigning values to the completely divergent series $G(x) = \sum_{n=0}^{\infty} n!x^n $. I guess the question COULD end here if you already have an idea of how to ...
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Endpoint Calderon-Zygmund inequality of nonlocal fractional laplacian
For $s\in(0,1],$ consider the following non-local fractional laplacian:
$$(-\Delta)^sv= f ~~\text{on } \mathbb{R}^n.$$
Then how to use "the standard elliptic estimate" to obtain:
for $p\in[...
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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 ...
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Existence of a limit of alpha-difference quotient for Hölder functions
Let $f:\mathbb{R}\to \mathbb{R}^d,d\geq 1,$ be an Hölder function with exponent $\alpha\in (0,1)$, meaning that
\begin{equation}
\sup_{x, y \in \mathbb R, \,x\neq y}\frac{|f(x)-f(y)|}{|x-y|^\alpha}<...
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Precise decay of solution fo fractional Schroedinger equations
Let us consider the time-independent fractional Schroedinger equation $$(-\Delta)^s u + u = \vert u \vert^{p-1}u$$ in $\mathbb{R}^N$, where $0<s<1$, $N>2s$ and $1<p<\frac{N+2s}{N-2s}$.
...
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Gagliardo-Nirenberg type inequality for fractional relativistic Laplacian operator?
In [1], authors note that by the seminal approach of M. Weinstein in [2] and [3], there is a non-trivial solution $Q\in H^s(\mathbb{R})$ which optimizes next Gagliardo-Nirenberg type inequality:
$$\...
6
votes
2
answers
805
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Morse index in PDEs
I have encountered with the term "Morse index" multiple times while reading papers in PDEs (e.g. [1] and [2]). However the definition differs for each context. As far as I know this came ...
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Convergence of the best constant in the $s$-fractional $L^p$ Sobolev inequality
It is known that the fractional $L^p$ Sobolev inequality
$$
\|f\|_{L^{p^*_s}(\mathbb R^n)}^p \leq \sigma_{n,p,s} (1-s) \int_{\mathbb R^n}\int_{\mathbb R^n} \frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}} dx dy
$$
...
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prove spectral equivalence bounds for inverse fractional power of matrices
The question is an extention to the answered question prove spectral equivalence bounds for fractional power of matrices.
Let $A, D \in \mathbb{R}^{n \times n}$ be two symmetric,positive definite and ...
- 117k
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prove spectral equivalence bounds for fractional power of matrices
Let $A, D \in \mathbb{R}^{n \times n}$ be two symmetric,positive definite and tri-diagonal matrices for that we know that they are spectrally equivalent, thus ist holds
$$ c^- x^\top D x \le x^\top A ...
- 117k
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votes
1
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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 $u=u(x,y)$,
$...
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votes
0
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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 ...
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Difference between summation for "$\aleph$" terms and summation for "$\aleph_0$" terms
Addition: Could we say that the dimension of a space is "$\aleph_0$" or"$\aleph$"? I guess that every elementary functions can be uniquely expanded as integer order power series ...
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1
answer
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Property about the fractional Laplacian
Let $s \in \mathbb{R}$ such that $0<s<1$. Consider the fractional Laplacian $(-\Delta)^s$ in the real line defined via Fourier series as follows: if $f:[-\pi,\pi] \subset \mathbb{R} \...
CommunityBot
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Poisson Kernel and solution formula for fractional elliptic problem
$$
k (-\Delta)^s u + u = 0, \qquad x \in U, \\
u(x) = f(x), \qquad x \in \mathbb R^n \setminus U,
$$
with $f \in L^\infty(\mathbb R^n)$, $k>0$, and $(-\Delta)^s$ is the singular integral ...
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