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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)=...
Mathlover's user avatar
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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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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 ...
Jingeon An-Lacroix's user avatar
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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 ...
Physics user's user avatar
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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 ...
charlesalexanderlee's user avatar
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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 ...
Elio Li's user avatar
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2 answers
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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 ...
Ilya Kossovskiy's user avatar
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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( |...
Takieddine Zeghida's user avatar
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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 ...
Jingeon An-Lacroix's user avatar
2 votes
2 answers
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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 ...
CBBAM's user avatar
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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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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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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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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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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}, ...
Dante's user avatar
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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\...
Davidi Cone's user avatar
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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 \...
eN.meshok's user avatar
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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+}^\...
Joegin 's user avatar
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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(\...
José's user avatar
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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 ...
Ali's user avatar
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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. ...
Ali's user avatar
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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 ...
Kanghun Kim's user avatar
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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 $\...
Lorenzo Pompili's user avatar
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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 ...
pxchg1200's user avatar
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3 votes
1 answer
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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[...
sorrymaker's user avatar
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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 ...
Matt Rosenzweig's user avatar
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55 views

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 $1<p<\frac{N+2s}{N-2s}$. ...
Siminore's user avatar
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6 votes
0 answers
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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}<...
Paz's user avatar
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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: $$\...
Jingeon An-Lacroix's user avatar
5 votes
3 answers
326 views

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 ...
Sidharth Ghoshal's user avatar
6 votes
2 answers
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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 ...
Jingeon An-Lacroix's user avatar
5 votes
0 answers
178 views

Convergence of the best constant in the $s$-fractional $L^p$ Sobolev inequality

It is known that the fractional $L^p$ Sobolev inequality $$ \|f\|_{L^{p^*_s}(\mathbb R^n)}^p \leq \sigma_{n,p,s} (1-s) \int_{\mathbb R^n}\int_{\mathbb R^n} \frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}} dx dy $$ ...
julian haddad's user avatar
4 votes
1 answer
145 views

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 ...
Luna947's user avatar
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3 votes
1 answer
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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 ...
Luna947's user avatar
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-1 votes
1 answer
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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 $u=u(x,y)$, $...
Mr. Proof's user avatar
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Difference between summation for "$\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 ...
Astroichthys's user avatar
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140 views

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 ...
Riku's user avatar
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4 votes
1 answer
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How to validate the exponentiality of fractional calculus?

Is it true $$ \frac{1}{\Gamma(1-\nu)}\frac{1}{\Gamma(\nu)} \int_{0}^{x}(x-y)^{-\nu}dy\int_0^y (y-t)^{\nu-1}f(t)dt = \int_0^x f(u)du$$ for any continuous function $f(x)$ such that $f(0)=0$ and $0<\...
Watheophy's user avatar
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1 vote
1 answer
470 views

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+...
Student's user avatar
  • 655
2 votes
0 answers
43 views

Nonlocal perimeter of level sets

Let $u \in W^{s,1}(B)$ be given and $k < l$ be two numbers, then I am looking for a way to bound the following term from above. Here $B$ is the euclidean ball. $$ \int_{B: u < k} \int_{B:u>l} ...
Adi's user avatar
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0 answers
118 views

Fractional Laplacian of smooth cut off functions

Suppose we have a smooth compactly supported function $\phi\in C^{\infty}_c(B_\epsilon(0))$ such that $0\leq \phi \leq 1$, $\phi\equiv 1$ on the unit ball and $\phi$ vanishes outside $B_\epsilon(0).$ ...
Student's user avatar
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1 vote
0 answers
199 views

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 $[m,n]$ : ...
Max Muller's user avatar
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1 vote
0 answers
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Multivariable fractional calculus

I recently started to learn about nonlocal PDEs. Sorry if I am missing something really standard in this field. I know Riesz derivative is defined through Fourier transform: $$\mathcal{F}(\partial^\...
Jingeon An-Lacroix's user avatar
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0 answers
59 views

How to prove this estimate involving the Stein Derivative?

Recall the Stein Derivative, $$\mathcal{D}^{s} f(x)=\left(\int \frac{|f(x)-f(y)|^{2}}{|x-y|^{n+2 s}} d y\right)^{1 / 2}.$$ I want to show that, $$\left\|\mathcal{D}^{s}(f g)\right\|_{2} \leq\left\|f \...
Student's user avatar
  • 655
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0 answers
65 views

Normal vector to a level set and fractional Laplacian

Let $U=\{u\le0\}$ and $\partial U=\{u=0\}$. Suppose $\nabla u$ does not vanish on $\partial U$. Then the (canonical extension of the) normal vector field to $\partial U$ (pointing to the interior of $\...
user173196's user avatar
1 vote
1 answer
325 views

Why should we model infectious diseases with fractional differential equations?

With COVID19 becoming a pandemic I saw some researchers trying to model it with fractional differential equations (FDE) instead of ordinary differential equations (ODE). From a technical standpoint I ...
SetofMeasureZero's user avatar
0 votes
1 answer
114 views

Inequality involving 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} \...
Guilherme's user avatar
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