Questions tagged [fractional-calculus]

Questions regarding derivatives and integrals of non-integer order.

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2
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1answer
62 views

regularity of the solutions of Prandtl equation on the segment

Let $p(x)$ be a positive measurable function on $(-1,1)$. Consider the Prandtl equation $$ u(x)-\frac{p(x)}\pi \int_{-1}^1 \frac{u'(t)}{t-x}dt=p(x)h_0(x),\quad u(1)=u(-1)=0.\quad\quad(\star) $$ What ...
0
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0answers
131 views

Fourier series and fractional calculus

We already know that riemann-zeta function on the critical band $$\zeta(\alpha) \propto \sum_{k=1}^{\infty} (-1)^{k+1}k^{-\alpha}, \Re(\alpha) \in ]0, 1[ $$ Is it possible to say that $$ \zeta(\...
1
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1answer
99 views

Fourier analysis and fractional calculus

Do Fourier transform properties still hold in the case of fractional derivatives ? i.e I have seen many times that some lectures define fractional derivative as : $$\frac{d^{\alpha}}{dx^{\alpha}}f=...
2
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1answer
52 views

fractional moments of multivariate normal distributions

is there an analytic formula for fractional moments of multivariate normal distribution? $E(\prod_{i=1}^k x_i^{\nu_i})=?$ where $X=(x_1,\ldots,x_k) \sim N_k(\mu, \Sigma)$, $\nu_i\in \mathcal{R}$ and $\...
0
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0answers
54 views

About generalized binomial theorem and Grünwald-Letnikov fractional derivative

I have run into a problem while computing the fractional derivatives of order $\alpha$ for the Riemann zeta function. My Theorem states Let $s\in\mathbb{C}$, $\mathfrak{Re}(s)>1$, then the ...
0
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1answer
94 views

Fractional Sobolev norm of characteristic function of an interval?

Is there an explicit expression giving a fractional Sobolev norm of the characteristic function of some interval $I=[a,b)$? I believe it is true that $\chi_{I} \in W^{s,1}(\mathbb{R})$ for $s < \...
1
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1answer
119 views

Expected value of absolute value of shifted binomial distribution

Recently my research needs to calculate the close form of $\mathsf{E}[|X-\frac{n}{2}|]$ where $X$ follows binomial distribution with parameter $(n,p)$. When $p=\frac{1}{2}$, this is just the mean ...
5
votes
1answer
146 views

Domain of $(-\Delta)^\alpha$, $\alpha \in (0,1)$

Is there any simple way to characterize explicitly the domain of fractional powers for a given operator? For example, the domain of Dirichlet Laplacian on a bounded nice domain $\Omega \subset \mathbb{...
2
votes
1answer
118 views

Courant nodal domain theorem for fractional Laplacian

Let $\lambda_k$ and $\varphi_k$ be the $k$-th eigenvalue and a corresponding eigenfunction of the fractional Laplacian in a bounded domain $\Omega \subset \mathbb{R}^N$, $N \geq 2$. That is, $\...
6
votes
1answer
228 views

Fractional powers of an operator

What is the large class of operators for which one can define fractional powers? For example, we can consider an operator $A: D(A) \subset X \rightarrow X$, generator of an analytic semigroup on a ...
3
votes
2answers
214 views

Question about the regularity of fractional Heat equation

Let $\Omega$ be a bounded smooth domain of $\mathbb{R}^n$, $0<s<1$ and $(-\Delta)^s$ denotes the restricted fractional Laplacian. Let consider the following fractional Heat equation: ‎$‎‎$‎ ‎\...
1
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0answers
64 views

Equivalence of Cauchy Type Problems and Volterra Integral Equations

The following Theorem is from the book Theory and Applications of Fractional Differential Equations by Kilbas, Srivastava and Trujillo. It's Theorem 3.10 from page 163. Theorem. Let $ \alpha \in \...
1
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1answer
117 views

Optimal Sobolev regularity for $(-\Delta)^{-s}$ on domains

In the paper Chen, Huyuan; Véron, Laurent, Semilinear fractional elliptic equations involving measures, (J. Differ. Equations 257, No. 5, 1457-1486 (2014). ZBL1290.35305) in the proof of Proposition ...
0
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1answer
96 views

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 ...
6
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2answers
185 views

For which tempered distributions is the fractional derivative well-defined?

Let $\gamma \geq 0$ and consider the fractional derivative operator defined in Fourier domain by $$\mathcal{F} \{\mathrm{D}^{\gamma} \varphi \} (\omega) = (\mathrm{i} \omega)^{\gamma} \mathcal{F}\{\...
0
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0answers
57 views

Green's Function for Fractional Laplacian on the Union of Two Balls

I have two disjoint open intervals $B_1, B_2 \subset \mathbb{R}$, and variables $0 < s < 1$ and $t \in B_1 \cup B_2$. I want to solve: $$r_{B_1 \cup B_2}(\Delta^{s} f) = \delta_t$$ for $f$. ...
4
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0answers
97 views

Covergence of fractional Taylor series

Let $f(x)$ be a function that is continuous and infinitely smooth on entire $\mathbb R$. Let's consider Taylor-Maclaurin series for this function: $$f(x) = \sum_{0}^{∞}\frac{f^n(x_0)(x-x_0)^n}{n!}$$ ...
2
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1answer
244 views

Solving Fractional Laplacian Equations with Boundary Condition

I'm attempting to solve a simple Dirichlet problem on the fractional Laplacian with boundary conditions: $r^{+}(\nabla^s) v = f$ where $0 \leq s \leq 1/2$, $v$ is zero outside of $[0,1]$, $r^{+}$ ...
3
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2answers
136 views

Why we use Caputo fractional derivative in application?

I'm working on some papers which use Caputo fractional evolution equation (see on Wikipedia) as application for thier main result: For example: $$\left\{\begin{matrix} ^CD^{\sigma}_tx(t)+Ax(t)=&...
5
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1answer
166 views

Anderson localization for fractional Laplacians

There is a vast literature on Anderson localization, namely, the study of decay of eigenfunctions of operators on $l^2(\mathbb{Z}^d)$ such as $$ -\Delta+\lambda V $$ where $\Delta$ is the discrete ...
1
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0answers
46 views

Limit contration rates and expansion rate solenoid map

Let M:=$S^{1}\times \mathcal{D}^1$ where $\mathcal{D}=\{v\in \mathcal{R}^2 | |v|<1\}$ carries the product distance and suppose $f:M\rightarrow M$,$(x,y,z)\rightarrow (\gamma x, \lambda y+v(x), \mu ...
-6
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1answer
109 views

Behavior of $f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r \log \left(1+\frac{x}{x+a+r} \right)$ [closed]

Consider the following function defined on $x \in \mathbb{R}^+ \cup\{0\}$ $$ f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r \log \left(1+\frac{x}{x+a+r} \right), $$ ...
6
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1answer
649 views

How to define transfinite derivatives of a function?

There are all manners of theories generalizing the notion of derivative. Amongst them is the fractional calculus, a rich theory which gives a sense to the derivation and integration of non-integer (i....
1
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0answers
95 views

formula of spectral fractional laplacian

In fact, if $U$ is a solution of \begin{equation} \ \ \left\{\begin{aligned} (-\Delta)^s U &=f &&\text{in } \Omega \\ U &= g &&\text{on } \partial \Omega \\ \end{...
2
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3answers
324 views

Reference on spectral fractional Laplacian

Are there Harnack type inequalities and Schauder type estimates for the spectral fractional Laplacian. References are welcome.
4
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1answer
253 views

Divergence form of the fractional Laplacian

Can I write the fractional Laplacian $$(-\Delta)^{\alpha/2} u(x) : = c_{\alpha,d} \mathrm{P.V.}\int_{\mathbb{R}^2} \frac{u(x) - u(y)}{|x-y|^{d+\alpha}}dy$$ in the divergence form $$(-\Delta)^{\...
3
votes
1answer
116 views

Fractional Sobolev spaces of order 0

For $1\leq p <+\infty$, $0<s<1$ and $\Omega\subset R^n$ domain, the fractional Sobolev space $W^{s,p}$ is defined as $$W^{s,p}(\Omega):=\big\{f \in L^p(\Omega)\colon \int_{\Omega} \int_{\...
2
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0answers
109 views

Eigenvalues and kernel of the the fractional laplacian in the $d$-dimensional torus

Let $\mathbb{T}^d$ be the $d$-dimensional torus. Consider the operator $$ \Delta^{(\alpha/2)}u(x):= \int_{\mathbb{T}^d} \frac{u(x+y)+u(x-y)-2u(x)}{(d_{\mathbb{T}^d}(x,y))^{d+\alpha}} dy$$ Where $u$ ...
1
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1answer
232 views

Dirichlet fractional Laplacian and zero boundary conditions

Does there exists a non-zero function $$f\in C_0([0,1]):=\{f:[0,1]\to \mathbb R:\ f\text{ is continuous and } f(0)=f(1)=0\},$$ such that $(-\Delta)^{\frac\alpha 2}f\in C_0([0,1]) $, where $(-\Delta)^{\...
4
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0answers
90 views

Normalisation in fractional integration and Brownian motion

Fractional Brownian motion comes in two forms (following Marinucci and Robinson 1998) for fraction $\alpha$ and Brownian motion $W_s$: Type II (Levy, Volterra, Riemann) $$ \tilde W^\alpha_t = \int_0^...
2
votes
1answer
325 views

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 version 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 ...
5
votes
2answers
438 views

Fractional moments of Poisson distribution

I wonder if there is a formula to calculate the fractional (0,1) moments of a Poisson distribution? Thanks in advance.
30
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3answers
2k views

Why are there so many fractional derivatives?

I have been interested in fractional calculus for some time now, and I have seen "lots" of definitions of the $\frac {d^\alpha} {dx^\alpha}$ operator. I started with the book The Fractional Calculus ...
1
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0answers
42 views

Solution of two coupled fractional differential equations

The following arose in a physical problem: Solve the two equations \begin{eqnarray} p(t)+D^\alpha x(t)+\omega x(t)=0\\ _T^- D^\alpha p(t)-\omega p(t)+k x(t)=0 \end{eqnarray} subject to the ...
4
votes
1answer
214 views

integration for fractional laplacian

Is it possible to integrate by parts the fractional laplacian $(-\Delta)^su+ u=f(u)$ in $\mathbb{R}^N$, or is it true that $\int_{\mathbb{R}^N}u= \int_{\mathbb{R}^N} f(u) $ with $s\in (0, 1)$, $u\in ...
6
votes
1answer
509 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 ...
6
votes
2answers
408 views

Reference for LIL for fractional Brownian motion

(Cross-posted to https://math.stackexchange.com/questions/2377810/law-of-iterated-logarithm-for-fractional-brownian-motion.) It seems strange but, even after consulting several books, and hours ...
2
votes
1answer
95 views

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 ...
1
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0answers
49 views

Fractional Leibniz rule with Lorentz spaces

The "fractional Leibniz rule" asserts that $$\Vert D^s(fg)\Vert_{L^r}\lesssim\Vert D^sf\Vert_{L^{p_1}}\Vert g\Vert_{L^{q_1}}+\Vert f\Vert_{L^{p_2}}\Vert D^sg\Vert_{L^{q_2}}$$ where $$\frac{1}{p_i}+\...
4
votes
1answer
226 views

Uniqueness of a SDE with positivity constraint

We start by fixing some notation. If $x\in\Bbb R^N$, we denote the usual euclidean norm in $\Bbb R^N$ with $\|x\|$: we omit the reference to the space $\Bbb R^N$ or to the dimension $N$ since it ...
10
votes
1answer
614 views

The Hölder inequality for fractional order Sobolev seminorm?

This question is post on MSE a week ago. I move it here to draw more attention. Let $u\in C^\infty(\bar I)$ be given where $I=(0,1)$. Define $$ t(\alpha):=\left(\int_I\int_I \frac{|u(x)-u(y)|^\alpha}{...
1
vote
1answer
90 views

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 $0<\alpha<1$. ...
2
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0answers
134 views

Fractional derivative of the Wright function

It is mentioned in some papers (Appendix in this paper, for example) that the (formal) solution of the fractional drift (or transport) equation $$ \partial_{t}^{\alpha}u(t,x)+\partial_{x}u(t,x)=0\quad\...
1
vote
2answers
240 views

How to evaluate the following integral

Would anyone please let me know how to compute the following integral: $$\int_{-\infty}^{+\infty}\frac{a\log(t^2+1)}{t^2 + a^2}dt,$$ here $a > 0$.
1
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0answers
127 views

Another proof of the comparison principle for PDEs by the theory of viscosity solutions

In my research I study (fully) nonlinear PDEs with fractional derivatives. For an analysis of these, I use the theory of viscosity solutions. So I am now at the end of my tether becasuse I can not ...
7
votes
3answers
810 views

A definition of the fractional derivative

I was investigating the idea of fractional derivatives and devised the following definition. WHich definition is it equivalent to and can I have a reference for it? $$\frac{d^n}{dx^n}f(x) = \lim_{h \...
2
votes
0answers
92 views

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 ...
2
votes
2answers
308 views

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 ...
4
votes
1answer
207 views

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 ...
1
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0answers
610 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 ...