# Questions tagged [fractional-calculus]

Questions regarding derivatives and integrals of non-integer order.

79
questions

**2**

votes

**1**answer

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**

votes

**0**answers

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**

vote

**1**answer

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**

votes

**1**answer

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**

votes

**0**answers

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**

votes

**1**answer

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**

vote

**1**answer

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

**1**answer

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

**1**answer

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

**1**answer

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

**2**answers

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**

vote

**0**answers

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**

vote

**1**answer

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**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**0**answers

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**

votes

**0**answers

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**

votes

**1**answer

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**

votes

**2**answers

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**

votes

**1**answer

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**

vote

**0**answers

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**

votes

**1**answer

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**

votes

**1**answer

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**

vote

**0**answers

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**

votes

**3**answers

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**

votes

**1**answer

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

**1**answer

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**

votes

**0**answers

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**

vote

**1**answer

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**

votes

**0**answers

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

**1**answer

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

**2**answers

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**

votes

**3**answers

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**

vote

**0**answers

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

**1**answer

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

**1**answer

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

**2**answers

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

**1**answer

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**

vote

**0**answers

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

**1**answer

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

**1**answer

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

**1**answer

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**

votes

**0**answers

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

**2**answers

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**

vote

**0**answers

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

**3**answers

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

**0**answers

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

**2**answers

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

**1**answer

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**

vote

**0**answers

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 ...