# Questions tagged [fractional-calculus]

Questions regarding derivatives and integrals of non-integer order.

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votes

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45 views

### Fractional differential equation and inverse Laplace transform

I am writing my bachelor thesis to prove that a certain linear fractional differential equation of order $(n,q)$ has $N$ linearly independent solutions, where $N$ is the smallest possible integer ...

**5**

votes

**1**answer

122 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 ...

**0**

votes

**0**answers

39 views

### explicit formula for fractional laplacian.

Let $u$ be a smooth positive bounded function. Define $v(x)= \log u(x)$. Is it possible to compute $ (-\Delta)^{s} v(x)$ in terms of $u$ explicitly where $0<s<1$.

**1**

vote

**0**answers

41 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

95 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 $x \in \mathbb{R}^+ \cup\{0\}$
$$
f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r \log \left(1+\frac{x}{x+a+r} \right),
$$
...

**5**

votes

**1**answer

625 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

79 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

180 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

159 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

88 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_{\...

**1**

vote

**0**answers

53 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

146 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

86 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^...

**1**

vote

**1**answer

212 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

235 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

1k 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

41 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 ...

**3**

votes

**1**answer

150 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 ...

**5**

votes

**0**answers

379 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

318 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

73 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

43 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

194 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 ...

**0**

votes

**0**answers

44 views

### Solving Fractional Nonlinear Schroedinger Equation (System of Nonlinear Equations)

I also posted this on Math StackExchange last night. I'm solving the fractional nonlinear (time-independent) Schroedinger equation of the form
$$\frac{1}{2}u-\frac{1}{2}\frac{\partial ^{\alpha}u}{\...

**8**

votes

**1**answer

449 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}{...

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vote

**1**answer

83 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

128 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

235 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 $a > 0$.

**2**

votes

**0**answers

109 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

771 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

85 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

211 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

205 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

534 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 ...

**1**

vote

**1**answer

136 views

### What is the fractional derivative smoothness of functions from the Zygmund class?

Let $\Lambda([0,1])$ be the Zygmund class of continuous on $[0,1]$ functions for which $$\sup h^{-1}|f(x+2h)-2f(x+h)+f(x)|<\infty.$$ What would be the exact smoothness class for the fractional ...

**1**

vote

**1**answer

84 views

### positively invariant set respec to fractional system

In my research I need to show that the set
$$M := \{X \in \mathbb{R}^4,X≥0\}$$
where
$$X(t)=(x_1(t),x_2(t),x_3(t),x_4(t))^T$$
is positively invariant with respect the following system of ...

**6**

votes

**2**answers

559 views

### Fractional Laplacian and stereographic projection

The ordinary Laplacian on $\mathbb{R}^N$ behaves nicely under a stereographic projection onto $\mathbb{S}^N\setminus\{P\}$. (Here $P$ is either the north or south pole of the unit sphere $\mathbb{S}^N$...

**3**

votes

**1**answer

456 views

### Prove that these two definitions of “natural” integration constant coincide when both converge

These are two possible definitions of antiderivative (integral) incorporating a supposedly natural choice of an integration constant (see this question for further details).
The first one is based on ...

**-4**

votes

**1**answer

609 views

### Why calculus textbooks do not include the natural integration constants in the tables of integrals? [closed]

The formulas for integrals in the textbooks usually define indefinite integral up to a constant term. Yet the natural integration constant for antiderivative can be fixed from the following formula ...

**1**

vote

**1**answer

129 views

### Is the fractional integral of order 1/2 of an L_2 function continuous

Let $R_\alpha f(t) = \int_0^t (t-s)^{-\alpha} f(s)\,ds$ the fractional integration operator. If $f \in L_q(0,1)$ for some $q>2$ then $R_{1/2} f$ is (even Hölder) continuous on $[0,1]$.
My ...

**0**

votes

**1**answer

488 views

### Fractional Laplacian on compact hypersurface/manifold via harmonic extension?

Let $M$ be a sufficiently smooth compact hypersurface of dimension $n-1$ in $\mathbb{R}^n$.
In pages 10-11 of this paper, the authors define $\mathcal{M} = M \times (0,\infty)$ and consider the ...

**0**

votes

**2**answers

746 views

### Fractional laplacian of radially symmetric functions

For a "good" function $u$, I consider its (Gagliardo) fractional laplacian ($0<s<1$)
$$
(-\Delta)^s u(x) = \int_{\mathbb{R}^N} \frac{u(x)-u(y)}{|x-y|^{N+2s}}dx\, dy,
$$
at least as a principal ...

**1**

vote

**0**answers

222 views

### Fractional Derivatives Of Sums

I have a question regarding the definition of a fractional derivative. I've searched, but I can't find a definition of fractional derivatives that explain the concept in terms of an operator on some ...

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vote

**0**answers

337 views

### A fractional calculus eigenvalue problem

One set of eigenfunction for the following fractional integral operator is $f(z)=e^{-bz}$ for any constant Re$b>0$, with eigenvalue $\lambda=\frac{\Gamma(\alpha)}{b^\alpha}$,
$$\int_z^\infty (\zeta-...

**35**

votes

**1**answer

4k views

### 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, (...

**3**

votes

**0**answers

101 views

### A question about fractional integrals

I am starting from Theorem 3.4 in Struwe's book Variational methods. The authors proves an existence result for a problem with critical growth. I would like to replace the laplacian with the ...

**2**

votes

**1**answer

1k views

### Chain rule for fractional derivative defined via Fourier transform

It is well known that in the case of integer order differentiation the formula $\partial_{x}f(x,u(x))=\partial_{u}f\cdot \partial_{x}u+\partial_{x}f\cdot u$ holds. If we define fractional derivative ...

**3**

votes

**1**answer

505 views

### What is the (fractional) half-derivative of zeta at $s=0$ (and how to compute it)?

(I asked this in MSE before but there was only a general reference which did not help for my specific question)
I think I understood the concept of fractional derivatives applied to ...

**0**

votes

**1**answer

223 views

### On the fractional Schrödinger equation

I wonder if there is any theory about what we can call the fractional Schrödinger equation:
$$
\mathrm{i}\frac{\partial \psi}{\partial t} = (-\Delta)^s \psi + g(|\psi|^2)\psi \quad\hbox{in $\mathbb{R}^...

**1**

vote

**2**answers

712 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 ...