# Questions tagged [fractional-calculus]

Questions regarding derivatives and integrals of non-integer order.

174 questions
Filter by
Sorted by
Tagged with
1 vote
50 views

33 views

42 views

### 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 ...
73 views

### 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 ...
38 views

263 views

41 views

### 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 ...
112 views

### 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. ...
12 views

### How to solve numerically nonlinear fractional differential equation containing at least two derivatives of non integer order?

Let's consider differential equation: $$x^{(n)}(t) = f(t,x(t),x'(t),\ldots,x^{(n-1)}(t))$$ with initial conditions $x(0) = x_0, x'(0) = x_1, \ldots, x^{(n-1)}(0) = x_{n-1}$. It is easy to solve this ...
217 views

302 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 ...
627 views

### Morse index in PDEs

I have encountered with the term "Morse index" multiple times while reading papers in PDEs (e.g.  and ). However the definition differs for each context. As far as I know this came ...
135 views

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

It is known that the fractional $L^p$ Sobolev inequality $$\|f\|_{L^{p^*_s}(\mathbb R^n)}^p \leq \sigma_{n,p,s} (1-s) \int_{\mathbb R^n}\int_{\mathbb R^n} \frac{|f(x)-f(y)|^p}{|x-y|^{n+sp}} dx dy$$ ...
141 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 ...
65 views

37 views

58 views

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, $$\left\|\mathcal{D}^{s}(f g)\right\|_{2} \leq\left\|f \... 0 votes 0 answers 58 views ### Normal vector to a level set and fractional Laplacian Let U=\{u\le0\} and \partial U=\{u=0\}. Suppose \nabla u does not vanish on \partial U. Then the (canonical extension of the) normal vector field to \partial U (pointing to the interior of \... 1 vote 1 answer 243 views ### Why should we model infectious diseases with fractional differential equations? With COVID19 becoming a pandemic I saw some researchers trying to model it with fractional differential equations (FDE) instead of ordinary differential equations (ODE). From a technical standpoint I ... 0 votes 1 answer 100 views ### Inequality involving the fractional Laplacian Let s \in \mathbb{R} such that 0<s<1. Consider the fractional Laplacian (-\Delta)^s in the real line defined via Fourier series as follows: if f:[-\pi,\pi] \subset \mathbb{R} \... 0 votes 1 answer 416 views ### When is a product of two two-parameter Mittag-Leffler functions a Mittag-Leffler function? I am studying properties of the two-parameter Mittag-Leffler function.$$ E_{\alpha,\beta}(z)=\sum_{k=0}^\infty \dfrac{z^k}{\Gamma(\alpha k+\beta)}. I am particularly interested in recurrences and ...
Cross-post from math.sx. Let $\eta:\mathbb R^n\to[0,1]$ be a smooth and compactly supported function and assume $f:\mathbb R^n \to\mathbb R$ is measurable. I want to bound the commutator of the ...
### Explicit expression for $(-\Delta)^s (|x|^2)$ where $x\in \mathbb{R}^n$ ($n>2s$) and $s\in (0,1)$
Do we know an explicit expression for $(-\Delta)^s (|x|^2)$ where $x\in \mathbb{R}^n$ ($n>2s$) and $s\in (0,1)?$