# Questions tagged [fractional-calculus]

Questions regarding derivatives and integrals of non-integer order.

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### 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 ...
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### 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 ...
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### 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 ...
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### 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: ‎$‎‎$‎ ‎\...
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### 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 $x \in \mathbb{R}^+ \cup\{0\}$ $$f(x)= \log\left(1+\frac{r}{x+a}\right) + \log\left(1+\frac{r}{2x+a}\right) - 2r \log \left(1+\frac{x}{x+a+r} \right),$$ ...
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### 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....
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### 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{...
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### Reference on spectral fractional Laplacian

Are there Harnack type inequalities and Schauder type estimates for the spectral fractional Laplacian. References are welcome.
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### 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 $u$ ...