(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 …

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 …

I am interested in literature/results characterizing the dual of the fractional Sobolev space $W^{s,p}(\Omega)$, where $\Omega \subset \mathbb{R}^N$ is open, bounded, and smooth, $ …

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 …

Riemann-Liouville fractional derivative is a nonlocal fractional derivative that doesn't vanish in general on differentiable functions. Kolwankar-Gangal fractional derivative is lo …

A fractional derivative of distributions is usually introduced using definition of fractional integral as a convolution of two distributions. However there is another approach base …

Let $E_{\alpha}(x^{\alpha})$ be a Mittag-Leffler function, $\alpha \in (0,1)$. It is an eigenfunction for nonlocal fractional derivative, defined as a convolution with $$ \Phi_{ …

We want to know if there exists a fundamental theorem of fractional calculus for the Riesz Derivative (a type of fractional Laplacian), e.g. there exists an operator $L$ such that …

Let $T=(-\triangle)^{\frac{1}{2}}$,Can we have similar estimates below hold in $L^p$ ? $\| T^{\alpha}(fg)-(T^{\alpha}f)g-f(T^{\alpha}g) \|_p \leq \|T^{\alpha-1}f\|_p \|T^{\alpha-1 …

when consider the fractional laplacian $(-\triangle)^\alpha$,is there an essential difference between $0<\alpha<1$ and $\alpha>1$ ? As far as I'm concern,the higer order la …

I wonder if there any established formula for fractional derivative of a function $\pi \cot (\pi x)$. I derived the following expression: $(\pi \cot (\pi q))^{(p)}=-\frac{\zeta'( …

Dear Friends, Define the kernel functions for $a\ge 1$, $$ G_a(t,x) := \frac{C_a t}{t^{1+1/a}+|x|^{1+a}}, \qquad \forall t>0,\: x\in R\;, $$ where the constant $C_a$ is some nor …

Consider the following two norms: The interpolation norm: 1) $\|u; [L_2,\dot H_1^{\infty}]_{1/3,\infty}\| := \sup_{s > 0} \inf_{u=u_0+u_1} \frac{\|u_0\|_{L^2}}{s^{1/2}} + s \|\ …

Hello everyone. I am trying to establish a fractional integration lemma of this form. For $\alpha\geq 0$, and $1\leq p,q<\infty$ and $0\leq \frac{1}{q}-\frac{1}{p}=\frac{\alp …

I would like to know if the the following exist or are defined The Fourier transform $\mathcal{F}\left(\frac{d^{\frac{1}{2}}y}{dx^\frac{1}{2}}\right)$ of a fractional differentia …