Questions regarding derivatives and integrals of non-integer order.

**3**

votes

**1**answer

178 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 ...

**3**

votes

**0**answers

93 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

**0**answers

34 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

57 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\...

**2**

votes

**0**answers

56 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 ...

**1**

vote

**0**answers

109 views

### Can infinitely many orbifolds be “added up” to form a fractal space?

Disclaimer: this question is rather vague and thus might not be suitable for this site. If so, feel free to tell me and I'll delete it.
Intuitively, an orbifold, from what I understand, is a "...

**1**

vote

**0**answers

45 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 ...

**1**

vote

**0**answers

316 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

**0**answers

180 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 ...

**1**

vote

**0**answers

227 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-...

**1**

vote

**0**answers

404 views

### Relation between interpolation spaces and besov spaces

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 \|\partial_x u\|_{L^{\...

**0**

votes

**0**answers

11 views

### Can I study a global stability of equilibrium point in a fractional order system?

I have a system of fractional-order differential equations in the sense of Caputo’s derivatives. For this model, I obtained the equilibrium points and proved that some equilibrium points are locally ...