All Questions
Tagged with fractional-calculus differential-equations
10 questions
0
votes
0
answers
97
views
Generator of an analytic semigroup
Perhaps I have a naive question. My question is as follows:
When we consider a Cauchy proposition of the following form:
$$
\begin{cases}
x'(t)= -Ax(t)+ F(t,x(t)) &\text{for}\ t> 0 \\
x(0)=...
- 39
2
votes
0
answers
58
views
Is there a fractional derivative that preserves the composition of the one-parameter Mittag-Leffler function with $x\mapsto x^{\alpha}$?
Let $\alpha\in (0,1)$. The Riemann-Liouville fractional derivative of order $\alpha$ is defined by
$$ \sideset{_0^R}{}{D^{\alpha}f(t)}
=\frac{1}{\Gamma{(1-\alpha)}}
\frac{d}{dt}\left(\int_{0}^{t}
\...
- 852
2
votes
0
answers
44
views
Fractional Laplacian in higher order case
Let $n\geq 2$ and $\sigma \in (0,\frac{n}{2})$. Denote the critical Sobolev exponent $2_{\sigma}^*:=\frac{2n}{n-2\sigma}$, consider Sobolev space $E$ which is the space of real-valued functions $u\...
- 471
1
vote
0
answers
89
views
Domain where the fractional Laplacian operator is a closed operator
Consider the fractional Laplacian defined by
$$(-\Delta)^s u(x) = P.V. \int_{\mathbb{R}^N} \frac{u(x) - u(y)}{|x - y|^{N + 2s}}dy, \ s \in (0,1).$$
Also consider that
$$D((-\Delta)^s) = \{u \in H^s(\...
- 111
1
vote
1
answer
355
views
Why should we model infectious diseases with fractional differential equations?
With COVID19 becoming a pandemic I saw some researchers trying to model it with fractional differential equations (FDE) instead of ordinary differential equations (ODE). From a technical standpoint I ...
0
votes
0
answers
83
views
Difference between two fractional Schrödinger equations
Let us consider the fractional Schrödinger equation with periodic boundary conditions
$$
\begin{cases}
iu_t\mathbf{+}(-\Delta)^{\alpha}u= \pm |u|^2u,\; x \in \mathbb{T}, t \in \mathbb{R}_+\\
u(x,0)=...
- 205
2
votes
1
answer
512
views
Solving Fractional Laplacian Equations with Boundary Condition
I'm attempting to solve a simple Dirichlet problem on the fractional Laplacian with boundary conditions:
$r^{+}(\nabla^s) v = f$
where $0 \leq s \leq 1/2$, $v$ is zero outside of $[0,1]$, $r^{+}$ ...
- 93
1
vote
1
answer
447
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)^{\...
- 213
4
votes
3
answers
910
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 ...
- 329
3
votes
2
answers
1k
views
Do these kernel functions satisfy the semigroup property?
Define the kernel functions for $a\ge 1$,
$$
G_a(t,x) := \frac{C_a t}{t^{1+1/a}+|x|^{1+a}}, \qquad \forall t>0,\: x\in \mathbb{R}\;,
$$
where the constant $C_a$ is some normalization constant ...
- 1,649