# Questions tagged [fractional-calculus]

Questions on the differentiation/integration of functions to fractional order. Fractional calculus is a branch of mathematical analysis that studies the possibility of taking real number powers or complex number powers of the differentiation and integration operators.

438 questions
Filter by
Sorted by
Tagged with
1 vote
28 views

### What is the domain of Caputo fractional derivative? Is zero included or excluded?

Caputo fractional derivative is defined as $$^CD^\alpha f(x)=\frac{1}{\Gamma(n-\alpha)}\int_0^x\frac{f^{(n)}(t)}{(x-t)^{\alpha-n+1}}dt$$ Where $n$ is the ceiling of $\alpha$. I found some papers ...
• 1,240
15 views

• 308
40 views

### A doubt in the Sobolev space.

Sobolev space $H^{2,p}(\mathbb{R}^n)=\left\{u\in \mathcal{S}': \mathcal{F}^{-1}((1+|\xi|^2)\widehat{u})\in L^p(\mathbb{R}^n)\right\}$ (Wong definition in introduction pseudo differential operators) If ...
• 3,574
1 vote
48 views

### Product rule fractional laplacian

It is well known the product rule for the Laplacian: $$\Delta(uv) = u\Delta v +2\nabla u\cdot\nabla v + v\Delta u.$$ I was wondering if there is any similar product rule for the fractional laplacian, ...
• 3,418
29 views

• 3,418
1 vote
55 views

• 111
65 views

28 views

### Substitutions for integrals involving branch cuts

I have the definition of the Liouville fractional derivative $$\frac{d^{-\delta}}{d x^{-\delta}} f(x) = \int_{0}^x f(t)(t-x)^{\delta - 1} dt$$ where $\delta > 0$ and for ...
• 737
47 views

• 488
20 views

### Functional Application of the Differential Operator: Can the Order of Differentiation be a Function?

I've been contemplating the traditional differential operator ( D ) used in calculus, and I'm interested in a potentially broader application. Instead of having a fixed real or fractional order for ...
1 vote
68 views

### If $u\in D^{s, p}(\mathbb R^n)$, is that true that $u\in C(\mathbb R^n)$?

Let $0<s<1$, $p>1$ and $n>sp$. For $u\in C_0^{\infty}(\mathbb R^n))$ let $$[u]_{s, p} =\left(\iint_{\mathbb R^n} \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\right)^{\frac1p}$$ be the ussual ...
• 3,418
1 vote
29 views

### If $\widetilde{u}(x) = u(-x)$, what is $[\widetilde u]^2_s$?

Let $s\in (0, 1)$ and $u:\mathbb R^n\to \mathbb R$ be a measurable function. The Gagliardo seminorm of $u$ is defined as $$[u]^2_s =\int_{\mathbb R^{2n}} \frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}} dx dy.$$ I ...
• 3,418
102 views

### Show that $G_t(x)=(2\pi)^{-n/2}\int_{\mathbb{R}^n}\mathbb{e}^{ix\cdot\xi}\mathrm{e}^{-t(1+|\xi|^2)^{\alpha}}\,d\xi$ is uniformly bounded in $L^1$.

I am studying properties of the kernels of different operators and the following question arose. Given the operator $-(-\Delta)^{\alpha}u=\mathcal{F}^{-1}(|\xi|^{2\alpha}\widehat{u}(\xi))$, this ...
• 3,574
36 views

### Understanding some calculations about the kernel of the fractional laplace operator

Hello. I am trying to understand an estimate on the kernel of the fractional laplace operator in the following article "Well-posedness of the Cauchy problem for the fractional power" by Miao,...
• 3,574
95 views

• 3,418
146 views

### Defining a fractional derivative operator over Laplace transforms

$$\Large{ \text{Introduction} }$$ I'm interested in solving methods for fractional-differential-equations (FDEs) without specifying the nature of the fractional derivatives. In doing so, I keep coming ...
• 1,897
76 views

54 views

### Central difference approach on Riemann-Liouville fractional derivative

Take a look at the simple fractional differential equation (FDE Initial Value Problem): $$_0D_x^{\frac{1}{2}}y(x)+2y(x)=0,~~x>0,~\text{and}~_0D_x^{-\frac{1}{2}}y(0)=1.$$ I solved the IVP using ...
• 765
1 vote