# 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.

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### How to derive a recursive formula from the following formula

How to derive a recursive formula from the following formula, $$u_{n}=a_{n-1}u_{0}+\sum_{k=1}^{n-1}(a_{n-1-k}-a_{n-k})u_{k}+\Gamma(2-\alpha)h^{\alpha}f(t_{n},u_{n})?$$ P.S.: Consider the following ...
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### comparison property for fractional derivative

I am studying fractional derivative. In the case of normal derivative, it is known that for $u,v \in C^1[0,1],$ if $u(0)=v(0)~\hbox{and} ~u'(0) < v'(0)$, then $u(t) < v(t)$ for all small $t>0$...
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### what is the partial derivative ot the following function?

Can you help me to compute the partial derivative $\dfrac{\partial F(t,\varrho_{1} ( t ) ,\varrho_{2}( t ) ,^{RL}\mathcal{D}_{0^{+}}^{\frac{1}{2}}\varrho_{1} (t) )}{\partial \varrho_{1}}$of the ...
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### Imaginary order differential equations

I would like to find the solution of the imaginary order differential equation $y^{(2i)}+y^{(i)}+y=0$ I started with the Fourier transform differintegral as it seemed more suitable than the Riemann-...
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### Prove that $\int_U u^2\le C\int_U\int_U\frac{(u(x)-u(y))^2}{|x-y|^{n+2s}}dx\,dy$ for some $C>0$

Recently I am working on some kind of fractional Sobolev inequalities, and I would like to prove that, for all $u\in W^{s,2}(U)$, $$\|u\|_{L^2(U)}\le C[u]_{W^{s,2}(U)}\qquad (\star)$$ for some $C>0$...
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### Commutator of pseudodifferential operator and multiplication operator

Let $\eta:\mathbb R^n\to[0,1]$ be a smooth and compactly supported function and assume $f:\mathbb R^n \to\mathbb R$ is measurable. I want to bound the commutator of the operators $\eta(-i\nabla)$, ...
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### Why is $W^{s,p}(\mathbb R^n)$ continuously embedded in $L^q(\mathbb R^n)$ for $q\in[p,\frac{np}{n-sp}]$, given the fractional Sobolev inequality?

In Theorem 6.5 of Hitchhiker’s guide to the fractional Sobolev spaces, it is stated that: Let $s\in(0,1)$ and $p\in[1,+\infty)$ be such that $sp<n$. Then, there exists a positive constant $C$ ...
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