# 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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### Riemann-Liouville fractional derivative of order $\alpha$ of the Bessel function

I'm trying to compute the Riemann-Liouville fractional derivative of order $\alpha$ of the Bessel function $J_{0}(\sqrt{t})$ with $0 < \alpha < 1.$ $\mathbf{Some\,\,terminology}$ The Riemann-...
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### What is the relation of fractional calculus to fractional function decomposition?

Background : Fractional function decomposition can be defined like $$x\to f^{\circ (1/k)}(x) \text{ s.t. } x\to (f^{\circ (1/k)})^{\circ k}(x) = f(x)$$ This is in general a hard inverse problem unless ...
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### What techniques can be used to find fused global-local estimators of fractional derivatives?

The history of fractional calculus is a long one. A question on this topic appeared earlier today. This is a nice example of a minimal version of a global fractional derivative in a Fourier sense. But ...
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### Fractional derivative with linear algebra

I have this question in linear algebra that asks us to represent $\frac{d}{dx}\sin x$ and $\frac{d}{dx}\cos x$ as a matrix in the basis of $\binom{1}{0}=\sin x, \ \binom{0}{1}=\cos x$ using ...
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### Fractional exponentiation of functions

I run into a simple question about the fractional exponentiation of a function. Suppose that $f(x)$ is a real valued function which is always positive definite on its domain. Is it possible to obtain ...
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### A problem about Construct auxiliary function？

problem: f has the second differential function on $[a,b]$, prove:$\exists \alpha∈(a,b)$ such that : $f(a)-2f(\frac{a+b}{2})+f(b)=\frac{1}{4} (b-a)^2 f^{''}(\alpha)$ My attempt:we choose $λ$ ...
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### Multivariate fractional derivative of monomial

Given a vector $\mathbf x \in \mathbb{R}_{+}^d$, a multi-index $\mathbf k \in \mathbb{N}^d$ and a sctrictly positive real $\alpha \in \mathbb{R}_+$, i think the expression i have below is some ...
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### What is the $X^{th}$ fractional derivative of a function $f(X)$ where $X$ belongs to the complex or real domain and is changing along the axis?

I know this question sounds strange and I'm not exactly sure how to explain however I will try my best. I do not think this question will have any mathematical uses; however, personally I find it ...
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### Convergence of fractional laplacian in $\mathcal{S}_s$

Let $s\in(0,1)$, let: $$\mathcal{S}_s=\biggl\{ f\in C^\infty(\mathbb{R}^n): \sup_{x\in\mathbb{R}^n}(1+|x|^{n+2s})|\partial^\alpha f(x)|<\infty,\,\forall \alpha\in\mathbb{N}_0^n\biggr\}.$$ The ...
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### Integral Inequality for $u\in\mathcal{S}(\mathbb{R}^n)$

Let $s\in(0,1)$, $u\in\mathcal{S}({\mathbb{R}^n})$, $x\in\mathbb{R^n}$ with: $|x|\geq1$, i have to prove that: $$\int_{B_{|x|/2}(0)} \frac{|u(x+y)+u(x-y)-2u(x)|}{|y|^{n+2s}}\,dy\leq c|x|^{-n-2s},$$ ...