# 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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### Justification for a substituion that turns a finite sum to infinite - constructing the Grunwald-Letnikov fractional derivative (Fractional Calculus)

Steps in question These steps raise numerous questions. What is the reasoning behind choosing $\delta _Nx\equiv [x-a]/N$ ? This seems almost arbitrary. I understand that $a$ and $x$ eventually ...
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### fractional integral with integral measure on a fractional power

I got a bit familiar with fractional calculus, and it seems that the Riemann-Liouville differintegral formula (and all other similar formulas) are defined for a function $f$ that is one dimensional, ...
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### Fractional Bernoulli equation and logistic function

I'm investigating the solution of the special case of the Bernoulli differential equation $$\dfrac{dy}{dt} = \dfrac{y(1-y)}{\tau}, \tag{1}$$ with $\tau$ a time constant, and which models innovation ...
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### Riemann–Liouville fractional integral from measure theoretic point of view

Does there exist a definition of the Riemann–Liouville fractional integral with respect to any general measure? Could there be perhaps a relation between the Riemann–Liouville fractional integral and ...
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### Numerical method for space fractional derivative in 1 dimension

I am very new to the subject of fractional derivatives which arise while characterizing the anomalous transport of passive scalar in turbulence. I have found an equation of the following form, to ...
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### Leibniz type rule for the fourier based fractional derivative

Given $f(t) \in L^2(\mathbb{R})$, $\alpha \in \mathbb{R}$ and the definition \begin{align} D^\alpha f(t) = \mathcal{F}^{-1}\left( (i\omega)^\alpha \mathcal{F}\left(f(t);\omega\right);t\right) \end{...
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### Line element, metric tensor, integral and the sphere in fractional dimensions

I have a question regarding fractional calculus, namely, what is the line element "$$ds^2 = g^{\mu\nu} dx_\mu dx_\nu$$ in fractional "n" dimensions? I am aware of some formulas that ...
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### Stuck on proving fractional integral is continuous

I haven't proved continuity in ages, but I found a situation where it would be helpful. I'd like to prove the Riemann-Liouville differintegral https://en.wikipedia.org/wiki/Riemann%E2%80%...
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### Prove fractional integral at $\alpha = 0$ converges to the original function

https://en.wikipedia.org/wiki/Riemann%E2%80%93Liouville_integral Under the properties section, it says $I^{\alpha}f \rightarrow f$ as $\alpha \rightarrow 0.$ But it offers no proof, is there a proof ...
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### How to understand the notion of a weak fractional derivative in a fractional Sobolev space

recently I have been studying the article named "Hitchhiker’s guide to the fractional Sobolev spaces" (I leave the external link here: https://arxiv.org/pdf/1104.4345.pdf) and as the ...
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### $\int_0^t (t-r)^{1/3} r^{5/3} dr$

The integral $$\int_0^t (t-r)^{1/3} r^{5/3} dr$$ came up in Miller's Introduction to Differential Equations (1987), (Sec 6.6, Convolutions, in the chapter on Laplace transformations, problem 5a on p. ...
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### Questions about iterating the Euler-Maclaurin summation formula

Introduction The Euler–Maclaurin summation formula is as follows for a positive integer $p$ and a continuous function $f(\cdot)$ that is $p$ times continuously differentiable on the interval $[m,n]$ : ...
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Let $b >0$ be arbitrary but fixed and let $\alpha >0$ if $$g_{\alpha}(x)=\frac{1}{\Gamma(\alpha)}\frac{1}{x^{1-\alpha}} \chi_{]0,b]}(x)$$ If $f:[0,b] \rightarrow \mathbb{R}$ is continous in [0,b]...