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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4
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1answer
90 views

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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17 views

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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12 views

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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3answers
58 views

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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3answers
163 views

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 ...
4
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1answer
49 views

On solving the fractional Laplacian

My current research has lead to me solving a fractional Laplacian equation on $\mathbb{R}^d$. For Laplace's equation $-\Delta u = f$, I know we can solve it by an integral, $$ u(x) = \int_{\mathbb{R}^...
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31 views

Fractional derivative such that $\partial_x^{\alpha} e^{\lambda x} = \lambda^\alpha e^{\lambda x}$

An eigenfunction $f$ of a linear operator $T$ satisfies $T(f) = \lambda f$ for some $\lambda$, which we call its eigenvalue. If we let $T = \partial_x$ be the derivative, then $f(x) = e^{\lambda x}$ ...
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11 views

Inverse Fourier transform (in general sum of two linear operators)

What is the inverse Fourier transform of the functions $\frac{1}{(2\pi |\xi|)^{2s_1} + (2\pi |\xi|)^{2s_2}}$ for $s_1, s_2 \in (0,1].$ Any ideas, how to proceed? We know the inverse Fourier transform ...
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1answer
50 views

Is there a function $f\in L_{1}[0,1]$ but $ \frac{d}{dx}\Big(\int_{0}^{x}\frac{f(s)}{(x-s)^{\alpha}}ds\Big)\not\in L_{1}[0,1]? $

Let $\alpha\in (0,1).$ For a given $f\in L_{1}[0,1]$ consider $$ \phi(x)=\int_{0}^{x}\frac{f(s)}{(x-s)^{\alpha}}ds, \,\,\,\,\,x\in [0,1]. $$ It is clear that if $\phi\in AC[0,1]$ then $\phi$ is ...
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On properties of Riemann–Liouville integral and derivative.

I am reading this book. My question is related to the following theorem. where $I^{\alpha}_{a+},$ $D^{\alpha}_{a+}$ Riemann–Liouville fractional integral and derivative respectively. My question is, ...
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1answer
45 views

Algebraic derivatives of a given function [closed]

I was studying about fractional derivatives and after that did some own work on imaginary order derivatives. Now I'm curious to know are algebraic order derivatives possible? Like the xth derivative ...
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64 views

Bounds for derivatives of fractional heat kernel

Consider the following equation $$ p(0,x) = \delta_{\{x\}} \\ \partial_t p(t,x) + (-\Delta)^s p(t,x) = 0 $$ where $t\in\mathbb{R}$, $x\in\mathbb{R}^n$, $s\in(0,\infty)$ and $(-\Delta)^s$ is the ...
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Return time of the fractional brownian motion in two dimensions

Let $X$ be a two dimensional fractional brownian motion with Hurst parameter $h<\frac{1}{2}$. In `Notes on the two-dimensional fractional Brownian Motion', the authors state that it is a recurrent ...
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30 views

Euler–Lagrange Equation in Fractional Calculus

I found an example in https://doi.org/10.1016/S0022-247X(02)00180-4 Example 1. As the first example, consider the following unconstrained fractional variational problem: $$\text{minimize}\quad J[y]=\...
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1answer
16 views

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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38 views

How do we define fractional derivatives for complex argument functions?

NOTE: This question is not about fractional derivatives of complex order. That topic has already been discussed on this site, here, for example. No - this question is more simple. How do we define $\...
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30 views

Visualization of multiple integral/derivative

I've recently gotten into researching fraction integrals and derivatives. One of the foundational pieces of math for this is the Cauchy formula for repeated integration. I am wondering if there's a ...
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2answers
57 views

Decay of fractional derivative of Schwartz function

Let $\phi$ be a Schwartz function and let $\alpha>0$. I want to analyze the decay as $x\rightarrow\infty$ of: $$\int_\mathbb{R}e^{2\pi i x\xi}|\xi|^\alpha\phi(\xi)\,d\xi$$Heuristically, for $\xi\ll ...
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20 views

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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26 views

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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0answers
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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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35 views

Integrating a function to a fractional power over the sphere

I'm interested in evaluating the following integral (over the unit sphere, $S^2$): $$ \iint_{S^2} \frac{1}{(c_0 + c_1(\mathbf{M}:\mathbf{\hat{r}}\otimes\mathbf{\hat{r}})^2 + c_2(\mathbf{M}\cdot\...
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14 views

Invert Fractional Differencing - Fractional Integrating?

I'm using fractional differencing to make a signal stationary and do regression. The output of the regression model now follows the fractional differenced input I gave it. However, I want to invert ...
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9 views

A question about first eigenfunction of fractional laplacian

Let $\Omega$ be bounded and smooth domain in $\mathbb{R}^n$, $s\in(0,1)$, $e_1\in \mathbb{H}^s(\Omega)$ the first eigenfunction of fractional laplacian $(-\Delta)^s$ with eigenvalue $\lambda_1>0$, ...
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1answer
58 views

Solving the Integral: $f(x) = \int_{-\infty}^{\infty} \left[ \frac{1}{1 + \sigma^{4} t^{2}} \right]^{\frac{L}{2}} e^{-jtx} dt$ when $L$ is odd

I want to solve the following integral when $L$ is odd: $$ f(x) = \int_{-\infty}^{\infty} \left[ \frac{1}{1 + \sigma^{4} t^{2}} \right]^{\frac{L}{2}} e^{-jtx} dt $$ which can be simplified to: $$ f(...
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1answer
34 views

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: $$ \int_{B_{|x|/2}(0)} \frac{|u(x+y)+u(x-y)-2u(x)|}{|y|^{n+2s}}\,dy\leq c|x|^{-n-2s}, $$ ...
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36 views

An inequality involving fractional laplacian

I have to prove that for $s\in(0,1)$ and $\phi\in\mathcal{S}(\mathbb{R}^n)$, ($\phi$ is a Schwartz's function): $$|(-\Delta)^s \phi(x)|\leq c_{n,s}|x|^{-n-2s}, \quad\forall x\in\mathbb{R}^{n}\setminus ...
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48 views

Fractional Cauchy Integral formula (generalization ?)

We have the Cauchy's integral formula in complex analysis Let $\Omega\subset\mathbb{C}$ open. If $f:\Omega\longrightarrow\mathbb{C}$ is analytic and suppose $B(z_{0},r)\subset\Omega$. If $\gamma(t)=...
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0answers
58 views

The semi-group property of Riesz potentials in higher dimensions

I'm trying to wrap my head around the Riesz potential in the sense of a higher dimensional generalization of Riemann-Liouville fractional integrals but some things are coming across as somewhat ...
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5 views

derivative of a fractional derivative with respect to an extra parameter

so we know that we can define the fractional derivative of any function in the fashion $$ \frac{d^{s}}{dx^{s}}f(x) $$ but then my question is how would we define the derivative with respect the ...
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0answers
100 views

A sequence of fractal sine wave curves

I am trying to find a more general answer to the question What is the function for a 'fractal sine wave'?, i.e. an intuitive generalized definition for a sequence of curves, where $$ \gamma_n(t) = (...
2
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0answers
30 views

Double half-derivative of a constant

Just as the title says, how does performing the half derivative of a constant twice work out? I haven't looked at the Riemann-Liouville method yet, but in terms of using the generalized power rule. $D^...
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31 views

Are there any relations between fractional calculus and fractional iteration?

I wonder how the field of fractional calculus and the study of fractional iterates of functions relate to one another. I'm interested in cases in which techniques from the former field can be used to ...
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1answer
59 views

“Differential equation” but we are solving for the order of integration

I know and I have studied some differential equation and, normally, we want to solve for a function or a family of functions that do indeed solve the equation for example: $$y'' - y' = y \sin(x)$$ But ...
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77 views

Are fractional Sobolev spaces in dimension one embedded in $L^{\infty}$?

Consider the space \begin{equation*} X(\Omega) := \left\lbrace u \in L^{2}(\Omega) \iint_{\Omega\times \Omega}\frac{(u(x)-u(y))^{2}}{|x-y|^{2}}dxdy ; u= 0 \textrm{ a.e in } \mathbb{R} \setminus \...
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1answer
58 views

A general theory of fractional operations?

This post is motivated by observing that certain operators have "fractional analogs". For example, let $f: R\to R$ be a differentiable function. The differential operator $D^{1}(f)$ produces ...
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66 views

p-adic Fractional Differentiation

The other day I had a fun thought that we can have p-adic "fractional" derivatives by extending the usual integer order derivatives to p-adic orders in some special cases. $$\frac{d^n}{dx^n} ...
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27 views

How do I calculate $\mathcal{L}^{-1}\left[\frac{s}{s^{\alpha}(s^{2}+1)}\right]$?

I was solving some problems of fractional calculus, and needed this intermediate result. How do I calculate $\mathcal{L}^{-1}\left[\frac{s}{s^{\alpha}(s^{2}+1)}\right]$?
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2answers
696 views

Is there any meaning to this “Super Derivative” operation I invented?

Does anyone know anything about the following "super-derivative" operation? I just made this up so I don't know where to look, but it appears to have very meaningful properties. An answer to ...
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1answer
92 views

Fractional Laplacian when n < 2s

Let $(-\Delta)^s $ be the fractional laplacian. Consider the space dimension to be $n = 1$, so that $$ (-\Delta)^s u = C_{1,2s}p.v.\int_{\mathbb{R}}\frac{u(x)-u(y)}{|y-x|^{1+2s}}dy $$ Do you know some ...
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1answer
38 views

Are fractional derivatives defined analogously for a complex parameter?

The Riemann-Liouville integral is defined for $q<0$ by $$ \left[ \frac{d^q f}{d(x-a)^q}\right]_{RL} = \frac{1}{\Gamma(-q)} \int _a^x (x-y)^{-q-1}f(y)\,dy $$and for $q\leq 0$ by analytic ...
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31 views

How to prove $I_{0}^{\alpha}( I_{1}^{\alpha}(1)) = \frac{t^{\alpha}}{\Gamma(\alpha+1)} - \frac{t^{\alpha+1}}{\Gamma(\alpha+2)}?$

Riemann - Liouville fractional Integral: $I_{0}^{\alpha}x(t) = \displaystyle\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-s)^{\alpha-1}x(s)\rm{d}s$ and $I_{1}^{\alpha}x(t) = \displaystyle\frac{1}{\Gamma(\...
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0answers
89 views

Lie group theory's connection to fractional calculus?

My friend and I were discussing the following identity: $$ M = \exp\big{(}\log(M)\big{)} = \exp\Big{(}\frac{\log(M)}{2}\Big{)} \exp\Big{(}\frac{\log(M)}{2}\Big{)} $$ for any $M$ in a Lie group and $\...
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1answer
75 views

Formula for Repeated Derivatives

Does there exist a formula akin to Cauchy's Repeated Integration formula, but for derivatives? Cauchy's formula doesn't seem ideal for finding, say, the 100th derivative of a function as factorials ...
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0answers
127 views

What is the value of $\int_{0}^{t} (t-s)^{\alpha-1} (1-s)^{\alpha} \rm{d}s$?, where $0<t<1$ and $0 <\alpha \leq 1$.

I saw the value of this integral in one research article. They wrote $-\frac{1}{\Gamma{(\alpha+1)}\Gamma{(\alpha)}}\displaystyle\int_{0}^{t} (t-s)^{\alpha-1} (1-s)^{\alpha} \rm{d}s = -\frac{t^{\alpha}}...
2
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1answer
45 views

Solvability of a simple nonlocal elliptic equation.

I am looking for a theory to handle an elliptic equation with square root of Laplacian involved. Namely, $$ (-\Delta)^{1/2} v(x) + c(x) v(x) = f(x),$$ with relevant boundary conditions. Here I can ...
2
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0answers
41 views

mixing fractional Brownian motions

[first posted here] Given two Brownian motions $W_t^1, W_t^2$, we can have them correlated by $$W_t^1 = \rho W_t^2+\sqrt{1-\rho^2}Z_t$$ where $W_t^{2}$ and $Z_t$ are independent of each other. My ...
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0answers
10 views

fractional integral of a distribution - reference request

Suppose I have $f:[0,T]\to\mathbb{R}$ which is $\beta$-Holder continuous, $\beta\in (0,1)$, $\alpha\in (0,1)$ and I want to consider the fractional integral $$ g(t):=\Gamma(\alpha) \int_0^t (t-s)^{\...
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0answers
69 views

Find $\left(\frac{\,d}{\,dx}\right)^{\pi} x^{\mathrm{e}}$ [duplicate]

As you can the question in the title, that is what I need to find $$\left(\frac{\,d}{\,dx}\right)^{\pi} x^{\mathrm{e}}$$ But I have never encountered anything in which we need to find the irrational ...
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1answer
58 views

Proof of Caputo fractional derivative at zero and first order

Currently I am working with fractional derivative and encountered Caputo's definition: $\frac{\partial^\alpha}{\partial x^\alpha}y(x = x_0) = \frac{1}{\Gamma(1-\alpha)}\int_{-\infty}^{x_0} (x_0-x)^{-\...

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