Questions on the differentiation/integration of functions to arbitrary 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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52 views

Simplify into partial fractions $\frac{x^8}{x(x^2+1)^2}$

Simplify into partial fractions $$ \frac{x^8}{x(x^2+1)^2}$$ I tried many times but after a point a $"x"$ of denominator vanishes. Can't all fraction be shown in its partial fraction? If not ...
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32 views

Evaluation of $\exp\left(a\frac{d^2}{dx^2}\right)f(x)$

I know that \begin{align*} \exp\left(a\frac{d}{dx}\right)f(x)=f(x+a)\,, \end{align*} by comparing the Taylor expansions of both sides ($f(x)$ is an arbitrary function). However, if I have, where $f(...
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1answer
34 views

Calculating $\sum_{n=1}^x\frac{r^n}{n^k}$ with integrals

Through some work, I've managed to solve the following sum in the form of integrals: $$\sum_{n=1}^x\frac{r^n}{n^k}=\int_0^r\frac1{a_{k-1}}\int_0^{a_{k-1}}\frac1{a_{k-2}}\int_0^{a_{k-2}}\dots\int_0^{...
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0answers
14 views

Fourier coefficients of $\;\log\log$

I was curious if there is an effective way to compute (the asymptotic of) the Fourier coefficients of $$ F(x)= \log\log\left(\frac{1}{\left\lvert x\right\rvert}\right) \cdot \chi\left(\left\lvert x\...
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0answers
23 views

Do different methods of calculating fractional derivatives have to be equal?

Do different methods of calculating fractional derivatives have to be equal? Or do they sometimes end up differently? An example would be nice, and if possible, an explanation as too why such ...
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0answers
85 views

General Leibniz Rule to fractional differ-integrals

The General Leibniz Rule is given as $$(uv)^{(n)}=\sum_{k=0}^n\binom nku^{(n-k)}v^{(k)}$$ Where $u^{(n)}$ means the $n$th derivative of $u$ with respect to $x$ and $\binom nk$ is a binomial ...
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0answers
10 views

Necessary condition to be what the integral is finite

I'm quitly confusing so please give advices. Consider a limit of an integral $$ \limsup_{k\to0}\int_{0}^{a-k}(a-x)^{-\mu}(f(a-k)-f(x))dx, $$ where $1<\mu<2$ is a constant and $f$ is a $\mu-1$-...
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1answer
54 views

What is the fractional constant of integration?

In fractional calculus, one usually tends to ignore the constant of "differ-integration" if you will, but when I attempted a problem with some fractional calculus, I found the result was somewhat off, ...
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1answer
45 views

Fractional anti-derivatives and derivatives of the logarithm

Anti-derivatives and derivatives of the natural logarithm are well defined until we attempt to evaluate the fractional derivative and anti-derivatives. The background to this problem was that I was ...
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0answers
14 views

Why is (x-xi)^n still a linear factor (Partial Fraction Decomposition)?

When we perform a Partial Fraction Decomposition and one of the solutions of the denominator is a multiple solution (let's say quadratic), we write: $$\frac{A_{1}}{(x-x_{i})} + \frac{A_{2}}{(x-x_{i})^...
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2answers
37 views

Problems Calculating Fractional Derivative

I have been trying to calculate the fractional derivative of $e^{ax}$ using the Liouville Left-Sided derivative, which states that, for $x>0$ and $0<n<1$, $D^n f(x) = \frac{1}{1-n} \frac{d}{...
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1answer
74 views

n-th derivative where $n$ is a real number?

We know that $$\frac{d^n}{dt^n} e^{at}= a^n e^{at}; \, n\in \mathbb N.$$ I want to know if the result is true if $n$ is a real number, i.e., $n\in \mathbb R$ ?
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1answer
26 views

Partial Fraction Decomposition - Multiple Answers-Question

Now I do understand how partial fraction decomposition works and why you can do it, but there is one case that I don´t understand. And that is, the following: $$\frac{A_{1}}{(x-x_{1})} + \frac{A_{2}}{(...
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1answer
77 views

Definite integral problem of $\frac{x^n}{n!}$

I want to evaluate the following definite integral. $$\int_0^\infty\frac{x^n}{n!}dn$$ Where we have $n!=\Gamma(n+1)=\int_0^\infty t^ne^{-t}dt$ so that we can have $n\in\mathbb{R}$. I don't think ...
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0answers
28 views

Relationship between delay and fractional order differential equation

There are control systems with delay [1] There are differential equations with fractional order: $$D^\alpha x(t)=f(t,x(t))$$ I am wondering why we see control systems with delay and fractional ...
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1answer
65 views

Is there a notion of a complex derivative or complex integral?

While reading about fractional calculus in http://arxiv.org/pdf/math/0110241.pdf , I came across the following quote: Fractional integration and fractional differentiation are generalisations of ...
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0answers
32 views

The solutions of differential equations for a fractional order of differentiation that is a function of x

The fractional derivative of $x^k$, for $k>0$, is $$\frac{d^a}{dx^a}x^k = \frac{\Gamma{(k+1)}}{\Gamma{(k-a+1)}}x^{k-a}$$ Variabilising the order of differentiation by making it a function of $x$, ...
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20 views

Nonlinear Maximum Principle estimate

Im interested in the in the 2D Boussinesq equations given by $$\begin{cases}\partial_{t}u+u\cdot\nabla u+\nabla p+\Lambda^{\alpha}u=\theta e_{2}\\ \nabla\cdot u=0\\ \partial_{t}\theta+u\cdot\nabla\...
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40 views

Some pecular fractional integrals/derivatives of the natural logarithm

(Be prepared for a very long post) I have deduced the following formula: $$D^{-n}\ln(x)=\frac{x^n(\ln(x)-n)}{(-n)!}=\frac{x^n(\ln(x)-n)}{\Gamma(-n+1)}$$ Where $$D^{-1}f(x)=\int f(x)dx$$ $$D^{-...
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1answer
72 views

The $n$th integral of $\ln(x)$ and fractional derivatives

For a related question, I need to know the $n$th integral of $\ln(x)$ and the fractional derivative of $\ln(x)$. A break down of how fractional derivatives may be found on the Wikipedia. In ...
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1answer
65 views

How to add and multiply on fractional vector space

How to add and multiply on fractional vector space Please, answer in layman terms. I don’t understand the notation of Supersimetry and Super vector spaces. If a fractional “vector space” (or his ...
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0answers
16 views

When is $f(x,y,z)= \frac{x \cdot y}{z}$ convex?

I would like to know under what conditions $f(x,y,z)= \frac{x \cdot y}{z}$ is convex, pseudo-convex, or quasi-convex. I know that $g(x,y)= \frac{x^2}{y}$ $ \text{when } y >0 $ is convex and ...
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2answers
37 views

Taylor Series in Fractional Calculus

I recently studied fractional calculus, namely the possibility to define fractional derivatives of some functions, like $$\frac{\text{d}^{1/2}}{\text{d}x^{1/2}}\ f(x) ~~~~~~~~~~~~~ \frac{\text{d}^{2/...
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0answers
20 views

Can a Brownian motion be defined for negative time?

I was just looking at fractional brownian motions on this page. The definition of $B_H(t)$ requires integrating on a negative time domain on $dB(t)$ where $B(t)$ is a Brownian motion! Could you please ...
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0answers
20 views

Riesz potential of a set and its complement

Let $F\subset [0,1]$ be a closed set, $G = [0,1]\setminus F$, $\alpha \in(1,2)$. Is there a simple condition on $F$ under which the integral $$ \int_F\int_G \frac{dx\,dy}{|x-y|^{\alpha}} $$ is finite? ...
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0answers
17 views

Input a matrix with integral element in matlab

I am working on stochastic fractional differential equation , like below $$ D^2y+D^{\frac{3}{2}}y+y=1+t ,\space y(0)=0 ,y(1)=2$$I use RBF's (radial basis function) to solve it .so $y \sim \lambda_1 \...
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24 views

Reference for Fractional calculus and Differential Operators

I`ve been struggling with Fractional calculus and differential operators while studying special functions, and got to the conclusion that I need some references for them. So I ask for as many ...
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41 views

New method to solve this fractional differential equation

this is my problem \begin{align} & D{{\text{ }}^{2}}\text{ }y\left( x \right)\text{ }+\text{ }D{{\text{ }}^{\frac{3}{2}}}\text{ }y\left( x \right)\text{ }+\text{ }y\left( x \right)\text{ }=\...
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2answers
61 views

How do you take the fractional derivative $J^\frac{1}{2}f(x)$ where $f(x)=w\sin(x)$, and where $w$ is a constant? [closed]

What is $y$ in $$J^\frac{1}{2}f(x)=y$$ $$f(x)=w\sin(x)$$ where $w$ is a constant?
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0answers
45 views

An Article on an Application of Fractional Calculus

Just need some clarification on a few things from this article that I am reading: Solutions of the Telegraph Equations using a Fractional Calculus Approach. This article goes briefly over the ...
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1answer
61 views

Calculate derivative: $\frac{d^\beta}{d\alpha^\beta}\frac{d^\alpha}{dx^\alpha}\sin(x)$

Is it possible to "calculate" / simplify this expression? If it is, how can it be done? $$ \frac{d^\beta}{d\alpha^\beta}\frac{d^\alpha}{dx^\alpha}\sin(x) $$ for $ \alpha,\beta\in\mathbb{R}_{\ge0} $ ...
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1answer
59 views

What is( are) the advantage(s) of caputo's to Riemann-Liouville derivation?

I am new in fractional calculus. I see most of articles uses Caputo's derivation instead of Riemann-Liouville derivation. Is there some advantage? Can someone make some basic (simple) example for ...
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1answer
33 views

What condition on $f$ makes the formula $(−\Delta)^sf(x)=c_{n,s}\int_{\mathbb{R}^n}\frac{f(x)−f(y)}{|x−y|^{n+2s}}dy$ true?

I'm trying to understand the concept of fractional Laplacian, and I found the page https://www.ma.utexas.edu/mediawiki/index.php/Fractional_Laplacian,and the formula $$(−\Delta)^sf(x)=c_{n,s}\int_{\...
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0answers
34 views

Fractional logarithm operator

Does there exist a fractional logarithm operator? Something like this: $$L_0(x) = x$$ $$L_1(x) = log(x)$$ $$L_{0.5}(x) = ???$$ This is the motivating situation: consider perception of sound, which ...
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0answers
27 views

Another proofs of the comparison principle for PDEs with nonlocal term (derivative)

In my research I study (fully) nonlinear PDEs with fractional derivatives. For an analysis of these, I use the theory of viscosity solutions. So I am now at the end of my tether becasuse I can not ...
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0answers
40 views

Caputo Fractional Derivative

Can any one explain how to find Caputo fractional derivative of translated function.If we know Caputo fractional derivative of f(t) how can we find Caputo fractional derivative of f(a-t), a is a real ...
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3answers
160 views

Derivative of a negative order?

Below, $\Delta$ means taking the derivative, $\frac{d}{dx}$. For $n\in\mathbb{Z}$, $n\geq 0$, we have $$\Delta^n\sin{x}=\sin{(x+n\tau/4)} \\ \Delta^n\cos{x}=\cos{(x+n\tau/4)}$$ I found that out while ...
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1answer
26 views

Positivity of a polynomial with a fractional order term

I'm trying to show this problem. Show that $$ f(x):=2(2-a)(3-a)x^{2}-5(3-a)x+6-9(1-x)^{3-a} $$ is positive on $\{x\in\mathbb{R} \mid \frac{2}{3}<x\le1\}$ for all $a\in(0,1)$. I have tried ...
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32 views

Proof of a certain inequality in two-dimensional Euclidean space

Please think it easy because it is not an assignment. I'm trying to show the following problem. Show that the inequality $$ 12(x^{5/2}+y^{5/2})+15(\sqrt{x}y^{2}+x^{2}\sqrt{y})>25(\sqrt{x}+\...
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53 views

Form of the fractional Laplacian in polar coordinates

Can anybody point out the form of the fractional Laplacian $ (-\Delta)^s f (\rho, \varphi) $ in polar co-ordinates on a restricted domain? I am trying to solve an equation defined in a semi-unbounded ...
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24 views

Domain specification of derivative extension.

Given the definition of Taylor expansion: $$f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$$ We can find the $m$'th derivative of $f(x)$ quite easily: $$\frac{d^m}{dx^m} f(x) = \sum_{n=0}^{\...
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0answers
12 views

How is the uncertainty of fractional computation 4 events per 10 million?

I have observed a systematic four-event uncertainties in a series of particle physics computations among selection size of up to 10 000 000 events. Now, this term came up with the computation of the ...
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56 views

How to solve an integral with a fractional order.

How should I find a value of these integrals: $$ A:=\int_{0}^{\infty}\frac{\sin(x)}{(x+1)^{2-\nu}x^{\nu}}dx \quad\text{and}\quad B:=\int_{0}^{\infty}\frac{\sin(x)}{(x+1)^{1-\nu}x^{\nu}}dx, $$ where $\...
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1answer
52 views

Is this derivative thing I found a defined mathematical concept? If so, what is it called?

I'm sure you're aware that $\frac{d^n}{dx^n}\frac{1}{x}=\frac{(-1)^nn!}{x^{n+1}}$ Well, what if $n=\frac{1}{2}$? $$\frac{d^\frac{1}{2}}{dx^\frac{1}{2}}\frac{1}{x}=\frac{(-1)^\frac{1}{2}(\frac{1}{2})!...
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26 views

Generalised Taylor series to fractional order derivatives and special functions

A year ago or so I read this papar which was wonderfully illuminating link. For example the author seduces the reader with wonderfully compact representations like that of the bessel $J_v(z)$ function ...
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23 views

A question on Riesz potential (Leibniz rule) : $ (-\partial_x^2)^{1/2} (fg) = f(-\partial_x^2)^{1/2} g + g(-\partial_x^2)^{1/2} f $?

I am wondering if I can regard the Riesz potential $$ (-\partial_x^2)^{1/2} = (-1)^{1/2}\partial_x $$, where the Riesz potential $(-\partial_x^2)^{1/2} = \mathscr{F}^{-1}|\xi|\mathscr F$ with the ...
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32 views

Continuity of Fractional Derivative

Here's a continuity result that I believe to be true, but I don't know if my assumptions are minimal (i.e. does it still go through with just assuming continuity or something slightly weaker ...
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0answers
42 views

Difference between terminal value problem and initial value problem of ODE r FDE?

What is difference between terminal value problem and initial value problem of differential equations. Kindly give an example.
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1answer
155 views

Fractional order Riemann Stieltjes integral

The definition of fractional order integral is well-known. Is there any definition for fractional order Riemann Stieltjes integral?
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2answers
331 views

Can the following trick be expanded upon?

Main Question What is the expansion of $d^{1+\epsilon}?$ Background I noticed the following trick (sometimes more laborious) to directly differentiate $ f(x) $ twice without differentiating it even ...