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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Limits with exponential functions [on hold]

First prob: ( I tried with $e^{\ln}$ property but doesn't work) Here is the first problem pic Second prob:(at this if I get stuck at the part where I got $\frac{e^1}{x^2} - \frac{e^1}{x^2}$) second ...
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20 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 ...
4
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1answer
59 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
27 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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18 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\\ ...
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35 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 $$D^{-1}f(x)=\int f(x)dx$$ ...
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49 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
61 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
13 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 ...
1
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1answer
24 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) ~~~~~~~~~~~~~ ...
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16 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
17 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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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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20 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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36 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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1answer
39 views

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

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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37 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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51 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
32 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 ...
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31 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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21 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
34 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 ...
3
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2answers
109 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 ...
3
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1answer
25 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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0answers
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 $$ ...
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43 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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23 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) = ...
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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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54 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 ...
4
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1answer
51 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 $n=\frac{1}{2}$? ...
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25 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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22 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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0answers
27 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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38 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
141 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
330 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 ...
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46 views

Is the Riemann–Liouville fractional derivative holomorphic in order?

If my understanding of complex analysis is correct then the arbitrary order generalization of Cauchy's formula for repeated integration $$(J^\alpha f) ( x ) = { 1 \over \Gamma ( \alpha ) } \int_0^x ...
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1answer
49 views

Non-integer order derivative

I do not know much about fractional calculus, except what I have read in a few short posts at MSE and https://en.wikipedia.org/wiki/Fractional_calculus. I know that order of a derivative can be ...
4
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1answer
58 views

Example of a function that has fractional derivatives of order less than 1 but not 1

I have recently learned that some fractals can have fractional derivatives of order less than 1, say of 1/2 even if they are not differentiable (have no derivative of order 1). I wonder if there is a ...
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0answers
55 views

Generating fractional taylor series

I was considering the notion of taylor series which posit that the sum $$ \sum_{i=0}^{\infty} \frac{1}{i!} a_ix^i $$ Where: $$ a_i = \frac{d^if}{dx^i}_{x= a} $$ Converge to the function f in a ...
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35 views

Geometric and physical significance of differintegrals

Consider a fractional integral or derivative, what can we associate to them in geometrical or physical terms, I have seen that for fractional derivatives the greater the order of derivative the less ...
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4answers
218 views

Taking the half-derivative of $e^x$

While attempting to teach myself the fractional calculus, I encountered a tragically early roadblock. For non-power rule fractional derivatives, I am having a lot of trouble evaluating for a closed ...
3
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1answer
47 views

What is the integral of $e^{a\cdot x+b\cdot y}$ evaluated _under_ the Koch Curve

Grew out of frustration about this question; just replace "over" by "under": What is the integral of $e^{a\cdot x+b\cdot y}$ evaluated over the Koch Curve What is $$ \iint_{K} e^{a \cdot x + b ...
3
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1answer
43 views

References for the operator $(I-\Delta)^{\alpha /2}$

I am studying PDEs involving fractional differential operators, and I have found a few properties for the operator $(I-\Delta)^{\alpha /2}$ scattered through scientific papers. I wonder if there is a ...
6
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1answer
227 views

Does Fractional Calculus define the derivative over the Weirstrass Function?

I recently read this paper on defining the fractional derivative for the Wierstrass function. This seems very interesting since derivatives over fractals are generally not well defined. Yet, this ...
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1answer
77 views

Relation between Laplace and Fourier transform

I have a function that has the property $\tilde f(s) = \tilde{f}(abs(s))$. For this function, I need the inverse Fourier transform. I actually know the inverse Laplace transform of $\tilde f$ and I ...
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1answer
31 views

Derivative of a function which is defined as a derivative

I'm new to this kind of stuff so maybe this is a stupid question but I don't even know what to search on the internet. My problem is that: find the derivative of the following function on $\Bbb R^3$ ...
4
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2answers
93 views

Geometric Interpretation of Fractional Derivatives

I was looking for a geometrical interpretations of fractional derivatives and fractional integrals. I would be glad to see any kind of intuitive and preferably visual interpretation of the objects ...
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0answers
65 views

Fractional derivative definition

Suppose that $f(x) \in C^1$ for a $x \in [a, x]$. Then a regularization of Riemann-Louisville fractional derivative is defined as: $ \frac{1}{\Gamma(1-b)} \frac{d}{dx} \int_{a}^{x}\left( ...