Questions on the differentiation/integration of functions to arbitrary order.

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3
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1answer
32 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
votes
1answer
35 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 ...
7
votes
1answer
144 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 ...
1
vote
1answer
37 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 ...
0
votes
1answer
26 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 $\Bbb R^3$ ...
1
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0answers
35 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 ...
2
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0answers
59 views

Does Fractional Calculus have a real connection with Fractals? (or is it just an extra variable trick)

The fractional derivative and integral (operators that let you differentiate or integrate a fractional number of times) have drawn a lot of attention from people outside the field. Yet, after reading ...
0
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0answers
12 views

Fractional-order Halanay-type inequality

If there is any research article or books are available for Fractional-order Halanay-type inequality, if yes means, Please provide the proof or references list.
0
votes
0answers
11 views

The boundness of the solution of a fractional differential equation

D is the notation of Riemann-Liouville derivative. When I did the common first order differential equation and prove the same thing, I am able to first solve it and then prove the statement. But ...
0
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0answers
46 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( ...
0
votes
0answers
15 views

Understanding fractional-differencing formula

I have a time series $y_t$ and I would like to model it as an ARFIMA (a.k.a. FARIMA) process. If $y_t$ is integrated of (fractional) order $d$, I would like to fractionally-difference it to make it ...
0
votes
0answers
42 views

Show that a complex number's set is no empty

Consider $\alpha \in \mathbb{C}$ such that $Re (\alpha) > |\alpha|^2.$ Why is the set $$\Omega_{\alpha}=\mathbb{C}^{*} - \{\lambda^{\alpha}e^{i\theta\alpha}; \lambda > 0 \ \mbox{and}\ -\pi \leq ...
2
votes
1answer
45 views

Integral over Fractals with respect to fractal dimension

I understand that there is type of integral with respect to measures that can return values when evaluated over an integral. But is there an Integral that returns d dimensional volume where d is the ...
0
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0answers
19 views

fractional system fractional order equations

As resolved and how is the graph the phase plane of $D^{\alpha}y_{1}(t)=2y_{1}(1-\dfrac{y_{1}}{2y_{2}}-\dfrac{y_{1}}{2})$ $D^{\alpha}y_{2}(t)=3y_{2}(1-\dfrac{y_{2}}{2y_{1}}-\dfrac{y_{2}}{2})$ ...
2
votes
1answer
40 views

What can I do with half-derivative?

If I want to know the slope of the tangent of curve function I just have to find the derivative of this function and if I want to know the area under this curve I can integrate it (the function of ...
0
votes
0answers
29 views

properties of fractional calculus using grunwald letnikov

First definitions of fractional integral and derivative I 'm using $I^{\alpha}f(x)=\displaystyle\frac{1}{\Gamma(\alpha)}\displaystyle\int_{0}^{x}(x-t)^{\alpha-1}f(t)dt$ y ...
0
votes
2answers
88 views

Imaginary-Order Derivative

I would like to find the imaginary-order derivative of a function (let's just focus on a simple function for now). There is the Riemann-Liouville fractional-derivative: $$ _{a}D^{i}_{t} f(t) = ...
1
vote
0answers
59 views

Computing the fractional derivative of a fractional integral

I know that $D^{\alpha}I^{\alpha}f(x)=f(x)$ and $D^{\alpha}I^{\beta}f(x)=D^{\alpha-\beta}f(x)$ but How can prove this? ...
1
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0answers
56 views

Is $(-\Delta)^{s}$ c0incident with $(-\Delta)^{s/2}$?

We already know the following facts´╝Ü $$\displaystyle (-\Delta)^su(x):=c_{n,s}\text{P.V.}\int_{\mathbb{R}^N}\frac{u(x)-u(y)}{|x-y|^{N+2s}}\,dy, $$ where $s\in (0,1)$. $$\int_{\mathbb{R}^N} ...
2
votes
2answers
40 views

Is the Fractional integral operator well-defined?

How to prove the fractional integral operator $J_{\alpha}:L^p(\Bbb R^+)\rightarrow L^p(\Bbb R^+)$ (of order $\alpha>0$) which is defined for each $f\in L^p(\Bbb R^+)$ by $$J_{\alpha}f(x):={1\over ...
1
vote
0answers
48 views

Can a Local Fractional Differential Operator exist?

Let $f: \mathbb{R} \rightarrow \mathbb{R}$. The derivative of $f$ is defined pointwise, and we say that $f$ is differentiable if the derivative exists in each point. Higher order derivatives are ...
0
votes
1answer
50 views

Formula for tangent derivatives, how to prove?

How to prove? $$(\tan x)^{(s-1)}=\pi^{-s}\Gamma(s)\left(\zeta\left(s, \frac12-\frac x\pi\right)+(-1)^s\zeta\left(s, \frac12+\frac x\pi\right)\right) $$
6
votes
1answer
100 views

What level of math is needed to learn fractional calculus?

I was skimming through wikipedia pages and stumbled upon the fractional calculus page. My interest increased when I noticed it has applications in physics. I was wondering as an undergraduate who's ...
3
votes
1answer
53 views

Is fractional order Sobolev spaces reflexive?

Let $0<s<1$, we define $$ W^{s,p}(\Omega):=\left\{u\in L^p(\Omega),\,\frac{|u(x)-u(y)|}{|x-y|^{\frac{N}{p}+s}}\in L^p(\Omega\times\Omega)\right\} $$ with norm $$ \|u\|:=\left(\int_{\Omega} ...
0
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0answers
40 views

Solutions of fractional linear dynamical systems

The Mittag-Leffler function is defined as: $$ E_\alpha(\tau) = \sum_{k=0}^{\infty}\frac{\tau^k}{\Gamma(\alpha k + 1)}, $$ which can also be defined, analogously, for matrices $A\in\mathbb{R}^{n\times ...
9
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0answers
232 views

Is this similarity just a coincidence?

Here is the function $-1/x$: If we add infinitely many similar functions with a shift of pi/2 each in both directions, we get $\tan x$. But if we do the same only in one direction, we get ...
1
vote
2answers
63 views

n-th Derivative

It can be proven the for a function $h(x)=f(x)g(x)$, letting $f^{(k)}(x)=\frac{d^k}{dx^k}f(x)$ and $g^{(k)}(x)=\frac{d^k}{dx^k}g(x)$ then the n-th derivative, for n is an integer is: ...
0
votes
1answer
31 views

Simplifying general formula for fractional derivative by removed derivative of integral.

On the wikipage about fractional calculus, there's a general formula for the fractional derivative: $D^\alpha$ is the derivative operator. $$D^\alpha ...
3
votes
1answer
52 views

Do smooth functions have fractional derivatives of all orders?

Suppose $\nu > 0$ and $n$ is such that $\lceil\nu\rceil = n$. The Riemann-Liouville definition of the fractional derivative would be $$f^{(\nu)}(x) = ...
1
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0answers
49 views

Geometric mean of fractional derivative

What is the geometrical mean of the fractional derivative (of order $\alpha \in (0,1)$) for a function $f:\Bbb R \rightarrow \Bbb R$? For example $f$ is increasing on $\Bbb R$ if $f'$ is positive.
0
votes
1answer
40 views

fractional derivative of test function

I have a small questions concerning the fractional derivative of a test function. Is it true that if $u \in C^{\infty}_{c}(\mathbb{R})$ and we define the fractional derivative of this function as ...
1
vote
3answers
71 views

Non integer derivative of $1/p(x)$

I need to find the $k$'th derivative of $1/p(x)$, where $p(x)$ is a polynomial and $k\in\mathbb{R}$ It dosen't have to be an explicit formula, an algorithm which finds a formula for some $k$ is fine. ...
2
votes
1answer
71 views

Fractional powers of the operator $B: L_2(\mathbb{R}) \mapsto L_2(\mathbb{R})$, $Bf = f-f^{''}$.

Consider the linear operator $B: L_2(\mathbb{R}) \mapsto L_2(\mathbb{R})$ defined by the following mapping: $Bf = f-f^{\prime\prime} \equiv (I-\Delta)f$, where $\Delta$ is the Laplace operator that ...
0
votes
1answer
46 views

Fractional Derivatives on a function with bounded Support

I have a question about functions that have bounded support in $\mathbb{R}$. In particular, suppose that I have a function $f$ with support $A\subset \mathbb{R}$ so that $A$ is compact. Without loss ...
4
votes
0answers
72 views

Fractional derivatives of Gamma function

For integer $n \geq 0$, we have $\dfrac{d^n}{ds^n} x^s = (\ln x)^n \,x^s$. From this it follows, for example, that $$\int_0^{\infty}e^{-x}\ln^n x \,dx= \Gamma^{(n)}(0)$$ Question: is there a way of ...
2
votes
0answers
104 views

Fractional Derivatives

If we define the (forward) difference operator as $$\Delta f(x)=f(x+\Delta x)-f(x)$$ we can break it up using the "shift" operator $E\,f(x)=f(x+\Delta x)$ and the "identity" $1\,f(x)=f(x)$. Then ...
1
vote
1answer
142 views

Definition of fractional Laplacian on a compact manifold?

How does one define the fractional Laplacian operator $(-\Delta)^s$ on a compact Riemannian manifold? In $\mathbb{R}^n$, it is defined $$ (-\Delta)^s f(x) = c_{n,s} \int_{\mathbb{R}^n} \frac{f(x) - ...
1
vote
0answers
38 views

fractional derivaitve of logarithm function $x^ {a} log(x) $

Given the function $ x^{a}\log(x) $ natural logarithmic Could someone tell me how to evaluate the fractional derivative $$ \frac{d^{b}}{dx^{b}}x^{a}\log(x) $$ for positive $a$ and $b$
21
votes
4answers
767 views

Fractional Calculus: Motivation and Foundations.

If this is too broad, I apologise; let's keep it focused on the basics if necessary. What's the motivation and the rigorous foundations behind fractional calculus? It seems very weird & ...
2
votes
0answers
40 views

How should I interpret this function notation?

I'm trying to implement an FDGD Algorithm from a paper and I'm a little stuck how to interpret a piece of function notation. See page 7, equations 2 and 3 in this document: In there we have ...
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vote
0answers
37 views

Convergence of a limit

everyone. I have a question regarding the convergence of a certain limit. I've been fiddling with it but its been proving quite evasive. What I am trying to calculate is the Grundwald-Letnikov ...
3
votes
1answer
138 views

Do fractional derivatives maintain the $[fg]'=f'g+g'f$ and $f(g(x))'=f'(g(x))\cdot g'(x)$ rules?

Of course, I'm not really familiar with all fractional derivative methods, but is it a necessary rule that they all should comply with? If not, which ones, for example, do and which don't ? ( ...
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0answers
76 views

A question about fractional derivatives

What would be the fractional derivative of any order 'b' of the function $ (a-x) $ ? My guess is: $$ \frac{d^{s}}{dx^{s}}(a-x)^{-1}= \frac{\Gamma(s+1)}{(a-x)^{s+1}} $$ Is this correct?
5
votes
2answers
177 views

Are all fractional deriviatives/integrals of $e^x$ equal to $e^x$?

I have learned through calculus that the derivatives and the indefinite integrals of the exponential function are the same (at integer arguments) but was wondering if this holds true for fractional ...
3
votes
0answers
54 views

Characterization of functions with fractional expansion near zero

I would like to understand if it is possible to completely characterize real-valued functions with an expansion of this type: $f(x)=f'(0)\cdot x + o(x^{\alpha})\qquad \alpha \in (1,2)$ I am not ...
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vote
0answers
61 views

Fourier Transform of Fractional Laplacian

I'm trying to solve a PDE with a spectral method. The PDE has a fractional Laplacian... $\Delta^s$. In regards to a numerical implementation, will the "s" term simply become the exponent of the ...
11
votes
1answer
198 views

Could you explain me the use of fractional derivatives?

For first time in my (loooong !) life, I heard, thanks to a question posted on SE, about fractional derivatives. In Wikipedia, I found very interesting material. But, being a physicist and not a real ...
2
votes
0answers
244 views

Fractional Derivative of a Taylor Series?

I have a function defined only by it's taylor series: $f(x) = \sum_{k=0}^\infty \frac{f^{(k)}(0)}{k!}x^k$ Obviously, integer derivatives can be defined as $\frac{d^n}{dx^n} f(x) = \sum_{k=0}^\infty ...
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0answers
113 views

How can we interpret the coefficients of Laurent series?

The coefficients of a Taylor series of a function about a given point are related to the nth derivatives of the function at that point. Can we make a similar statement about what the (negative-index) ...
2
votes
1answer
786 views

Solving double integral numerically in matlab

In the paper "The fractional Laplacian operator on bounded domains as a special case of the nonlocal diffusion operator". Where the author has solved a fractional laplacian equation on bounded domain ...