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

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Fractional derivative of sine function

I try to reproduce the results of a paper. The authors are dealing with the fractional diffusion equation \begin{align} \partial_t^\alpha u - \partial_x^2 u = f \end{align} on the domain ...
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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 ...
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2answers
65 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) = ...
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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? ...
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56 views

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

We already know the following facts: $$\displaystyle (-\Delta)^su(x):=c_{n,s}\text{P.V.}\int_{\mathbb{R}^N}\frac{u(x)-u(y)}{|x-y|^{N+2s}}\,dy, $$ where $s\in (0,1)$. $$\int_{\mathbb{R}^N} ...
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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 ...
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37 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 ...
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42 views

Formula for tangent derivatives, how to prove?

How to prove? $$(\tan x)^{(s-1)}=\pi^{-s}\Gamma(s)\left(\zeta\left(s, \frac12-\frac x\pi\right)+(-1)^s\zeta\left(s, \frac12+\frac x\pi\right)\right) $$
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73 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 ...
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36 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} ...
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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 ...
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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 ...
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2answers
61 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: ...
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29 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 ...
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46 views

Intensity of fractional brownian noise

Having a White noise driven SDE $dX = f(X)dt + \sqrt{2D}dW$, the noise intensity is equal to D. What is the noise intensity, if I consider fractional brownian noise, instead of white one?
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40 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) = ...
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43 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.
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35 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 ...
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3answers
67 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. ...
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66 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 ...
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37 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 ...
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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 ...
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Interpreting the area between two integer order derivatives.

Below is a rough figure of the fractional derivative of $\,f(t) = 1$ where the horizontal axis represents the order of the derivative ranging from the from the 0th to the 3rd derivative. The vertical ...
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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 ...
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Does the inequality $\int_{\Omega}(-\Delta)^{\frac 12}G(w(x))(u(x)-C)^+ \geq 0$ hold? If not, can we bound it from above in a particular way?

Let $G$ be a locally Lipschitz function such that $G(0)=0=G'(0)$ and $G$ is also increasing. I want to know if $$\int_{\Omega}(-\Delta)^{\frac 12}(G(w(x))(u(x)-C)^+ \geq 0$$ where $C$ is a constant. ...
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109 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) - ...
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36 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 $b$
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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 & ...
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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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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 ...
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1answer
128 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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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?
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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 ...
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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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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 ...
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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 ...
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214 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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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) ...
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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 ...
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Fourier transformation on a torus and the definition of fractional Laplacian

as we know, in $R^n$, for a function $f$, we can define its Fourier transform as $$\hat f(\xi)=\int_{R^3}f(x)e^{-ix\cdot \xi}d x,$$ with this, the Laplacian of $f$ can be elegently defined by ...
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Problem with the application of the fractional integral (as in wikipedia) , example $f(x)=\exp(x)-1$

I am still fiddling with the understanding and application of the fractional integration/differentation. I've tried the wikipedia-formula for the Cauchy's iterated integration: $$ (J^{\alpha} f)(x) = ...
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Proprieties of the Fractional Laplacian on unbounded domain

I'm interesting to the stochastic PDE $$\left\{\begin{array}{l}\dfrac{\partial u}{\partial t}(t,x)=\Delta_{\mathbf{\alpha }}u(t,x) + {\dot{W}}(t,x), \\u(0,x)=u_{0}(x),\,\,\,\, ...
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I can Euler-sum $\sqrt{-\ln(1)}-\sqrt{-\ln(2)}+\sqrt{-\ln(3)}-\cdots$. But how can I do $\sqrt{-\ln(1)}+\sqrt{-\ln(2)}+\sqrt{-\ln(3))}+\cdots$?

This is also related to an older thread in MSE ("what is the half derivative of zeta at zero?") . One of the possible steps in the problem of that thread was to evaluate the series ...
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150 views

Fractional calculus

I have this exercise : "Consider the Cauchy problem's : $$ ^C D^{\alpha}y(t)=f(t,y(t),y'(t)), t\in [0,T] ....(1) $$ $$ y(0)=y_0, y'(0)=y_1 .... (2) $$ Where ...
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What does a “half derivative” mean?

I was looking at fractional calculus on Wikipedia, specifically this section and came across the half derivative of the function $y=x$ which is $y=\frac{2\sqrt{x}}{\sqrt{\pi}}$ . The derivative tells ...
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60 views

Are fractional calculus differintegrals with arbitrary algebras for the order possible?

I confess, I'm a bit of a dilettante with respect to mathematics; But one thing I've been interested in is generalizations of abstractions. So naturally when I heard about the possibility of getting ...
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288 views

What is the intuitive or geometric explaination of fractional derivatives?

I'm starting to study more advanced solid mechanics, particularly understanding elastomers' stress strain relationships and creep. A common way of describing the variation in the aforementioned ...
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why does a fractional differential equation have a unique solution?

Why must there be a unique solution to a linear constant-coefficient fractional differential equation of order $(n,q)$ with $\lceil\frac{n}{q}\rceil$ initial conditions? (All notation is as in Miller ...
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62 views

I need clarification on $\delta$ - derivative

Please can someone tell me more about $\delta$ -derivative ($\delta=x\dfrac{d}{dx}$) as it appears in the Hadamard definition of frational derivative or elsewhere. Why, when or where we use it. Does ...
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167 views

$\frac{d^{\frac{1}{2}}}{dx^{\frac{1}{2}}}$, what does it mean?

I would like to know what these fractional exponents means in a derivative $\frac{d}{dx}$ operator. Like, I've seen $\frac{d^2}{dx^2}$ but I don't know what $\frac{d^{\frac{1}{2}}}{dx^{\frac{1}{2}}}$ ...