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

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1answer
22 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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0answers
40 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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1answer
28 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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0answers
32 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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1answer
27 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
56 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
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1answer
61 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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1answer
29 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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48 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
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0answers
35 views

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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0answers
83 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 ...
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18 views

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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1answer
84 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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0answers
31 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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4answers
563 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 & ...
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0answers
38 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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0answers
33 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 ...
2
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0answers
99 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
63 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?
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2answers
125 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 ...
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0answers
53 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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0answers
42 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 ...
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141 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 ...
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0answers
165 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
79 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) ...
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1answer
593 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 ...
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0answers
67 views

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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2answers
117 views

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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0answers
77 views

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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0answers
361 views

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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1answer
143 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 ...
2
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1answer
963 views

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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0answers
54 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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3answers
247 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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0answers
84 views

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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1answer
61 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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0answers
59 views

Difficult to see Leibniz's theorem for differentiating an integral in fractional calculus

The text book tell's: Consider the formula $$\dfrac{d^{-1}f}{[d(x-a)]^{-1}} = \int_a^xf(y)dy=\dfrac{1}{n!} \dfrac{d^n}{dx^n}\int_a^xf(y)dy$$ For general integer $n$ one need only notice that ...
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2answers
162 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 $\frac{d^{\frac{1}{2}}}{dx^{\frac{1}{2}}}$ ...
5
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1answer
184 views

What is the physical meaning of fractional calculus?

What is the physical meaning of the fractional integral and fractional derivative? And many researchers deal with the fractional boundary value problems, and what is the physical background? What ...
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3answers
410 views

Half order derivative of $ {1 \over 1-x }$

I'm new to this "fractional derivative" concept and try, using wikipedia, to solve a problem with the half-derivative of the zeta at zero, in this instance with the help of the zeta's ...
0
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1answer
79 views

Introduction to fractional calculus: problem with identity

I can't see the next step: $D^\alpha e^{ix} = i^{\alpha}e^{ix} = e^{i\alpha \frac \pi2}e^{ix}$
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0answers
79 views

How to predict order of a set of fractional differential equations?

I have a set of differential equations of the form: $$\frac{dv}{dt} = a[b-c*m-d*n-e*h]$$ $$\frac{dm}{dt} = p(v)$$ $$\frac{dn}{dt} = q(v)$$ $$\frac{dh}{dt} = r(v)$$ Using fde12 in MATLAB I can ...
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3answers
261 views

What is the half-derivative of zeta at $s=0$ (and how to compute it)?

[Update 3:] I gave a new partial answer following the ansatz in question Q3. I leave the other parts of the question untouched, they are also partially answered in specialized other questions in MSE. ...
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2answers
667 views

Software for Solving Numerically Fractional Differential Equations

I have been trying to find information on how to solve fractional differential equations numerically with the usual maths software (Mathematica, Maple, Matlab..). Or to find an alternative program to ...
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0answers
38 views

Analog of Picard's theorem for Fractional Differential equations.

I need an analog of Picard's theorem of existence and uniqueness of solutions. The theorem is to be applied to linear fractional order differential equations with constants coefficients. I don't want ...
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0answers
113 views

Importance of Riemann-Liouville fractional derivative from historical point of view

Why Riemann-Liouville fractional derivative is important from historical point of view than that of Caputo fractional derivative? As we know Riemann-Liouville fractional derivative is more theoretical ...
3
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0answers
112 views

Fractal derivative of complex order and beyond

Is there some precise definition of "complex (fractal) order derivative" for all complex number? I am aware of the Riemann-Liouville fractional definition given here: Complex derivative but I would ...
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0answers
83 views

Geometrical Interpretetion of Half Derivative [duplicate]

How would you understand in a intuitive way the meaning of: $$D^{\frac{1}{2}}x^2=\frac{\Gamma(3)}{\Gamma(\frac{5}{2})}x^{\frac{3}{2}}=\frac{8}{3\sqrt{\pi}}x^{\frac{3}{2}}$$ or ...
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0answers
137 views

Kernel of Fractional Differential Operator

Suppose we have a fractional differential equation: $$\left[D^{nv}+a_{1}D^{\left(n-1\right)v}+\dots+a_{n}D^{0}\right]y(t)=0$$ where $\nu=\frac{1}{q}$ and $q\in\mathbb{N}$ and y is an analytic ...
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3answers
222 views

Help in proof of theorem about Riemann-Liouville Fractional Calculus

Theorem: Let, $$\left[D^{nv}+a_{1}D^{\left(n-1\right)v}+\dots+a_{n}D^{0}\right]\left(y\right)=0$$ be a fractional differential equation of order $\left(n,q\right)$, where $v=\frac{n}{q}$, and let ...