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

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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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17 views

The semigroup of powers of the differential operator in fractional calculus.

In my ignorance I'm slightly wary of a follow-up question here as it might belong in MO. If so, I'm sorry. Motivated simply by curiosity and this question, I'd like to investigate the semigroup $S$ ...
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13 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. ...
1
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1answer
42 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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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
396 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
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0answers
28 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
29 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
80 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
46 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
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2answers
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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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0answers
46 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
24 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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0answers
84 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
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0answers
116 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
60 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
521 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
48 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
100 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
62 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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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
127 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
524 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
41 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 ...
3
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2answers
181 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
73 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
60 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
33 views

problem with a fractional derivative

I want to evaluate the half derivative function of $ 1/t $ by induction i get that for every 's' then $D_{x}^{s} \frac{1}{t}=\frac{\Gamma(s+1)(-1)^{s}}{t^{s+1}} $ however if i use the formulae from ...
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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
136 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 $\frac{d^{\frac{1}{2}}}{dx^{\frac{1}{2}}}$ ...
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1answer
141 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
302 views

(fractional) half 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 ...
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1answer
69 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
60 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
229 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
424 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
34 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
92 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 ...
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0answers
90 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
82 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
125 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
206 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 ...
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1answer
110 views

Existence of Riemann-Liouville Integral

The Riemann Liouville integral is defined as: $\frac{1}{\Gamma\left(\nu\right)}\int\limits _{h}^{t}\left(t-\xi\right)^{\nu-1}f\left(\xi\right)d\xi$ It is supposed it does exist for all $\nu>0$ and ...
3
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2answers
181 views

Taylor Expansion of the 1/2th Derivative

In trying to solve the problem $\sqrt D f(x)=g(x)$ I tried to expand the derivative as a Taylor series, and have encountered a lot of problems. Is there some reason that this shouldn't be possible? ...
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2answers
907 views

Fractional Derivative Implications/Meaning?

I've recently been studying the concept of taking fractional derivatives and antiderivatives, and this question has come to mind: If a first derivative, in Cartesian coordinates, is representative of ...
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1answer
148 views

Fixed memory principle in fractional calculus

I am trying to read this paper and have a certain doubt. On page 8 the author comments that First of all, there is a fundamental problem associated with all fractional differential operators ...
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0answers
103 views

Derivatives and integrals, normal and fractional, and their explanations and relations

Assuming, naively, that one acquires the nth derivative of a function by repeatedly differentiating and finding a pattern. Thus one gets $f^{(n)}(x)=g(x,n)$. I have a few questions about this ...
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63 views

nonlinear integral equation

let be the integral equation for two functions $ f(x) $ and $ g(x) $ in the form $$ g(s)= \int_{0}^{s}\sqrt{s-f(x)}dx $$ is valid to accept that in the sense of fractional calculus, the ONLY ...
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3answers
217 views

fractional derivative of a heaviside function

given the function $$ f(x)= \frac{H(x+1)}{\sqrt{x+1}} $$ how can i evaluate the fractional derivative $$ \frac{d^{1/2}}{dx^{1/2}}f(x) $$ if i use the standar definition for powers of 'x' i get a ...
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2answers
319 views

Fractional derivative of a constant (Riemann-Liouville Derivative)

In a book I read about Riemann-Liouville fractional derivative, it says, $$_0D_t^\alpha 1=\frac{t^{-\alpha}}{\Gamma(1-\alpha)},\alpha\geq0,t\geq0$$ which identically vanishes for ...