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Questions tagged [fractional-calculus]

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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Fractional derivative question

This might be a duplicate but I could not find the answer here. I want to prove the following relation for the fractional derivative $a \in \mathbb{R}$: $$ (i\partial_x)^a = (ix)^{-a} \frac{\Gamma(a-x\...
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1answer
48 views

Definition of Negative Half Derivative

I know the definition of positive fractional derivative, which is given by $$D^{\alpha} f(x)=\frac{1}{\Gamma(1-\alpha)} \frac{d}{d x} \int_{0}^{x} \frac{f(t)}{(x-t)^{\alpha}} dt,\quad\quad \alpha\in(0,...
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Compact embedding of fractional space

Is the space $H^\lambda((0,T); H^1(K))$ for $0 <\lambda <1$ where $K$ is compact subset of $\mathbb{R}^n$ compactly embedded in $L^2( (0,T) \times K)$? $H^\lambda((0,T); H^1(K))\hookrightarrow \...
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Fractional integration in terms of shift operator

I am trying to understand the fractional derivation and fractional integration. I found a representation of the fractional derivation operator in term of shift operator that is: $D^n=(1-S)^n = \sum_{...
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Definition of Riemann-Liouville fractional integral

A real function $f(x),\;x>0$ is said to be in the space $C_{\mu},\;\mu\in\mathbb{R}$ if there exists a real number $p(>\mu)$ such that $f(x)=x^pf_1(x)$, where $f_1(x)\in C[0,\infty)$ and is said ...
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1answer
41 views

About an equality of fractional Laplacian on a bounded domain

Let $0<s<1$. Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. We know that $$\|(-\Delta)^{s/2}u\|_{L^2(\mathbb{R}^n)}^2=\int_{\mathbb{R}^n}\int_{\mathbb{R}^n}\frac{|u(x)-u(y)|^2}{|x-y|^{n+...
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Is Fractional Calculus an important research topic “in pure mathematics” today?

Being a potential graduate student, I would like to know if fractional calculus is an actively developing research topic in the area of pure mathematics today.
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34 views

Integration against anti-symmetric kernel

Let $u\in C[0,T]$ and $\beta\in(0,1)$. Suppose that $$ \int_0^x u(r)k(x,r)\,dr = 0,\quad \text{for all }x>0, $$ where $$ k(x,r):= (x-r)^{-\beta}-r^{-\beta},\quad x>r>0. $$ I want to prove ...
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24 views

Derivative of constant by fractional derivative of Jumarie’s type?

The fractional derivative of a constant function is zero in Jumarie’s type? How can this be proven? The derivative formula of Jumarie is given by $$D^{\alpha} f(t) = \dfrac{1}{\Gamma(n-\alpha)} \dfrac ...
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Proof of laplace tansform

How to prove Laplace transform of Jumarie fractional derivative $$ L\{f^{(\alpha)}(t)\} =s^\alpha F(s) -s^{\alpha-1} f(0),$$ in which $f^{(\alpha)}$ is a fractional derivative. Using the paper with ...
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Inverse Laplace with fractional power

What is inverse laplace for $y=\{-16s^{v-1}+2s^{2v-1}/s^{2v}-4s^{v}+13\}$ where $v$ is fraction . The answer that I need to get after applying invers laplace is $y=[E(2t^v)][\cos(3t^v)-5\sin(3t^v)]$ ...
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Inverse Laplace transform for fractional power

enter image description here My question in the picture ; because im new i found that difficult to write the question here .can any one help me please -
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1answer
30 views

Inverse laplace with fractional order

enter image description here $$ L^{-1}\left[ \frac{s^{\alpha}}{s^2\alpha+a^2}\right]\\ L^{-1}\left[ \frac{a}{s^2\alpha+a^2}\right] $$ Can any one help me with those where alpha is fraction/rational ...
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Fractional Derivative and the Fourier Transform

I've recently came across the notion of a fractional derivative of a function $f$ that is defined as $$\big(D^{\frac{1}{2}} f\big)(x)= \frac{1}{\Gamma(\frac{1}{2})}\int_0^x (x-t)^{-\frac{1}{2}}f(t) ...
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$f(x)=x^2(2-x)^2$ How to calculate $\frac{d^{1.8}f(x)}{d_{+}x^{1.8}}$

In the book , I found and So for $f(x)=x^2(2-x)^2$ Ithink \begin{align*} \frac{d^{1.8}f(x)}{d_{+}x^{1.8}}&=(2-x)^2\sideset{_0}{_x^{1.8}}{\mathop{D}}x^2+\binom{1.8}{1}*(-2)*(2-x)\sideset{_0}{_x^...
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Laplace Transform of Riesz Fractional Derivative

I wanted to calculate the Laplace Transform of Riesz Fractional Derivative. But got some troubles in the middle. The Riesz Fractional Derivative is given by: $R^{\alpha}f(t)=-\frac{1}{2\pi} \int_{-\...
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Derivative of the fractional Brownian motion

From Jost (2008), we can see $$W_t^{H} = C_H \left\{ \int_{-\infty}^t \frac{dW_s}{(t-s)^{1/2-H}} - \int_{-\infty}^0 \frac{dW_s}{(-s)^{1/2-H}}\right\},$$ with $C_H = \sqrt{\frac{2H\times \Gamma(3/2- ...
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Eigenvectors of half-derivative operator

The eigenvectors of the derivative operator $D$ are $\exp(kx)$, as talked about in this post and computed with Wolfram Alpha here. The eigenvectors of $D^2 = D \circ D$ with eigenvalue -1 are linear ...
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81 views

Singularity of Riemann-Liouville Fractional Derivative

I have recently started studying fractional order calculus and it seems that the definition of the fractional order differentiation, in the Riemann-Liouville (RL) sense, has singularities. To explain ...
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2answers
102 views

Inverse Laplace transform of $\frac1{s^{1/2}(s^2 + 1)}$

I was studying about Caputo Fractional Derivative for a scientific project and I was trying determine the 1/2 order derivative in Caputo-Sense of $\sin(\omega t)$. Throughout the development of the ...
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Varying a fractioanl derivative order with respect to a function.

I wanted to find out what g(x) would be if $g(x)= \frac {d^{sin(x)}}{dx^{sin(x)}} cos(x)$, or using the differintegral operator $g(x)= \Bbb {D}^{sin(x)}cos(x)$. After some research on the internet I ...
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How to differentiate to first order by minus the order of a fractional differintegral?

How can we differentiate the Riemann-Liouville fractional differintegral $\Large\mathrm{D}_x^{-s}\LARGE(\Large{\frac{e^{2\pi ix}}{1-e^{2\pi ix}}}\LARGE)$ by minus the order to which it is fractionally ...
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The R-L fractional derivative interpretation

The Riemann-Liouville fractional derivative of order $\alpha$ of a continuous function $f:(0,\infty)\rightarrow \mathbb R$ is defined as: $$D^{\alpha}=\frac{1}{\Gamma(n-\alpha)}(\frac{\partial }{\...
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Time-fractional Navier-Stokes Equation

In the paper Analytical solution of a time-fractional Navier–Stokes equation by Adomian decomposition method by Shaher Momani and Zaid Odibat ($2006$), the Navier-Stokes equation was written in time-...
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71 views

Product of Mittag-Leffler functions $E_\alpha(x) \cdot E_\alpha(y)$

For the exponential function, the following relation holds: \begin{equation} e^x\cdot e^y = e^{x+y} \end{equation} The Mittag-Leffler function \begin{equation} E_\alpha(x)=\sum_{k=0}^\infty \frac{x^...
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Add Mittag-Leffler functions with same parameters

I have two functions in the form \begin{equation} f_1(t) = a_1 + b_1 E_\nu \left( c_1 \frac{t^{\nu}}{\tau^{\nu}} \right)\\ f_2(t) = a_2 + b_2 E_\nu \left( c_2 \frac{t^{\nu}}{\tau^{\nu}} \right) \end{...
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Theory for general fractional differintegral equations?

I am aware there exist ways to construct fractional calculus, fractional differential operators and integral operators, for example by using Cauchy integral theorem in complex analysis or by Fourier ...
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36 views

What's Fractional Partial Differential Equation and its application.

I read about fractional partial differential equation on wiki and I now know its an expansion of the usual integral power partial differential equation. I also read that it had some applications in ...
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On the origin of fractional Fokker Planck equation

I have recently encountered fractional Fokker Planck equation for the first time in literature to explain subdiffusive behaviour. However, I find it extremely uncomfortable since the classical ...
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How do discrete factional order functions look like?

A fractional order transfer function looks like this $$ \frac{Y}{U}=\frac{b_ns^{m\alpha}+\dots+b_1s^{\alpha}+b_0}{a_ns^{n\alpha}+\dots+a_1s^{\alpha}+a_0} $$ where $\alpha \in (0,1)$ and it is often ...
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Compute the solution of fractional differential equations?

Recently, there is the definition of the following the generalized fractional differential operator (GFDEO): \begin{equation} {}^{C}_{0}D^{\gamma,\lambda(t)}_{t}u(t) = \frac{1}{\Gamma(1 -\gamma)}\int^{...
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Fractional calculus reference for hypercyclicity of fractional derivative.

Two years ago in graduate school I came across an article that gave a gentle introduction to the fractional derivative $D^\alpha$ and derived some properties of the semigroup $(D^\alpha)_{\alpha\geq 0}...
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Mittag-Leffler function in matlab [closed]

I would like to write "Mittag-Leffler " function in MATLAB: Mittag-Leffler function: $$ E_{\mu}(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{\Gamma(n+\mu)} $$ where $\mu \in \mathbf{R}_{+}$. Thanks
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Is the following infinite series containing fractional derivative convergent?

Is the following series convergent? $\sum_{k=1}^{\infty} \frac{1}{\alpha^k k!} [(𝑻^{t_0}_\alpha ) 𝒇^{(𝒌)}(t) ]_{t=t_0} (t-t_0)^{k \alpha}$ where $(𝑻^{t_0}_\alpha ) 𝒇^{(𝒌)}(t)$ denotes the ...
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what is the conformable fractional differential transform of a constant.

You know, in 2014 a new fractional derivative is introduced which is the conformable fractional derivative. Conformable fractional derivative of $f: [a,∞)→ \mathbb{R}$ is defined as: $(T_\alpha f)(t)=...
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Fractional reaction diffusion equation

Consider the linear fractional reaction diffusion equation: $\begin{align} \frac{\partial u}{\partial t} + \alpha(-\Delta)^{s} u = 0, \\ \label{pr: fractionallineal2} u(x,0)=u_{0}(x) \end{align} ...
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1answer
64 views

complex integral / fractional derivative verification

Cauchy differentiation formula says $$f^{(n)}(z_0)=\frac{n!}{2\pi i}\int_{\gamma} \frac{f(z)}{(z-z_0)^{n+1}}dz $$ Would changing the $n$ by $\alpha\in \mathbb{R}$ and factorial by $\Gamma$ make it a ...
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Fractional derivatives of the power function between a=0 and a=-1

It's a well know fact that $$d(x^a)=ax^{a-1}dx\; \text{ for }\;a\in \Bbb{R}$$ It's a less well-known, but easily provable, fact that this generalizes to $$d^n(x^a)=\frac{a!}{(a-n)!}x^{a-n}\; \text{ ...
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Does taking a fractional derivative remove a fractional amount of Holder regularity?

We define the space $C^{n+\alpha}$ as functions who are $n$ times differentiable whose $n$th derivative is $\alpha$ Holder. That is, each time we take a derivative we remove one number of regularity. ...
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Existence of a function satisfying zero boundary conditions for fractional Laplacian (1d)

Does there exists a non-zero function $$f\in C_0([0,1]):=\{f:[0,1]\to \mathbb R:\ f\text{ is continuous and } f(0)=f(1)=0\},$$ such that $(-\Delta)^{\frac\alpha 2}f\in C_0([0,1]) $, where $(-\Delta)^{\...
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Series Derived from the Fractional Derivative of Geometric Series

With the geometric series:$$\sum_{n=0}^\infty r^n = \frac{1}{1-r} $$ It's normal to derive other formulas of series taking the derivative or integral of both sides of the formula. I wanted to see if ...
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Fractional derivative and Leibniz rule

I am trying to digest an old paper by Kermack & McCrea (see https://www.cambridge.org/core/journals/proceedings-of-the-edinburgh-mathematical-society/article/on-professor-whittakers-solution-of-...
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Can we derive the analytical expression for this discrete fractional differential operator?

Experimenting about on this question, I some days ago investigated the polynomial equation: $$p(x) = x^6 - d =0, p'(x) = 6x^5$$ I started with ${\bf X}_0{\bf = I}$ Then the iteration $$\begin{...
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1answer
89 views

Hadamard fractional integral of log function.

I do not know how to prove the following proposition: $$\frac{1}{\Gamma (\alpha)} \int_1^t \log(\frac{t}{x})^{\alpha -1} (\log (x)) ^{\beta-1} \frac{dx}{x} = \frac{\Gamma(\beta)}{\Gamma (\beta + \...
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1answer
132 views

In what function space is the fractional Laplacian $(-\Delta)^su$ essentially bounded?

Can it be said that if $u(x)=O(|x|^\beta)$, with $\beta\geq N+2s$, for $x\in\Omega\subset\mathbb{R}^N$ a bounded domain, then $(-\Delta)^su\in L^{\infty}(\Omega)$?. Is there a generalized result ...
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1answer
108 views

Is the order of a derivative not additive under the Riemann-Liouville definition?

I was doing some mathematics doodling today and I wrote down $$\frac{d^{n}}{dx^{n}}x^n = n!.$$ Looking at this made me wonder if when $n = 1/2$, if $D^{1/2}_0\sqrt{x} = \Gamma(1/2)$. Where $D^{1/2}_{...
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9 views

Construction of an integrablly bounded and unfirormally Lipchitz continuos function with respect spatial coordinate in a Banach space.

I want to Construct a function $f(t,x(t),I_{x}(t),J_{x}(t))$ on $t\in I=[0,T_{0}],$ where, $$I_x(t)=\int_{0}^{t}a(t,s,x(s))ds,$$ $$J_x(t)=\int_{0}^{T_{0}}b(t,s,x(s))ds$$ under the restriction that (...
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35 views

Reference to “famous Tchirnaus transformation”

On page 7 of André Galligo's paper Deformation of Roots of Polynomials via Fractional Derivatives, we find the following statement: Lemma 1. Let $f(x)$ be a polynomial of degree $n$, then $P_f (x, ...
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46 views

What is the eigenvector of the functional square root of the differentation operator?

The eigenvector of the differentation operator is the exponential function. But I wonder, what are the eigenvectors of operators in fractional calculus? For example, the "functional square root" of ...
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89 views

inverse Laplace transform of rational function of Euler gamma function

How do I calculate the inverse Laplace transform of the gamma function? I'd appreciate it if someone can help me. $$\sum_{n=0}^\infty(-1)^{a s^{\alpha}}\frac{Γ(b+a s^{\alpha})}{Γ(b+a s^{\alpha}-n)}$$ ...