Questions tagged [fractional-calculus]

Questions on the differentiation/integration of functions to fractional 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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What is the domain of Caputo fractional derivative? Is zero included or excluded?

Caputo fractional derivative is defined as $$^CD^\alpha f(x)=\frac{1}{\Gamma(n-\alpha)}\int_0^x\frac{f^{(n)}(t)}{(x-t)^{\alpha-n+1}}dt$$ Where $n$ is the ceiling of $\alpha$. I found some papers ...
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Discretizing Caputo Derivative

Have a look at the following article (Eqn 7), it is given an approximation that $$ \begin{aligned} & \left.\frac{1}{\Gamma(1-\alpha)} \int_0^\tau \frac{\partial u(x, \eta)}{\partial \eta}(\tau-\...
Riaz's user avatar
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Solving backwards fractional differential equation with transversality condition

Let's take $\alpha\in(0,1)$ and $f,g\colon[0,T]\rightarrow \mathbb{R}$ good enough functions. When using the chain rule in fractional calculus in the context of fractional optimal control $$ \int_0^T ...
Aner's user avatar
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A doubt in the Sobolev space.

Sobolev space $H^{2,p}(\mathbb{R}^n)=\left\{u\in \mathcal{S}': \mathcal{F}^{-1}((1+|\xi|^2)\widehat{u})\in L^p(\mathbb{R}^n)\right\}$ (Wong definition in introduction pseudo differential operators) If ...
eraldcoil's user avatar
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Product rule fractional laplacian

It is well known the product rule for the Laplacian: $$\Delta(uv) = u\Delta v +2\nabla u\cdot\nabla v + v\Delta u.$$ I was wondering if there is any similar product rule for the fractional laplacian, ...
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Extenstion of Caputo's fractional derivative to distribution.

Let us start with the definitions that $\frac{d}{dx}\theta(x)=\delta(x)$ where $\theta(x)$ and $\delta(x)$ are Heaviside and delta functions. Now, with the definition: $^c D^\alpha f(t)=\frac{1}{\...
user824530's user avatar
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ROC of a fractional order system in respective to its poles

My understanding about fractional order system is that it can have poles on the right hand side of imaginary axis in s-plane and yet being stable. (statement 1) There are other theorems about ...
Anders's user avatar
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Existence of half derivative of 1

I'm trying to find general properties for a half derivative operator, yet I encountered what seems to be a paradox. Let $H$ be a linear operator such that $H^2f = Df$ where $D$ is the first derivative....
ioveri's user avatar
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If $s\in (1/2, 1)$ and $[f]_s\le K$, does $||f||_{L^2}$ happen to be bounded by a constant depending on $K$?

Let $s\in (1/2, 1)$ and $f\in H^s(\mathbb R^n)$. By definition, $f\in L^2(\mathbb R^n)$ and $[f]_s<+\infty$, where $[\cdot]_s$ denotes the Gagliardo seminorm. Assume that there is a constant $K>...
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Proof that $\frac{d^a}{dx^a}\sin(x) = \sin(x+\pi a/4)$ iff these two infinite series are equivalent?

I am interested in analytically continuing the differentiation operation on functions, and I am currently focusing on trigonometric functions. Any high schooler can conjecture that $\frac{d^a}{dx^a}\...
Alexandra's user avatar
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Continuous embedding of the fractional space $H^{\frac12}(\mathbb R)$ into $L^q(\mathbb R)$

My question is about the Theorem 6.5 in the Hitchhiker's guide to the fractional Sobolev spaces. Consider the case $n=1$ and $p=2$. Theorem 6.5 states that if $s<\frac12$, then $H^s(\mathbb R)$ is ...
Physics user's user avatar
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stability region of fractional order system

I have a question about ellipsoid stability region of fractional order system with input saturation. In an integer system with input saturation we have the ellipsoid De as $ D_e=\left\{x(t)\in {\...
zahra's user avatar
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Closed form for $\psi^{1/k}(1)$, where $k$ is an integer

I have proven the identity $$ \sum_{k=1}^{\infty} \dfrac{\operatorname{_2F_1}(1, 2, 2-1/t,-1/k)}{{k}^{2}} = Γ(2-\dfrac{1}t){\psi^{1/t}(1)}+\psi(-\dfrac{1}t)(\dfrac{1}t(1-\dfrac{1}t))+\gamma(1-\dfrac{1}...
Aiden McDonald's user avatar
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Fractals by Fractional Integration

Since the nth order integral finds the volume enclosed by an n dimensional function, does that imply that fractional order integrals can be used to find the volume enclosed by rational (or irrational) ...
Anirudh Yamunan Govindarajan's user avatar
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Matrix-order derivatives (differentiating a function a matrix number of times)

I have been exploring methods of generalizing the order of derivatives to a broader range of inputs (such as real numbers, complex, and now matrices). We are very well familiar with integer-order ...
charlesalexanderlee's user avatar
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Any good book recommendations on discrete fractional calculus?

I am a math undergraduate student. Im interested in discrete fractional calculus topic, specifically about discrete fractional systems. Does anyone knows some book recommendations on the topic? P.S. I ...
Mohammad Fahrurrozy's user avatar
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Conditions on taking supremum over $t$ for $|f(t)|$?

Let $f: [0,T]\times X^{n} \to X$ is a continuous function given by $f \big(t, \mu_{1}(t), \mu_{2}(t), \dots,\mu_{n}(t) \big)$, $t\in [0,T]$. Here $X$ is a Banach space, and $\mu_{i}\in C([0,T],X),~ \...
Pratima Tiwari's user avatar
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Reference request: fractional heat equation $\partial_t u=(-\Delta)^s u$ with $s>1$

I was wondering if there are any good reference on the fractional heat equation $\partial_t u=(-\Delta)^s u$ with the fractional exponent $s>1$. I have found many references in the case $0<s<...
DDG's user avatar
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Fractional derivative according to Fourier

I am trying to show that $\frac{d^\alpha}{dx^\alpha}e^{ikx}=(ik)^\alpha e^{ikx}$ when using the following equations for fourier transforms: \begin{align} g(k)&=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{...
João Santos's user avatar
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How to obtain the standard Laplacian as a particular case of the Fractional Laplacian?

The definition of Fractional Laplacian is I am trying to understand how to obtain the classical case as a particular case. If $s=1$, there is a pole in the Gamma function. How can I prove the ...
Quiet_waters's user avatar
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Pre-compactness of $H^s(\mathbb R)$ when $s\in (\frac12, 1)$

Let $s\in (0, 1)$. By Corollary 7.2 in the paper https://arxiv.org/pdf/1104.4345.pdf, it is know that, if $s<1/2$, then $H^s(\mathbb R)$ is pre-compact in $L^p_{loc}(\mathbb R)$ for any $p\in [1, 2^...
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The space $H^s(\mathbb R)$ and the embedding in $L^p_{loc}(\mathbb R)$: for which $p$?

Let $s\in (0, 1)$ and consider the fractionl Sobolev space $H^s(\mathbb R)$ (see e.g. https://www.sciencedirect.com/science/article/pii/S0007449711001254). Let $(A, ||\cdot||)$ be a Banach space and ...
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Is $H^s(\mathbb R^n)$ continuously embedded in $L^2(\mathbb R^n)$ when $2s>n$?

Let $1/2 < s<1$. I got a question about the fractional Sobolev space $H^s(\mathbb R^n)$. It is well known that, if $2s>n$, then $H^s(\mathbb R^n)$ is continuously embedded in $L^\infty(\...
Physics user's user avatar
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Solving Differential Equations using Fractional Calculus

Are there any integer order differential equations or integral equations that can be solved using fractional calculus? For example, the Abel's Integral (like from the Tautochrone Problem): $$\int_{0}^{...
Anirudh Yamunan Govindarajan's user avatar
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Substitutions for integrals involving branch cuts

I have the definition of the Liouville fractional derivative \begin{equation} \frac{d^{-\delta}}{d x^{-\delta}} f(x) = \int_{0}^x f(t)(t-x)^{\delta - 1} dt \end{equation} where $\delta > 0$ and for ...
QFTheorist's user avatar
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A possible upper bound for a function that satisfies a singular integral inequality

I am currently working on an analysis problem in fractional calculus and after some work I have encountered the following inequality: $$ |v(s)|~\leq \epsilon+\beta \int_{0}^{1}|s-x|^{\alpha-1}\left( |...
Taki Zeg's user avatar
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1 answer
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Discrete Caputo fractional derivation

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f \in C_{1}(\mathbb{R})$ The Caputo fractional derivative is written as: \begin{equation} {C}{}{D}^{\alpha}_{a} f(x) = \frac{1}{\Gamma(1-\alpha)} \...
ayman lakehal's user avatar
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Fractional Fourier Transform by Fractional Integral

Is there any way to prove the equation of fractional Fourier transform of a function (the integral of the fractional kernel function), given that the nth Fourier transform is just the composition of ...
Anirudh Yamunan Govindarajan's user avatar
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Half Derivatives in Maxima

Is it possible to take the half derivative of a function in maxima? I tried the following: ...
Anirudh Yamunan Govindarajan's user avatar
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What do I need to know for deriving the limits of sequences of discrete convolutions?

This is for a recreational math project which I hope to turn into a video. If you start with a sequence $f_n$, finding the difference of that sequence $f_n-f_{n-1}$ is analogous to taking the ...
EmmaBellHelium's user avatar
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Does Green's formula for the ball contain the Poisson's kernel?

I am a bit confused with the definition of Green function (see Definition 1.9) here: https://arxiv.org/pdf/1502.06468.pdf Forget for a moment that the problem is for the fractional case. Even for the ...
Physics user's user avatar
3 votes
2 answers
77 views

Does $u$ solve $(-\Delta)^s u =0$ in $(-r, r)$ and $u=g$ in $\mathbb R\setminus (-r, r)$?

Let $g\in C(\mathbb R, \mathbb R)$ and let $r>0$. Consider the function $$u(x)=\begin{cases} 0 &\mbox{ in } (-r, r)\\ g &\mbox{ in } \mathbb R\setminus (-r, r) \end{cases}$$ Let $s\in (0, 1)...
Physics user's user avatar
4 votes
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What is the relationship between Hölder spaces and differentiability?

Let $C^{k,\alpha}$ be a Hölder space where $0 \leq \alpha \leq 1$. I have seen various sources informally describe these spaces as functions having at least "$k + \alpha$" derivatives. How ...
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Fractional differintegral for the reconstruction of time series

How do economists and climatologists reconstruct their time series from the available data? I know in particular that economists have built models to extimate the GDP per capita between the year 0 and ...
Beppe's user avatar
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Convolution and fractional operator

Let $h\in C_c^\infty(\mathbb R^n)$ and consider the equation $$(-\Delta)^s u = h,$$ where $(-\Delta)^s u$ denotes the fractional Laplacian of $u$. Let $\Gamma$ be its fundamental solution, i.e. $(-\...
Physics user's user avatar
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How to prove that $\frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\in L^1(\mathbb R^{2n})$?

Let $\Omega$ be an open bounded domain of $\mathbb R^n$. Let $s\in(0, 1)$, $p\in (1, \infty)$ and consider the Banach space $$X^{s, p}(\Omega)=\{u\in W^{s, p}(\mathbb R^n): u=0 \text{ in } \mathbb R^n\...
Physics user's user avatar
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Functional Application of the Differential Operator: Can the Order of Differentiation be a Function?

I've been contemplating the traditional differential operator ( D ) used in calculus, and I'm interested in a potentially broader application. Instead of having a fixed real or fractional order for ...
Wade Hunter's user avatar
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1 answer
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If $u\in D^{s, p}(\mathbb R^n)$, is that true that $u\in C(\mathbb R^n)$?

Let $0<s<1$, $p>1$ and $n>sp$. For $u\in C_0^{\infty}(\mathbb R^n))$ let $$ [u]_{s, p} =\left(\iint_{\mathbb R^n} \frac{|u(x)-u(y)|^p}{|x-y|^{n+sp}}\right)^{\frac1p}$$ be the ussual ...
C. Bishop's user avatar
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If $\widetilde{u}(x) = u(-x)$, what is $[\widetilde u]^2_s$?

Let $s\in (0, 1)$ and $u:\mathbb R^n\to \mathbb R$ be a measurable function. The Gagliardo seminorm of $u$ is defined as $$[u]^2_s =\int_{\mathbb R^{2n}} \frac{|u(x)-u(y)|^2}{|x-y|^{n+2s}} dx dy.$$ I ...
C. Bishop's user avatar
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3 votes
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Show that $G_t(x)=(2\pi)^{-n/2}\int_{\mathbb{R}^n}\mathbb{e}^{ix\cdot\xi}\mathrm{e}^{-t(1+|\xi|^2)^{\alpha}}\,d\xi$ is uniformly bounded in $L^1$.

I am studying properties of the kernels of different operators and the following question arose. Given the operator $-(-\Delta)^{\alpha}u=\mathcal{F}^{-1}(|\xi|^{2\alpha}\widehat{u}(\xi))$, this ...
eraldcoil's user avatar
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Understanding some calculations about the kernel of the fractional laplace operator

Hello. I am trying to understand an estimate on the kernel of the fractional laplace operator in the following article "Well-posedness of the Cauchy problem for the fractional power" by Miao,...
eraldcoil's user avatar
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2 votes
1 answer
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Are there a solutions for $\frac{\mathrm{d}^\alpha f}{\mathrm{d}x^\alpha} = xe^x$?

Are there solutions for the fractional differential equation $$\frac{\mathrm{d}^\alpha f}{\mathrm{d}x^\alpha} = xe^x$$ for $0 < \alpha < 1$? I have been using the Caputo definition $$ \frac{\...
655321's user avatar
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Equivalence norm in $W^{s, p}(\mathbb{R}^n)$ and Gagliardo seminorm $[u]_{s, p}$

I am reading a paper and the in the first pages it is written what follows. Let $s\in (0, 1), p>1$, $n> sp$ and $\Omega$ be a bounded domain in $\mathbb{R}^N$ with Lipschitz boundary. Let $$ [u]...
C. Bishop's user avatar
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2 votes
1 answer
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Defining a fractional derivative operator over Laplace transforms

$$\Large{ \text{Introduction} }$$ I'm interested in solving methods for fractional-differential-equations (FDEs) without specifying the nature of the fractional derivatives. In doing so, I keep coming ...
Kevin Dietrich's user avatar
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Does a closed form for this integral exist?

As part of a broader fractional integral calculation using the Riemann-Liouville operator, I encountered the following integral: $$f(z,h,a) = \int_0^z\, \frac{h^{1-u}}{u-1}\cdot(z-u)^{a-1}\, du \qquad ...
Agno's user avatar
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1 answer
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Differentiation of an improper parametric integral

In a textbook on fractional calculus, the following equality is stated without any comments or a proof: $$ \frac{d}{dt} \left( \int\limits_{-\infty}^{t} (t-s)^{-\alpha} f(s) ds \right) = \alpha \int\...
AnonymousUser's user avatar
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0 answers
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Central difference approach on Riemann-Liouville fractional derivative

Take a look at the simple fractional differential equation (FDE Initial Value Problem): $$_0D_x^{\frac{1}{2}}y(x)+2y(x)=0,~~x>0,~\text{and}~_0D_x^{-\frac{1}{2}}y(0)=1.$$ I solved the IVP using ...
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Proving the existence of a solution of the fractional heat equation using semigroup methods

I am trying to solve the following problem: $$u_t + (-\Delta)^su = 0$$ in $\Omega \subset \mathbb{R}^N$ with $N > 2s$, where $s \in (0,1)$ and Dirichlet Boundary conditions. Let my operator $A = (-\...
José's user avatar
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Using Semigroup Theory of Linear Operators to show that the operator $(-\Delta)^s$ is closed.

Consider the fractional Laplacian defined by $$(-\Delta)^s u(x) = P.V. \int_{\mathbb{R}^N} \frac{u(x) - u(y)}{|x - y|^{N + 2s}}dy, \ s \in (0,1).$$ Also consider that $$D((-\Delta)^s) = \{u \in H^s(\...
José's user avatar
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Show that the fractional Laplacian operator is closed

Consider the fractional Laplacian defined by $$(-\Delta)^s u(x) = P.V. \int_{\mathbb{R}^N} \frac{u(x) - u(y)}{|x - y|^{N + 2s}}dy, \ s \in (0,1).$$ Also consider that $$D((-\Delta)^s) = \{u \in H^s(\...
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