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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Justification for a substituion that turns a finite sum to infinite - constructing the Grunwald-Letnikov fractional derivative (Fractional Calculus)
Steps in question
These steps raise numerous questions.
What is the reasoning behind choosing $\delta _Nx\equiv [x-a]/N$ ?
This seems almost arbitrary. I understand that $a$ and $x$ eventually ...
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fractional integral with integral measure on a fractional power
I got a bit familiar with fractional calculus, and it seems that the Riemann-Liouville differintegral formula (and all other similar formulas) are defined for a function $f$ that is one dimensional, ...
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Integrating the fractional Laplacian over $\Omega$
Let $\Omega$ be a bounded regular open subset of $\mathbb{R}^N$, $N\geq 1$.
I want to know if this statement is true : for $u\geq 0$ in $\Omega$, and $u=0$ in $\mathbb{R}^N\setminus \Omega$,
$$
\int_\...
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The Grünwald–Letnikov fractional derivative of the function $x^{-\frac{1}{2}}$
Let $f(x) = \frac{1}{\sqrt{x}},$ then how can we find the Grünwald–Letnikov fractional derivative for this function. By definition,
$$\operatorname{D}_{0}^{p}f(x) = \lim_{h \to 0,\, n \cdot h = x - a}\...
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Fractional Bernoulli equation and logistic function
I'm investigating the solution of the special case of the Bernoulli differential equation
$$
\dfrac{dy}{dt} = \dfrac{y(1-y)}{\tau}, \tag{1}
$$
with $\tau$ a time constant, and which models innovation ...
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Riemann–Liouville fractional integral from measure theoretic point of view
Does there exist a definition of the Riemann–Liouville fractional integral with respect to any general measure? Could there be perhaps a relation between the Riemann–Liouville fractional integral and ...
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Can we define the fractional derivative by mapping a function to a sinusoidal?
For integer $n$ we have that:
$$\frac{d^n}{dx^n} \sin(x) = \sin\left(x+\frac{n \pi}{2}\right)$$
For any function $f(x)$ (ignoring domain restrictions for the time being), let:
$$f(x) = \sin(u) \ \ [1]$...
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Solving non-linear fractional differential equation
I want to solve the non-linear Caputo-type fractional equation of the form ($0 < \alpha < 1$)
$$ ^cD^{\alpha}_0 f(t) = af(t)^4 + bf(t) + c$$
I have found, the equation $^cD^{\alpha}_0 f(t) = af(...
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Relationship between fractional Poisson kernel for a region and its complement
Given a region $\Omega \in \mathbb{R}^d$, consider the Poisson problem for the fractional Laplacian $(-\Delta)^{2s}$ with $s>0$. Given a function $g: \Omega^C \to \mathbb{R}$, what is the function $...
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Some examples of Fractional Laplacian on $\mathbb{R}^{2}$ : $(-\Delta)^{\alpha/2}u(x,y)$
It is known that the fractional Laplacian of $u$ on $\mathbb{R}^2$ is defined as
\begin{align}\label{fraclap2dim}
(-\Delta)^{\alpha/2}u(x,y)=-c_{\alpha}\int \int_{\mathbb{R}^2}\dfrac{(p,q)\cdot \...
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Solving a fractional differential equation
I am working with a fractional Laplacian that is based on it's Fourier transform, namely
$$(-\Box)^{\alpha}f(t) := \int_{-\infty}^\infty d\omega e^{i\omega t} |\omega|^\alpha \int_{-\infty}^\infty d\...
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Find $a,b>0$ for which $||\langle x\rangle^{-b}|\partial_x|^{1/2}f||_{L^2}\lesssim||\partial_x f||_{L^2}+||\langle x\rangle^{-a}f||_{L^2}$
Consider the following problem.
Problem. Given $\alpha>0$, find all values of $\beta\geq 0$ such that the following estimate is true for all $\varphi\in \mathscr D(\mathbb R)$:
$$ ||\langle x\...
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A really really weird new(to my knowledge) kind of differential equation.
the equation $\dfrac {d^xf}{(dx)^x)} = f(x)$ where $ \dfrac{d^xf}{(dx)^x}$ means we are taking the xth derivative of f(x)(using fractional calculus, assuming the Riemann–Liouville fractional ...
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Reference request about the parabolic Sobolev spaces
I am looking for a reference where the following space is defined and studied :
$$
L^p\left(0, T ; \mathbb{W}_0^{s, p}(U)\right),
$$
where $T>0$, $p\geq 1$, $0<s<1$, $U$ is an open bounded ...
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Laplace transform of Riemann-Liouville fractional derivative vs Caputo fractional derivative
When comparing the differences between the Riemann-Liouville definition of the fractional derivative and the Caputo definition, one line from this paper confused me:
In his 1967 paper, Caputo ...
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Is it possible to write the exponential or logarithm of a derivative, in terms of a fractional derivative?
Consider a function of a derivative acting on a 'normal' function,
$f(\partial_t)g(t) \equiv \mathcal{F}^{-1}(f(i\omega)g(\omega); t)$
where certain properties are satisfied such that this converges ...
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Fractional integral related to stable process on half-space
I'm working with the isotropic $\alpha$-stable Lévy process in $\mathbb{R}^d$ $(\alpha \in (0,1) \text{ and } d \geq 2)$. I know that the distribution of the first hitting of this process into the ...
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On a solution of a fractional integral equation
I am looking for the solution of $$\frac{d^\alpha}{d x^\alpha}f(x)=g(x)f(x),$$ where $\alpha \in (0,1)$ and $\frac{d^\alpha}{d x^\alpha}$ is the Caputo derivative.
A series of Jumarie's papers, "...
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Caputo vs Riemann-Liouville fractional calculus computation
While stepping into the realm of fractional calculus, I have become confident on the RL fractional integral, defined as:
$$^{RL}_aI^p_tf(x) = _p\int^t_af(x)dx^p = \frac{1}{\Gamma(p)}\int^t_a(t-x)^{p-1}...
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Numerical method for space fractional derivative in 1 dimension
I am very new to the subject of fractional derivatives which arise while characterizing the anomalous transport of passive scalar in turbulence.
I have found an equation of the following form, to ...
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Leibniz type rule for the fourier based fractional derivative
Given $f(t) \in L^2(\mathbb{R})$, $\alpha \in \mathbb{R}$ and the definition
\begin{align}
D^\alpha f(t) = \mathcal{F}^{-1}\left( (i\omega)^\alpha \mathcal{F}\left(f(t);\omega\right);t\right)
\end{...
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Line element, metric tensor, integral and the sphere in fractional dimensions
I have a question regarding fractional calculus, namely, what is the line element "$$ds^2 = g^{\mu\nu} dx_\mu dx_\nu$$ in fractional "n" dimensions?
I am aware of some formulas that ...
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Stuck on proving fractional integral is continuous
I haven't proved continuity in ages, but I found a situation where it would be helpful.
I'd like to prove the Riemann-Liouville differintegral https://en.wikipedia.org/wiki/Riemann%E2%80%...
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Prove fractional integral at $\alpha = 0$ converges to the original function
https://en.wikipedia.org/wiki/Riemann%E2%80%93Liouville_integral
Under the properties section, it says $I^{\alpha}f \rightarrow f$ as $\alpha \rightarrow 0.$ But it offers no proof, is there a proof ...
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Proving that the Grünwald–Letnikov differintegral is homomorphic: deriving single limit from repeated limit
I'm trying to prove that the Grünwald–Letnikov differintegral is homomorphic over addition i.e. it composes as expected:
$${}^{GL}_{x_0}\mathbb{D}^a \circ {}^{GL}_{x_0}\mathbb{D}^{b} = {}^{GL}_{x_0}\...
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Laplace transform of Liouville Weyl Derivative:
I am trying to calculate Laplace transform of fractional derivative with $a=-\infty$ defined as:
$$ D_{-\infty,t}^\alpha V(t)= \frac{1}{\Gamma(1-\alpha)} \int_{-\infty}^{t} \frac{V'(\tau)}{(t-\tau)^\...
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How to understand the notion of a weak fractional derivative in a fractional Sobolev space
recently I have been studying the article named "Hitchhiker’s guide
to the fractional Sobolev spaces" (I leave the external link here: https://arxiv.org/pdf/1104.4345.pdf) and as the ...
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How to calculate this fractional integral?
Let $\alpha \in (0,1)$ and $y=(y_1,y_2), y_1>1 \text{ and } y_2 \in \mathbb{R}$. How can I calculate the following integral?
$$\int_{-\infty}^{\infty}\int_{-\infty}^{-1}(-1-u_1)^{-\alpha/2}|u|^{-2}|...
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Building Fractional Frequencies From Integer Frequencies
Given the ability to build any trigonometric polynomial of integer degrees:
$$T_d(\theta) = \sum_{n=-d}^d c_n e^{in\theta}$$
I wish to construct, or technically approximate:
$$e^{it\theta} \>\>\&...
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How to prove the existence and uniqueness of pde with Caputo fractional derivative in time
For any $\beta>0$ we denote the Riemann-Liouville fractional integral operator of order $\beta$ by
$$ I^{\beta}(f)(t)=\frac{1}{\Gamma(\beta)} \int_{0}^{t}(t-\tau)^{\beta-1} f(\tau) d \tau, \quad f \...
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How do I solve $\int \frac{20}{(x-1)(x^2+9)}dx$
I've been trying to solve the following integral:
$\int \frac{20}{(x-1)(x^2+9)}dx$
Sadly I'm kinda new to resolving fractional integrals and I'm not sure which method(s) I should use to solve it.
I've ...
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Fractional derivative calculator in Mathematical Softwares
Are there any fractional derivative calculators available in any Computer Algebra Systems or other Mathematical softwares?
I wish to find the value of $D^a\ln(1+e^x)$ for $x=0,a=0.9$. Is there a way ...
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A bound for Riemann- Liouville fractional
For two different function $u$ and $\tilde{u}$, we know that for every $\epsilon>0$
$$\Vert D^{\textbf{k}}u - D^{\textbf{k}}\tilde{u} \Vert _{L^{\infty}(\Omega)} < \epsilon ,$$
where $\textbf{...
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Identities for fractional laplacian.
We know that Green's third identity, is the following with $f$ a $C^2$ function on $D\subset\mathbb{R}^n$
\begin{equation}-4\pi f(r_0) = \int_D\frac{1}{|r-r_0|}\nabla^2f\,dV+\int_{\partial D}\left(\...
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Integration by parts for Bessel's operator.
I am having the following integral:
$$I = \int u\, J^s(\partial_x \overline{u})- \overline{u}\, J^s(\partial_x u))dxdy$$
where $J^S= (I-\Delta)^\frac{2}{2}$, $\mathbb{R} \ni s \geq 1$ and $u=u(x,y)$,
$...
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How to derive a recursive formula from the following formula
How to derive a recursive formula from the following formula,
$$
u_{n}=a_{n-1}u_{0}+\sum_{k=1}^{n-1}(a_{n-1-k}-a_{n-k})u_{k}+\Gamma(2-\alpha)h^{\alpha}f(t_{n},u_{n})?
$$
P.S.:
Consider the following ...
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140
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Fractional powers of the Laplacian
I have recently started to study fractional fractional powers of the Laplacian. In the books I've read, fractional powers are defined only for
$$-\Delta =-\sum_{j=1}^n\frac{\partial^2}{\partial x_j^2}....
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$\int_0^t (t-r)^{1/3} r^{5/3} dr$
The integral $$\int_0^t (t-r)^{1/3} r^{5/3} dr$$ came up in Miller's Introduction to Differential Equations (1987), (Sec 6.6, Convolutions, in the chapter on Laplace transformations, problem 5a on p. ...
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Are all fractional derivative definitions wrong except Grunwald Letnikov definition?
I'm trying to find out if there is a logical way to see which fractional definition is correct, so my attempt was as follows:
Suppose that we have the fact
$$\int_{0}^{\infty^{*}} \frac{\cos(ax)}{1-x^...
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56
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The Grünwald–Letnikov fractional derivative for the function $f(x)=e^{ax}$
The Grünwald–Letnikov fractional derivative for the function $f(x)=e^{ax}$ is known as follows,
$\displaystyle D_x^n f(x)=\lim_{h\to 0}h^{-n}\sum_{m=0}^n(-1)^m{}_nC_m f(x+(n-m)h)$ where $_nC_m=\frac{n!...
- 184
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Solving fractional differential equations using B-Spline Collocation
I am working on implementing the method shown in this paper to solve a particular fractional differential equation using the method of collocation (where the basis function used is fractional b-spline)...
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1
answer
74
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Cauchy principal in the definition of fractional laplacian
Let $\alpha \in \mathbb{R}$，$\ 0<\alpha <2 \ $, the fractional laplacian is defined as:
\begin{equation}\label{def}
(-\Delta)^{\frac{\alpha}{2}} u (x) := C_{n,\alpha} \lim\limits_{\epsilon \...
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0
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27
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Double sum in terms of generalized Mittag-Leffler functions
I have the series
$$S = \sum_{n=0}^\infty \sum_{m=0}^\infty \frac{x^n y^m}{\Gamma(1+\alpha n + \beta m)}\frac{(n+m)!}{n!m!}$$
which originates from a fractional calculus problem.
One can see that $S$ ...
- 1,184
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22
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Inverse Laplace transform of $ \frac{1}{s}\frac{1}{1 + (k/s)^a + (l/s)^b}$
I want to express the Inverse Laplace transform (arising from a fractional calculus problem)
$$ F(t) = \mathcal{L}^{-1}\Big\{ \frac{1}{s}\frac{1}{1 + (k/s)^a + (l/s)^b}\Big\}(t) $$
in terms of ...
- 1,184
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92
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Exact solution of a BVP of second order
Im solving a BVP which is $y^{\prime\prime}(t)=-y^{\prime2}(t)+y(t)(y^{2}(t)-\frac{3}{2}y(t)+\frac{1}{2})$ with boundary conditions $y(0)=1$ and $y(1)=2$. I need to find the exact solution for this ...
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Is there an integral transform formula for $(-\nabla^2 + m^2)^{\frac{1}{2}}$ in three dimensions? What about its one sided inverse?
I have come across the following formula for the positive square root of the (negative) 3D Laplacian
$$(-\nabla^2)^{\frac{1}{2}}[u](y) = C \text{ p.v. }\int_{\mathbb{R}^3}\frac{u(y)-u(x)}{\|y-x\|^4}dx$...
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What is the exact meaning of the "a" in the lower limit of the Riemann–Liouville integral?
$I^{\alpha }f(x)=\frac{1}{\Gamma(\alpha ) }\displaystyle\int_{a}^{x}f(t)(x-t)^{\alpha -1}dt$
In this definition of the Riemann–Liouville integral, wikipedia says that "$a$ is an arbitrary but ...
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0
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59
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Normal vector to a level set and fractional Laplacian
Let $U=\{u\le0\}$ and $\partial U=\{u=0\}$. Suppose $\nabla u$ does not vanish on $\partial U$. Then the (canonical extension of the) normal vector field to $\partial U$ (pointing to the interior of $\...
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Questions about iterating the Euler-Maclaurin summation formula
Introduction
The Euler–Maclaurin summation formula is as follows for a positive integer $p$ and a continuous function $f(\cdot)$ that is $p$ times continuously differentiable on the interval $[m,n]$ : ...
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1
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62
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Existence and calculation of a fractional integral
Let $b >0$ be arbitrary but fixed and let $\alpha >0$ if $$g_{\alpha}(x)=\frac{1}{\Gamma(\alpha)}\frac{1}{x^{1-\alpha}} \chi_{]0,b]}(x)$$
If $f:[0,b] \rightarrow \mathbb{R}$ is continous in [0,b]...
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