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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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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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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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!...
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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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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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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 $S$ ...
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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 ...
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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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Nonlinear ordinary fractional differential equation

Solve the following nonlinear ordinary fractional differential equation: \begin{equation} -au^{2}\frac{d^{\alpha}u}{d\zeta^{\alpha}}+bu\frac{d^{\alpha}u}{d\zeta^{\alpha}} +c\frac{d^{2\alpha}u}{d\zeta^{...
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Does the Riemann-Liouville integral always exist for any Lebesgue integrable function $f$?

Let $ J= [a, b],\, (-\infty<a<b<+\infty)$ be a finite interval on the real axis $\mathbb{R}$. In many monographes, i find that the Riemann-Liouville fractional integral of the function $f \in ...
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how to generalize fractional laplacian to be interdimensionally correlated?

sorry if my question sounds weird. struggling to find the appropriate vernacular. the laplace operator Δ can be generalized to have "cross-diffusion" ∇·M∇ where ∇ is the gradient operator, M ...
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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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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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An infinitesimal generator of fractional derivatives and integrals

Let $S$ be the unit circle in $\Bbb{R}^2$, and $F$ be the linear space $\{f \in C^{\infty}(S)|\int_{S}f \, d \mu = 0 \}$ where $\mu$ denotes the Hausdorff measure induced by the flat metric of $\Bbb{R}...
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Is argmin of a sum of two functions equal to sum of each argmin?

Problem: Given function $f: \mathbb{R}^n \to \mathbb{R}$ is differentiable, convex; function $g: \mathbb{R}^n \to \mathbb{R}$ is nongegative, convex and $h: \mathbb{R}^n \to \mathbb{R}$ is positive, ...
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Explanation of this integration passages

Let $$u(x)\in L^1_s(\mathbb{R}^n) = \left\{ u\in L^1_{\text{loc}}(\mathbb{R}^n; s\in(0,1); \int_{\mathbb{R}^n} \frac{|u(x)|}{1 + |x|^{n+2s}}\ \text{d}x < +\infty \right\} $$ And let $$A_r(y) = \...
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Why model infectious diseases with fractional differential equation

With COVID19 becoming a pandemic I saw some researchers trying to model it with fractional differential equations (FDE) instead of ordinary differential equations (ODE). From a technical standpoint I ...
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3 votes
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Caputo derivative to log Mittag-Leffler function

Suppose $0<\alpha<1$, I am wondering whether there is a closed form expression to Caputo derivative to log Mittag-Leffler function, i.e., \begin{align*} \frac{\partial^\alpha}{\partial x^\alpha}\...
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4 votes
1 answer
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Is $(1-x)^{\alpha} \log(1-x)$ a Sobolev function?

Let $f(x) = (1-x)^{\alpha} \log(1-x)$ be defined on $[0,1]$ with $\alpha > 0$ some real exponent. Does such a function belong to the Sobolev (or Bessel potential) space $H^{\beta}((0,1))$ with $\...
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comparison property for fractional derivative

I am studying fractional derivative. In the case of normal derivative, it is known that for $u,v \in C^1[0,1],$ if $u(0)=v(0)~\hbox{and} ~u'(0) < v'(0)$, then $u(t) < v(t)$ for all small $t>0$...
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what is the partial derivative ot the following function?

Can you help me to compute the partial derivative $\dfrac{\partial F(t,\varrho_{1} ( t ) ,\varrho_{2}( t ) ,^{RL}\mathcal{D}_{0^{+}}^{\frac{1}{2}}\varrho_{1} (t) )}{\partial \varrho_{1}}$of the ...
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3 answers
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general solution to fractional differential equation

I'd like to know the existence of the general solution to the following fractional differential equation $$D_{0+}^{\alpha} y(t)=0 \text{,}\label{1} \tag{1}$$ where $\alpha \in (1,2)$ and $$ D_{0+}^{\...
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Solving $y^{(x)}(x)=ax+b$ in closed form: What function equals $ax+b$ when you take the $n$th derivative at $x=n$? (with graphs)

Based on the fun of: Conjectured simple ODE solution: $$y^{(y(x))}(x)=f(x)\mathop \implies\limits^?(y(x))!+c_0Γ(y(x))=\int\limits_{c_1}^xf(t)(x-t)^{y(x)-1} dt $$ Imagine we had a function of which ...
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Fractional Powers in SDEs

First of all, apologies in advance for any imprecision in terms- it's been a while since I worked with these concepts and I'm quite rusty. Consider a stochastic system $X_t$ that's just a simple ...
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For which functions $f$ is $f^{(0)}(x) = f(x)$?

Classically, for $0<\alpha<1$ the fractional derivative of order $\alpha$ is given by: $$ D^{\alpha}f(x) = \frac{1}{\Gamma(1-\alpha)}\frac{d}{dx}\int_0^x \frac{f(t)}{(x-t)^{\alpha}}\,dt$$ When $...
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Explanation of solving a GL fractional differential equation using a numerical method

I'm not sure if this is the right place to post this but I'm a little lost with some work I'm doing at the moment. I'm looking to solve this equation: $$^{gl}D^{\alpha}x(t) = 5x(t), x(0) = x_{0}$$ ...
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7 votes
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How to solve this ODE: $y^{(y(x))}(x)=f(x)$?

$$\large{\text{Introduction:}}$$ This question will be partly inspired from: Evaluation of $$y’=x^y,y’=y^x$$ but what if we made the order of an differential equation equal to the function? Imagine ...
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Fractional stationary curve.

Define fractional stationary. Is there any relation between stationary curve and fractional stationary curve? please provide sufficient examples.
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What the normalization function in fractional derivative exactly is

I have some questions about fraction calculus. In Caputo-Fabrizio fractional derivative $$ ^{CF}_aD_t^\alpha f(t)=\frac{M(\alpha)}{1-\alpha}\int_a^t f'(\tau)exp(-\alpha\frac{t-\tau}{1-\alpha})d\tau $$ ...
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Prove that a given integral function has a sign

Let $u,v:\mathbb R \to \mathbb R$ and $\phi: \mathbb R \to \mathbb R_+$ be smooth bounded functions. Assume also $\phi'\le 0$. Assume that $u(0) - v(0) = 0$ and that $0$ is a strict global minimum of $...
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Fractional calculus and spectral theory

I've been trying to build a more solid understanding of fractional calculus, and the closer I look, the more it seems like there are some deep connections to Fourier analysis (and spectral theory in ...
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9 votes
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Functions whose Fractional derivative equal $0$

For $\alpha > 0$, let $n =\lfloor \alpha \rfloor + 1$ and $f : (0,\infty)\to\mathbb{R}$ be continuous. Define $$D^\alpha f(x) = \frac{1}{\Gamma(n - \alpha)} \left(\frac{d}{dx}\right)^n \int_0^x \...
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Imaginary order differential equations

I would like to find the solution of the imaginary order differential equation $y^{(2i)}+y^{(i)}+y=0$ I started with the Fourier transform differintegral as it seemed more suitable than the Riemann-...
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2 votes
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Prove that $\int_U u^2\le C\int_U\int_U\frac{(u(x)-u(y))^2}{|x-y|^{n+2s}}dx\,dy$ for some $C>0$

Recently I am working on some kind of fractional Sobolev inequalities, and I would like to prove that, for all $u\in W^{s,2}(U)$, $$\|u\|_{L^2(U)}\le C[u]_{W^{s,2}(U)}\qquad (\star)$$ for some $C>0$...
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3 votes
1 answer
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Commutator of pseudodifferential operator and multiplication operator

Let $\eta:\mathbb R^n\to[0,1]$ be a smooth and compactly supported function and assume $f:\mathbb R^n \to\mathbb R$ is measurable. I want to bound the commutator of the operators $\eta(-i\nabla)$, ...
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Why is $W^{s,p}(\mathbb R^n)$ continuously embedded in $L^q(\mathbb R^n)$ for $q\in[p,\frac{np}{n-sp}]$, given the fractional Sobolev inequality?

In Theorem 6.5 of Hitchhiker’s guide to the fractional Sobolev spaces, it is stated that: Let $s\in(0,1)$ and $p\in[1,+\infty)$ be such that $sp<n$. Then, there exists a positive constant $C$ ...
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1 vote
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Semi-integral of $f(x) = \frac{1}{x}$

I wish to know what the semi-integral (fractional integral of order one-half) of the function $f(x) = \frac{1}{x}$ is. From the definition of the Riemann–Liouville fractional integral one has $$D^{-\...
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2 votes
2 answers
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Asymptotics of $\int_{B(0,1)}\frac{dy}{|x-y|^{n+\alpha}}$ as $|x|\to 1^+$

In a study of Fractional Laplacian, I encounter the integral $$I(x):=\int_{B(0,1)}\frac{dy}{|x-y|^{n+\alpha}}$$ where $B(0,1)\subset\mathbb R^n$ is the $n$-dimensional unit ball centered at the ...
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Is the $n^\text{th}$ derivative of some function, where $n$ is a matrix, ever defined?

For the function $f(x) = x^a$, the $n^\text{th}$ derivative $\frac{d^n}{dx^n}x^a = \frac{a!}{(a-n)!}x^{a-n}$ . This can be extended to non integer values of $n$ thanks to the gamma function. I ...
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5 votes
1 answer
104 views

Integration by rational substitution

I was taking a look at a proof of the Riemann-Liouville Integral of consecutive integrations, and at some point I reached a step where it shows the following substitution: $$_{a}I_{x}^\alpha(_{a}I_{x}^...
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Separation of variables for the fractional laplacian

Consider the fractional Laplacian of $u\in H^{s}(\mathbb{R}^n)$ denoted as $(-\Delta)^s u$ with $s\in (0,1)$. I know that in general for the usual Laplacian we can write $\Delta = \Delta_r + r^{-2}\...
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Fractional Derivatives of $e^{ax}$

We know that for all $n\in\mathbb{N}$ and $a\in\mathbb{R}\setminus \{0\}$ $$ D^{n }e^{a x} = a^{n}e^{a x}$$ So I thought that the fractional derivative of this function would be $$ D^{\alpha }e^{a x} =...
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2 votes
1 answer
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Evaluating the Grünwald–Letnikov derivative of the power function

I recently came across the Grünwald–Letnikov derivative and I wanted to use it to evaluate fractional derivatives of various functions. Specifically I used this expression of the derivative $$D^qf(x)=\...
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2 votes
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Constraining many Gamma distribution

I'm working on a statistical model which involves many degrees of freedom $i=1...S$. Each degree of freedom is described by a gamma distribution with its own parameters, which we will assume to be all ...
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LOTUS for Laplace Space

Let $f(t)$ be the density of a non-negative, absolutely continuous random variable $T$. While $f(t)$ has no closed form representation, suppose its Laplace transform $$ \int_0^\infty e^{-st}f(t)\,\...
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