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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12 views

Proof of Caputo fractional derivative at zero and first order

Currently I am working with fractional derivative and encountered Caputo's definition: $\frac{\partial^\alpha}{\partial x^\alpha}y(x = x_0) = \frac{1}{\Gamma(1-\alpha)}\int_{-\infty}^{x_0} (t-x)^{-\...
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14 views

The Fractional Derivative and Stochastic Processes/ Fokker Planck equation

In this question I am interested in Fractional Derivatives and the Fractional Fokker Planck Equation. Note I only mention fractional in time, however if someone wants to reference to fractional in ...
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11 views

Converting fractional differential equation to integral equation

How could I convert a fractional differential equation of the form $$dx(t) = D^qx(t)dt + dW(t)$$ where $$D^qx(t)= \frac{1}{\Gamma(1-q)}\int\limits_{t_0}^{t}\frac{\dot{x}(\tau)}{(t-\tau)^q}d\tau$$ is ...
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8 views

Stochastic differential equation with fractional derivatives

In the book "Handbook of Stochastic Methods" by Gardiner, in page 93, it is shown that the stochastic differential equation (SDE) $$dx(t) = a[x(t),t]dt+b[x(t),t]dW(t)$$ is equivalent with the ...
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7 views

Relation between these two fractional integral operators.

Consider the composition of two functions $f(g(x))$ with $g(x)$ being a differentiable function with bounded derivative. Then is there any relation between the fractional integral of $f(g(x))$ with ...
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42 views

What is the fractional integral of the second derivative?

The following example illustrates the issue. The derivative of derivative is: $$ \frac{d}{dx}(\frac{dy}{dx}) = \frac{d^2y}{dx^2} $$ The derivative of square of derivative is: $$ \frac{d}{dx}(\frac{...
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18 views

Relation between the Bessel function of the first kind and fractional derivatives

Introduction The Bessel function of the first kind is defined as follows: $${\displaystyle J_{\alpha }(x)=\sum _{n=0}^{\infty }{\frac {(-1)^{n}}{n!\Gamma (n+\alpha +1)}}{\left({\frac {x}{2}}\right)}^...
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How to avoid division by zero in $\int_{0}^{t_i}(t_i-\tau)^{(m-\alpha-1)}\chi{_{N}^{m}}(\tau,\Omega)d\tau$

I want to discretize the above integral using trapezoidal rule. The problem is if $ m=1 $, $t_i$ and $ \tau $ are equal then their is infinity. where $\chi{_{N}^{m}}(\tau,\Omega)$ is a neural network. ...
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15 views

Immersion $H^s(\Omega) \hookrightarrow H^{s'}(\Omega)$ but with fractional laplacian defined on $\mathbb{R}^N$?

I know that it holds this embeding $$H^{s}(\Omega) \hookrightarrow H^{s'}(\Omega)$$ for $s>s'$ and for any $\Omega \subset \mathbb{R}^N$. In this case, anyway, the fractional laplacian is defined ...
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13 views

Computing the fractional Laplacian of ${\rm exp}(-|x|^\alpha)$

I would like to be able to give a good approximation for the following integral $$ [(-\Delta)^{s^\prime/2} \exp(-|\cdot|^s )] (x ) = c_{s^\prime} \int_{-\infty}^\infty \frac{-\exp(-|x-y|^s )-\exp(-|x+...
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30 views

MATLAB code for solving fractional order partial differential equation through q-homotopy analysis transform method (q-HATM)

Currently, I'm doing research about fractional order partial differential order and trying to solve it using homotopy analysis method with Laplace transform which is known as q-HATM. Then, solve the ...
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27 views

On compact imbedding of fractional Sobolev Space

Let $\Omega$ be a Lipschitz domain in $\mathbb{R^n}$. Let $s \in (\frac{1} {2},1)$. Then we know $H^1 (\Omega) \subset H^s (\Omega)$ with continuous imbedding. But can we also say that this imbedding ...
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24 views

Attempted formulation of the Riemann-Liouville (RL) Fractional Derivative.

I am a high school student studying fractional calculus. I recently came across a couple of issues regarding the formulation of the Riemann-Liouville (RL) Fractional Derivative from the Riemann-...
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33 views

Grunwald fractional derivative [duplicate]

I came across this maths statement \begin{align*} f(t+h) &= f(t) + h\frac{d}{dt}f(t) + \frac{h^2}{2!}\frac{d^2}{dt^2}f(t) + \frac{h^3}{3!}\frac{d^3}{dt^3}f(t) + \cdots \\ &= f(t) + hDf(t) + \...
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29 views

Analytic Continuation of Fractional Derivatives

I've been doing some work with fractional calculus, and I have been running into a problem. For a given meromorphic function $f: \mathbb{C}\to \mathbb{C}$, if a formula of the form $$(D^{\alpha}f)(0)=...
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26 views

Fractional Derivative and Laplace Transform

I've learned a lot with this lecture: https://www.youtube.com/watch?v=2FZlz4-pf-M. I have a doubt at 14:17. Why did the professor putted the minus sign? I also would like to know if the derivative ...
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29 views

Why is fractional stochastic integral no semimartingale?

I have problems to under stand the following thing: I totally see that the fractional Brownian Motion (https://en.wikipedia.org/wiki/Fractional_Brownian_motion) is no semimartingale hor $H\neq \frac{...
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66 views

How to find half derivative of $x^{-\frac{1}{2}}$?

I use this general definition to do fractional differentiation: $$(D^nf)(t)=\frac{1}{\Gamma(1-n)}\frac{d}{dx}\int_a^x (x-t)^{-n}\space f(t)\space\space dt,\space\space 0<n<1$$ However, when I ...
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19 views

What is the generalization of Taylor's expansion with the fractional number as a power index?

Given a classical Taylor expansion as $f(x)=\sum_{i=0}^{\infty} \frac{d^{i}f(x_{0})}{dx^{i}}\frac{(x-x_{0})^{i}}{i!}$ where i is a nonnegative integer. Can we generalize this expansion to fractional ...
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37 views

Differentiating using Cauchy's repeated integral formula [duplicate]

Can we use Cauchy's repeated integral formula to differentiate functions? $$(I^nf)(t)=\frac{1}{\Gamma(n)}\int_a^x (x-t)^{n-1}\space f(t)\space\space dt,\space\space n\notin \mathbb Z_{\leqslant 0}$$ ...
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57 views

Calculating half derivative of $x$ using Cauchy's repeated integral formula

As far as I know, Cauchy's repeated integral formula is defined as: $$(I^nf)(t)=\frac{1}{\Gamma(n)}\int_a^x (x-t)^{n-1}\space f(t)\space\space dt,\space\space n\notin Z_{\le 0}$$ I know that Gamma ...
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98 views

Solution to “Heat Equation” with Fractional Laplacian in 2 Dimensions

Statement of the Problem We consider the equation: $ \partial_t u + (- \Delta)^{1/2}u = 0 $ for $ u : \mathbb{R}^2 \rightarrow \mathbb{R} $. I would like to find a non-trivial solution to this ...
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How to use Pi and Gamma functions to find factorial of a number? [duplicate]

How can I find $\pi!$ or $e!$ or $(4-\pi)!$ etc. I have done it for $(\frac{3}2)!$ but Gaussian integral was used and it was easy but when I try for such values, it is very hard.
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29 views

N-th derivative of n-fold integral

I want to justify that the n-th derivative of an n-fold integral gives the original function. In other words that $$ \frac{d^n}{dx^n}\frac{1}{(n-1)!}\int_{0}^{x}(x-s)^{n-1}f(s)ds=f(x) $$ If I ...
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43 views

Is there a name for this kind of integral?

Just wondering if anyone knows the name of this kind of integral \begin{equation} X_t = \int_0^t \frac{f(s)}{(t-s)^\alpha} ds \end{equation} It's kinda unique as when $s \to t$ then $\frac{f(s)}{(t-...
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120 views

About generalized binomial theorem and Grünwald-Letnikov fractional derivative

I have run into a problem while computing the fractional derivatives of order $\alpha$ for the Riemann zeta function. My Theorem states Let $s\in\mathbb{C}$, $\mathfrak{Re}(s)>1$, then the ...
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27 views

Two questions on the Blancmange function.

I have two questions about the so-called Blancmange function (which I'll restrict to having domain $[0,1]$). That is, define: $$ B:[0,1]\to [0,2],\quad B(x):= \sum_{k = 0} 2^{-k}s\big(2^kx\big)$$ ...
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73 views

Factoring of Algebraic Fraction $\frac{x^4}{x^2-x+1}$

I came across an integral problem, it was a solved example, which goes something like this. $$\int \frac{x^4}{x^2-x+1} dx$$ They straight away factored above integral into $$\int x^2+x - \frac{x}{x^...
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17 views

Can it be defined something like $D^{f(z)}$ operator?

I've heard about fractional calculus where you can extend derivatives (and integrals) to complex numbers so that $\frac{d^{z}}{dx^{z}}\equiv D^z$ is meaningful where $z=a+bi$. So I was wondering if it ...
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33 views

Asymptotic expansion of integral involving cosine

How is the following expression for the asymptotic integral derived: $\int_{0}^{t} \frac{\cos\omega \tau}{\tau^\alpha} d\tau = \omega^{\alpha-1}\big[\Gamma(1-\alpha)\sin\frac{\pi \alpha}{2} + \...
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9 views

Truncated supersolution

Let $u\in L^2(t_1,t_2;H_{0}^1(\mathbb{R}^n))$ which is positive in $\mathbb{R}^n$ be a weak supersolution of the following nonlocal equation: $$ \partial_t u+(-\Delta)^s u=0\text{ in }\mathbb{R}^n\...
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30 views

About Mittag-Leffler function

We know that $e^{(a+b)t}=e^{at}e^{bt}$ but for Mittag-leffler function, can we say that $$\mathbb{E}_\alpha ((a+b)t^\alpha)=\mathbb{E}_\alpha (at^\alpha)\times \mathbb{E}_\alpha (bt^\alpha)$$ where $...
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92 views

Two definitions of the half-derivative of $e^x$ don't match. Who's right?

I find two main sources on how to compute the half-derivative of $e^x$. Both make sense to me, but they give different answers. Firstly, people argue, that $$\begin{align} \frac{\mathrm{d}}{\mathrm{d}...
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20 views

Treatment of fractional derivative inside an integral

I come from an engineering background, so please forgive my inaccurate math grammar. Currently I am looking at what we call a "spring-pot" system, where a material behaves somewhere between an ...
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41 views

Are there any applications of the “imaginary derivative” $D^i$ operator?

Where $D$ is the normal differential operator, $D f(x) = \frac{d}{dx} f(x)$, $D^n f(x) = (\frac{d}{dx})^n f(x)$, we can define, for example, the "half-derivative" $D^\frac{1}{2}$ such that $D^\frac{1}{...
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44 views

Integration involving Mittag-Leffler function

Consider the following function with $0<\alpha<1,\tau>0, E(t)$ an arbitrary continuous function for $t>0$: $f(t)=\int_0^t \int_0^t (\frac{t-s_2}{\tau})^{\alpha-1}E_{\alpha}(-(\frac{t-s_1}{...
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37 views

Calculus, Pattern

A quick question, anyone recognizes how they are related? \begin{equation} \lim_{\epsilon \to 0+} \left[\int_{\epsilon}^\infty (1-e^{-t})^{-\alpha-1} (-t)^\beta e^{-t} dt\right] = \sum_{j=1}^\infty \...
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22 views

Caputo Derivative multiplied with the integer time derivative

Let$f(t):=\frac{dy(t)}{dt}\frac{d^{\alpha}y(t)}{dt^\alpha}\\ \Rightarrow f(t)=\dot{y(t)}\int_0^t \frac{\dot{y(\tau)}}{(t-\tau)^\alpha}d\tau \qquad 1>\alpha>0, \quad where \quad \dot{y}=\frac{dy(...
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1answer
36 views

Oscillating reactions and fractional derivatives

I am wondering why in many oscillating reactions, as Lotka-Volterra, Brusscelator and Oregonator are used models with fractional derivatives. What is the advantage оver the integer derivatives?
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1answer
43 views

Properties of Riesz transform

I am studying something about the Riesz transform. The wikipedia page about Riesz transform https://en.wikipedia.org/wiki/Riesz_transform has something not clear for me. In particular, i refer to the ...
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98 views

Integration by parts for fractional Ornstein-Uhlenbeck process

So I have encountered a problem in a paper called Volatility is rough by Jim Gatheral et al. A stationary fractional Ornstein–Uhlenbeck process ($X_t$) is defined as the stationary solution of the ...
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26 views

Reference request: Calculating fractional integrals

I am trying to calculate/solve integrals of the type $f(x) = \int_0^x (x-t)^{\alpha-1} u(t) dt$ for a given $u(t)$ and $\alpha > 0$. Doing so by hand is pretty tedious, as you can imagine. This ...
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150 views

Find the $n$th derivative for the function $y=\sin(kx)$, $n\in\mathbb{R}$.

I created a question for a high school calculus exam: Find the $n$th derivative, $\dfrac{d^ny}{dx^n}$, for the function $y=\sin(kx)$, $n\in\mathbb{N}$. The solution: \begin{align} \dfrac{d}{dx}\sin(...
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38 views

Half-power of second derivative operator

I'm studying fractional powers of an operator and came across an exercise I could not solve. Let $A: D(A)\subset L^2(0,\pi) \to L^2(0,\pi)$ be a linear operator defined by \begin{gather*} D(A) = ...
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1answer
166 views

Cauchy Repeated Integral Formula with Root upper-bounds?

Cauchy's formula for repeated integration states that for any continuous function on $[0,1]$ we have that the $n$-fold integral can be represented by a single integral as follows $$ \int_a^x \int_a^{\...
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26 views

Derivation of $(iw)^{n}$ into the format $A+ib$

How to derive $(iw)^{n}$ into the format $A+ib$ ($A$ is the real part, $B$ is the imaginary part), where $i=\sqrt{-1}$, $w<0$, $n$ is a fractional number such as $\frac{1}{2}, \frac{2}{3}, ...$? ...
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1answer
66 views

Fourier transform of a polynomial of fractional degree

I'm trying to prove which polynomials $t^\alpha$ with $\alpha>0$ belong to $\hat{H}^{1+\beta}([0,T])$, $0<\beta\leq 1$, given by $$\hat{H}^{s}([0,T])=\left\{u \in L^2([0,T]): \int_{0}^T \left(1+|...
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62 views

Fractional derivative of fractional integral equal identity

Consider the definition of Riemann-Liouville fractional order derivative as follows $$^{RL}_a\mathcal{D}^\alpha_xf(x)=\frac{1}{\Gamma(1-\alpha)}\frac{d}{dx}\int_a^x \frac{f(u)}{(x-u)^{\alpha}}du,\...
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45 views

Finite range Caputo's fractional derivative in Matlab

I need to find a Matlab code that computes the finite range Caputo's fractional derivative, that is, in the definition, change the integration interval $(a, x)$ for $(x- \epsilon,x)$, where $\epsilon \...
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26 views

Fractional order derivative of integral with functions as bounds?

Consider the following integral with functions $u(t)$ and $v(t)$ as bounds $$\int_{u(t)}^{v(t)} f(\nu)d\nu$$ I need to compute the Riemann-Liouville fractional derivative of this integral as follows $...

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