Questions on the differentiation/integration of functions to arbitrary order.
9
votes
3answers
158 views
(fractional) half derivative of $ {1 \over 1-x }$?
I'm new to this "fractional derivative" concept and try, using wikipedia, to solve a problem with the half-derivative of the zeta at zero, in this instance with the help of the zeta's ...
0
votes
0answers
28 views
Introduction to fractional calculus: problem with identity
I can't see the next step:
$D^\alpha e^{ix} = i^{\alpha}e^{ix} = e^{i\alpha \frac \pi2}e^{ix}$
1
vote
0answers
22 views
How to predict order of a set of fractional differential equations?
I have a set of differential equations of the form:
$$\frac{dv}{dt} = a[b-c*m-d*n-e*h]$$
$$\frac{dm}{dt} = p(v)$$
$$\frac{dn}{dt} = q(v)$$
$$\frac{dh}{dt} = r(v)$$
Using fde12 in MATLAB I can ...
3
votes
1answer
72 views
What is the half-derivative of zeta at $s=0$ (and how to compute it)?
I'm trying to understand the concept of fractional derivatives and am fiddling with the examples at wikipedia. The a'th derivative of a monomial in x, where a can be fractional is accordingly $$ {d^a ...
1
vote
2answers
45 views
Software for Solving Numerically Fractional Differential Equations
I have been trying to find information on how to solve fractional differential equations numerically with the usual maths software (Mathematica, Maple, Matlab..). Or to find an alternative program to ...
2
votes
0answers
22 views
Analog of Picard's theorem for Fractional Differential equations.
I need an analog of Picard's theorem of existence and uniqueness of solutions. The theorem is to be applied to linear fractional order differential equations with constants coefficients.
I don't want ...
1
vote
0answers
28 views
Fractal derivative of complex order and beyond
Is there some precise definition of "complex (fractal) order derivative" for all complex number? I am aware of the Riemann-Liouville fractional definition given here: Complex derivative but I would ...
4
votes
0answers
68 views
Geometrical Interpretetion of Half Derivative
How would you understand in a intuitive way the meaning of:
$$D^{\frac{1}{2}}x^2=\frac{\Gamma(3)}{\Gamma(\frac{5}{2})}x^{\frac{3}{2}}=\frac{8}{3\sqrt{\pi}}x^{\frac{3}{2}}$$
or
...
1
vote
0answers
102 views
Kernel of Fractional Differential Operator
Suppose we have a fractional differential equation:
$$\left[D^{nv}+a_{1}D^{\left(n-1\right)v}+\dots+a_{n}D^{0}\right]y(t)=0$$
where $\nu=\frac{1}{q}$ and $q\in\mathbb{N}$ and y is an analytic ...
0
votes
2answers
145 views
Help in proof of theorem about Riemann-Liouville Fractional Calculus
Theorem:
Let,
$$\left[D^{nv}+a_{1}D^{\left(n-1\right)v}+\dots+a_{n}D^{0}\right]\left(y\right)=0$$
be a fractional differential equation of order $\left(n,q\right)$,
where $v=\frac{n}{q}$, and let
...
0
votes
0answers
49 views
Hadamard fractional derivatives in control theory or viscoelasticity
I am currently working on Caputo-Hadamard fractional differential equations and yet I can not figure out where and how to use Hadamard fractional derivatives in real phenomena such as ...
1
vote
1answer
90 views
Existence of Riemann-Liouville Integral
The Riemann Liouville integral is defined as:
$\frac{1}{\Gamma\left(\nu\right)}\int\limits _{h}^{t}\left(t-\xi\right)^{\nu-1}f\left(\xi\right)d\xi$
It is supposed it does exist for all $\nu>0$ and ...
1
vote
1answer
82 views
Taylor Expansion of the 1/2th Derivative
In trying to solve the problem $\sqrt D f(x)=g(x)$ I tried to expand the derivative as a Taylor series, and have encountered a lot of problems. Is there some reason that this shouldn't be possible? ...
9
votes
2answers
159 views
Fractional Derivative Implications/Meaning?
I've recently been studying the concept of taking fractional derivatives and antiderivatives, and this question has come to mind: If a first derivative, in Cartesian coordinates, is representative of ...
1
vote
1answer
74 views
Fixed memory principle in fractional calculus
I am trying to read this paper and have a certain doubt. On page 8 the author comments that
First of all, there is a fundamental problem associated with all
fractional differential operators ...
0
votes
0answers
57 views
Derivatives and integrals, normal and fractional, and their explanations and relations
Assuming, naively, that one acquires the nth derivative of a function by repeatedly differentiating and finding a pattern. Thus one gets $f^{(n)}(x)=g(x,n)$. I have a few questions about this ...
0
votes
0answers
40 views
nonlinear integral equation
let be the integral equation for two functions $ f(x) $ and $ g(x) $
in the form $$ g(s)= \int_{0}^{s}\sqrt{s-f(x)}dx $$
is valid to accept that in the sense of fractional calculus, the ONLY ...
1
vote
3answers
135 views
fractional derivative of a heaviside function
given the function
$$ f(x)= \frac{H(x+1)}{\sqrt{x+1}} $$
how can i evaluate the fractional derivative
$$ \frac{d^{1/2}}{dx^{1/2}}f(x) $$
if i use the standar definition for powers of 'x' i get a ...
0
votes
2answers
172 views
Fractional derivative of a constant (Riemann-Liouville Derivative)
In a book I read about Riemann-Liouville fractional derivative, it says,
$$_0D_t^\alpha 1=\frac{t^{-\alpha}}{\Gamma(1-\alpha)},\alpha\geq0,t\geq0$$ which identically vanishes for ...
1
vote
0answers
52 views
for what $\nu$ does Riemann-Liouville differintegral of digamma function $\psi(z)$ exist?
For what values of $\nu$ does the Riemann-Liouville differintegral $_{-\infty}D_{z}^\nu$ of the digamma function $\psi(z)=\frac{\Gamma'(z)}{\Gamma(z)}$ exist, with $c=-\infty$? All I've got so far is ...
5
votes
1answer
171 views
Fractional derivatives of delta function $ \delta (x) $
How can I define the fractional derivative of the Delta function?
I mean $D^{\alpha}= \frac{d^{\alpha}}{dx^{\alpha}} $ where $\alpha$ can be any real number, then if we define $D^{\alpha} \delta (x) ...
2
votes
2answers
202 views
Bessel and cosine function identity formula
by expanding into series ( sorry i have tried but get no answer) how could i prove that
$$ \sqrt \pi\frac{d^{1/2}}{dx^{1/2}}J_{0} (a\sqrt x) = \frac{\cos(a\sqrt x)}{\sqrt x}$$
4
votes
1answer
351 views
half-derivative of $x^2$
I was given this problem to challenge me.
$\frac{x^{1/2}}{dx^{1/2}}x^2$
I googled wikipedia, and tried to follow the steps shown.
I got an answer of $\frac{16\sqrt{ \pi x}}{9\pi}$ edited
2 part ...
7
votes
2answers
347 views
Fractional calculus in complex analysis
According to Fractional calculus, we know that $$(J^\alpha f) ( x ) = { 1 \over \Gamma ( \alpha ) } \int_0^x (x-t)^{\alpha-1} f(t) \; dt$$
It's in real analysis, but what about in complex analysis? ...
0
votes
1answer
83 views
Green's function for fractional operators
I am studying some papers about the fractional laplacian, and I am stuck on a formula that I do not understand. I would like to ask if anybody can give me some help.
In this paper, on page 12, there ...
1
vote
0answers
122 views
Decay of the fundamental solution of fractional laplacian equations
It is well known that the fundamental solution $\Gamma_1$ in $\mathbb{R}^n$ of the Schrödinger operator $-\Delta + 1$ decays exponentially fast, viz. $|\Gamma_1(x)| \leq C_1 \mathrm{e}^{-C_2|x|}$ as ...
5
votes
1answer
189 views
About fractional differentiation under the integral sign
$1.$ Does $\dfrac{d^n}{dx^n}\int_a^bf(x,t)~dt=\int_a^b\dfrac{\partial^n}{\partial x^n}f(x,t)~dt$ correct when $n$ is a positive real number?
$2.$ How about ...
0
votes
0answers
181 views
definition of fractional derivative integral
would be possible to define the fractional derivative (and integral) as
$$ D^{a}f(x)=F.P\frac{1}{\Gamma(-a)}\int_{c}^{x}dt \frac{f(x)-f(t)}{(x-t)^{1+a}}$$
here c ,a are real constant (a can be ...
5
votes
3answers
305 views
Calculating $\frac{\partial^{1/2}}{\partial x^{1/2}}\left( e^{-\alpha x^2 + \beta x} \right) $
I would like to calculate
$$\frac{\partial^{1/2}}{\partial x^{1/2}}\left( e^{-\alpha x^2 + \beta x} \right) $$
My intuition is that I would have to use some sort of fractional Leibniz formula to ...
2
votes
1answer
156 views
Is there any body of knowledge or study of the fractional calculus on definite integrals?
The fractional calculus is partly about nested indefinite integrals. Is there any study or body of knowledge on nested DEFINITE integrals? For example, the fractional calculus helps with this ...
2
votes
1answer
145 views
Complex derivative
The derivative of a function $f(x)$ is the limit of the quotient
$$\lim_{\Delta x\to 0}\frac{f(x+\Delta x)-f(x)}{\Delta x}$$
A formula defining the fractional derivative of the same function is for ...
0
votes
1answer
173 views
What is the fractional derivative of the function $\pi \cot (\pi x)$?
What is the fractional derivative of the function $\pi \cot (\pi x)$?
I derived the following expression:
$(\pi \cot (\pi q))^{(p)}=-\frac{\zeta'(p+1,q)+(\psi(-p)+\gamma ) \zeta (p+1,q)}{\Gamma ...
3
votes
3answers
169 views
Fractional differential equation
Does someone know how to solve this fractional differential equation? $$a\frac{d^2}{dx^2}u(x)+b\frac{d^\frac{1}{k}}{dx^\frac{1}{k}}u(x)+cu(x)=0$$ assuming $(a,b,c) =const$ and $k$ a parameter?
Thanks ...
0
votes
0answers
284 views
Integration by parts for fractional order
Let be the fractional derivative $ D^{a}f(x) $ for some real positive $a >0 $
My question is if $ \int_{a}^{b}dx D^{a}f(x)g(x) = C(a)\int_{a}^{b}dx D^{a}g(x)f(x) $
Provided that $ ...
2
votes
0answers
170 views
a problem in fractional calculus
One of the early applications of fractional calculus is the tautochrone problem set up by Abel in the integral form or its fractional derivative one. i wish to know its solution.
1
vote
1answer
129 views
Family of function with fractional derivatives
I would like a family of functions $f_a(x)$ so that $f_a$ is $a\in\mathbb{R}$ fractionally differentiable but not $a+\epsilon$ fractionally differentiable.
Does anyone know such functions which are ...
8
votes
5answers
1k views
Functions that are their Own nth Derivatives for Real n
Consider (non-trivial) functions that are their own nth derivatives. For instance
$\frac{\mathrm{d}}{\mathrm{d}x} e^x = e^x$
$\frac{\mathrm{d}^2}{\mathrm{d}x^2} e^{-x} = e^{-x}$
...
8
votes
4answers
346 views
Applications of Fractional Calculus
I've seen recently for the first time in Special Functions (by G. Andrews, R. Askey and R. Roy) the definitions of fractional integral
$$(I_{\alpha }f)(x)=\frac{1}{\Gamma (\alpha ...
13
votes
2answers
376 views
Is it meaningful to take the derivative of a function a non-integer number of times?
If I want to take the derivative of $ax^n$, I will get $anx^{n-1}$. If I were to take the derivative again, I get $an(n-1)x^{n-2}$.
We can generalize this for integer k easily to get the kth ...
