What is fractional: Definition and 221 Discussions
Fractional calculus is a branch of mathematical analysis that studies the several different possibilities of defining real number powers or complex number powers of the differentiation operator
D
{\displaystyle D}
D
f
(
x
)
=
d
d
x
f
(
x
)
,
{\displaystyle Df(x)={\frac {d}{dx}}f(x)\,,}
and of the integration operator
J
{\displaystyle J}
J
f
(
x
)
=
∫
0
x
f
(
s
)
d
s
,
{\displaystyle Jf(x)=\int _{0}^{x}f(s)\,ds\,,}
and developing a calculus for such operators generalizing the classical one.
In this context, the term powers refers to iterative application of a linear operator
D
{\displaystyle D}
to a function
f
{\displaystyle f}
, that is, repeatedly composing
D
{\displaystyle D}
with itself, as in
D
n
(
f
)
=
(
D
∘
D
∘
D
∘
⋯
∘
D
⏟
n
)
(
f
)
=
D
(
D
(
D
(
⋯
D
⏟
n
(
f
)
⋯
)
)
)
{\displaystyle D^{n}(f)=(\underbrace {D\circ D\circ D\circ \cdots \circ D} _{n})(f)=\underbrace {D(D(D(\cdots D} _{n}(f)\cdots )))}
.
For example, one may ask for a meaningful interpretation of
D
=
D
1
2
{\displaystyle {\sqrt {D}}=D^{\frac {1}{2}}}
as an analogue of the functional square root for the differentiation operator, that is, an expression for some linear operator that, when applied twice to any function, will have the same effect as differentiation. More generally, one can look at the question of defining a linear operator
D
a
{\displaystyle D^{a}}
for every real number
a
{\displaystyle a}
in such a way that, when
a
{\displaystyle a}
takes an integer value
n
∈
Z
{\displaystyle n\in \mathbb {Z} }
, it coincides with the usual
n
{\displaystyle n}
-fold differentiation
D
{\displaystyle D}
if
n
>
0
{\displaystyle n>0}
, and with the
n
{\displaystyle n}
-th power of
J
{\displaystyle J}
when
n
<
0
{\displaystyle n<0}
.
One of the motivations behind the introduction and study of these sorts of extensions of the differentiation operator
D
{\displaystyle D}
is that the sets of operator powers
{
D
a
∣
a
∈
R
}
{\displaystyle \{D^{a}\mid a\in \mathbb {R} \}}
defined in this way are continuous semigroups with parameter
a
{\displaystyle a}
, of which the original discrete semigroup of
{
D
n
∣
n
∈
Z
}
{\displaystyle \{D^{n}\mid n\in \mathbb {Z} \}}
for integer
n
{\displaystyle n}
is a denumerable subgroup: since continuous semigroups have a well developed mathematical theory, they can be applied to other branches of mathematics.
Fractional differential equations, also known as extraordinary differential equations, are a generalization of differential equations through the application of fractional calculus.
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The actual problem that i was looking at with my students was supposed to be
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##x^\frac{2}{3} - x^\frac{-3}{2}-6=0## on...
I am trying to compute the Peebles equation as found here:
I am doing so in Python and the following is my attempt:
However, I'm unable to solve it. Either my solver is not enough, or I have wrongly done the function for calculating the Equation.
# imports
from scipy.optimize import fsolve...
I have tried manipulating this to
$$1-\frac{8}{(\pi k_B T)^2}\mu^\frac12(\epsilon^\frac32-\mu^\frac32)=0\Leftrightarrow\left[1+\mu^\frac12(\epsilon^\frac32-\mu^\frac32)\right]^{-\frac{8}{(\pi k_B T)^2}}=0$$
but this doesn't seem to lead anywhere.
any hints please?
the solution is one of these...
I have to solve a certain numerical problem without using calculator and furthermore, there is a time limit for solving this problem.
The answer I have got so far is ## \sqrt{\frac{100}{99}}##
I know I can reduce the numerator to 10 but then I am stuck with square root of denominator which is...
Hi, PF
This is the quote:
"If ##m## is an integer and ##n## is a positive integer, then
6. Limit of a power:
## \displaystyle\lim_{x \to{a}}{\left[f(x)\right]^{m/n}} ## whenever ##L>0## if ##n## is even, and ##L\neq{0}## if ##m<0##"
What do I understand?
-whenever ##L>0## if ##n## is even: ##m##...
Hello! (Wave)
I found the following algorithm for the fractional knapsack problem.
Why at the case else, we do not change the variable w to w+(W-w)/S.weight? (Thinking)
(If I should have posted this in the Math thread instead of the Homework thread, please let me know.)
I have three questions which I will ask in sequence. They all relate to each other.
I've typed my questions and solutions attempts below.
I've also attached a hand-written version of this...
So I got the answer through a little addition i.e 9^(1/2) multiplied by 9^(1/2) = 9^1 or 9
3 x 3 = 9 so 3 is the answer to what is 9^(1/2)
I've tested this out with a few other numbers and have made this generalization, x^(1/2) = √x
It seems to make the equations orderly and consistent but is...
Why do fractional errors have a different error value from subbing in the raw values?
e.g. 10 +- 1 divided by 10 +- 1
fractional error yields 20%
11/9 - 9/11 yields 40/99
In cases like these do we use the original values and attempt to find the maximum error distance or use fractional errors?
Does anyone know any good research on this topic? I'm basically looking for information on what would be solving integral and differential equations in which the unknown you need to solve for is the level of a integral or derivative in the equation. For example F'1/2(u)+F'x(u)=F'1/3(u) where the...
I know how to find integral solutions of linear equations like x+y=C or x+y+z=C where C is a constant.
But I don't have any idea how to solve these type of questions.I am only able to predict that both x and y will be greater than 243554.Please help.
Introduction
This bit is what new thing you can learn reading this:) As for original content, I only have hope that the method of using the sets
$$C_N^n: = \left\{ { \vec x \in {\mathbb{R}^n}|{x_i} \ge 0\forall i,\sum\limits_{k = 1}^n {x_k^{2N}} < n - 1 } \right\}$$
and Dirichlet integrals to...
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I was just thinking about this earlier and couldn't come up with a good enough resolution. I'm guessing it's a matter of convention more than anything. If we have ##x^{2} = a##, taking the principle root of both sides gives ##\sqrt{x^{2}} = \sqrt{a} \implies |x| = \sqrt{a}##.
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Hi.
I would like to check that my understanding is correct. For ##f(x)=x^{1/n}## where n is an integer. If n is odd then f(x) is an odd function while if n is even then f(x) is neither odd or even as it involves the square root function which is only defined for non-negative x.
For ## f(x) =...
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https://www.chemguide.co.uk/physical/phaseeqia/idealpd.html#top
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Homework Statement
Determine whether there exist ##A## and ##B## such that:
$$\frac{1}{3x^2-5x-2} = \frac{A}{3x+1} + \frac{B}{x-2}$$Homework Equations
None
The Attempt at a Solution
[/B]
First I divided the polynomial ##3x^2-5x-2## by ##3x+1## and got ##x-2## as a result without a...
Why is fractional uncertainty not affected by systematic error? For example à vernier calipers measures the diameter of a coin:
(5.06+-0.04) mm
Can taking more readings, say 6, and taking average, reduce fractional error?
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Homework Statement
What would have caused humans to come up with fractional exponent notations?
Homework EquationsThe Attempt at a Solution
I understand that it makes sense to use the exponent notation when we have to multiply the same number a number of times. For example, 10^8 is the short...
Most QFT texts, such as Peskin&Schroeder and D. Tong's lecture notes, contain a mention that the renormalizability of an interacting theory requires the coupling constants to have correct dimensions, making scalar fields with ##\phi^5 , \phi^6, \dots## interactions uninteresting. Maybe there are...
Homework Statement
a3/2a5/4
Homework EquationsThe Attempt at a Solution
I'm hoping you can help. My solution to this problem would be:
a3/2+5/4=a8/6=a4/3
But the answer in the back of my book is given as a11/4
I'm confused!
Homework Statement
D+8/D-2 = 9/4
See image, original equation in black.
Homework EquationsThe Attempt at a Solution
See image.
Having a little trouble with this.
Ive attempted to solve it two ways. The first was to multiply both sides by ##d-2## which gave me the correct answer of...
How can this code be coded by another way?
## \frac {300-270.1}{T-130}## = ## \frac {313.-270.1}{135-130}##
And it is also very strange that the right side O.K but left side is not but everything is the same for left and right.
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I've been thinking about it since yesterday and have noticed this pattern:
We have, the first order derivative of a function ##f(x)## is:
$$f'(x)=\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h} ...(1)$$
The second order derivative of the same function is:
$$f''(x)=\lim_{h\rightarrow...
<Moderator note: Thread moved from General Physics hence no formatting template shown>
The fractional uncertainty is defined as:
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Homework Statement
Find critical numbers of the function: F(x)=t^3/4 - 2t^1/4
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Homework EquationsThe Attempt at a Solution
I have found the derivative and I understand I must pull out a t in order to find critical numbers, and run across this...
Hello!
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Homework Statement
I want to solve:
$$ x+a(x^2-b)^{1/2}+c=0$$
Homework Equations
The Attempt at a Solution
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I am trying to work through a simplication of this factorial with variables:
(n/2)!/[(n+2)/2]!
I get,
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cancelling the 2[n(n-1)]
leaves me with 1/[(n+2)(n+1)]
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Thanks
Homework Statement
If ##\{ x \}## denotes the fractional part of x, then solve:
## \{ x \} + \{ -x \} = x^2 + x -6##
It's provided that there are going to be 4 roots of this equation. And two of them will be integers.
Homework Equations
## 0 \lt \{ x \} \lt 1~~\text{if}~~x \not\in I##
## \{...
Homework Statement
So, I'm solving a dipole thing and I have these vectors:
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Homework Equations
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