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E ( ϕ ) = 0 2 π 0 h α ( ϕ ( θ , t ) , t ) + β ( t ) ϕ t ( θ , t ) 2 + γ ( t ) ϕ θ ( θ , t ) 2 + κ ( t ) d t d θ
E ( ϕ ˜ ) = 0 2 π 0 h α ( ϕ ( θ 0 , t ) , t ) + β ( t ) ϕ t ( θ 0 , t ) 2 + κ ( t ) d t d θ

so that

E ( ϕ ) - E ( ϕ ˜ ) = 0 2 π 0 h α ( ϕ ( θ , t ) , t ) + β ( t ) ϕ t ( θ , t ) 2 + κ ( t ) d t - 0 h α ( ϕ ( θ 0 , t ) , t ) + β ( t ) ϕ t ( θ 0 , t ) 2 + κ ( t ) d t + 0 h γ ( t ) ϕ θ ( θ , t ) 2 d t d θ

The expression

0 h α ( ϕ ( θ , t ) , t ) + β ( t ) ϕ t ( θ , t ) 2 + κ ( t ) d t
- 0 h α ( ϕ ( θ 0 , t ) , t ) + β ( t ) ϕ t ( θ 0 , t ) 2 + κ ( t ) d t

is always nonnegative, since θ 0 is a minimum f . Similarly,

0 h γ ( t ) ϕ θ ( θ , t ) 2 d t

is positive, since γ ( t ) is positive and ϕ θ is somewhere nonzero by assumption. Thus E ( ϕ ) - E ( ϕ ˜ ) > 0 .

Non-uniqueness

In section 2.3, we prove that there is a unique ϕ ( θ , t ) : [ 0 , 2 π ] × [ 0 , 1 ] R that is periodic in θ and satisfies Δ ϕ + sin ( 2 ϕ ) 2 = 0 with given boundary conditions. (The proof actually holds for t [ 0 , h ] , where 0 < h < 8 .) Furthermore, at least with horizontal boundary conditions, the short cylinder admits a vector field which attains the minimum energy. However, this is not the case on a sufficiently tall cylinder: ϕ ( θ , t ) = 0 is an unstable critical function.

Given a cylinder of height h , consider ϕ ϵ , h ( θ , t ) = ϵ t ( h - t ) . (Since this satisfies ϕ ϵ , h ( θ , 0 ) = ϕ ϵ , h ( θ , h ) = 0 , it describes a vector field with horizontal boundary conditions.) We can compute

E ( ϕ ϵ , h ) = 0 2 π 0 h cos 2 ϵ t h - t + ( ϵ h - 2 ϵ t ) 2 d t d θ
= 2 π 0 h cos 2 ϵ t h - t + ( ϵ h - 2 ϵ t ) 2 d t

Plotting E ( ϕ ϵ , h ) - 2 π h with respect to ϵ and h yields the graph shown in the following figure:

There is a clear region on which E ( 0 ) = 2 π h is greater than E ( ϕ ϵ , h ) . It is simple to construct specific examples where ϕ ϵ , h has lower energy than the horizontal field; a more difficult task is to find the lowest h 0 that admits a lower-energy field. By Theorem 2, such an h 0 is bounded below by 8 . A series of examples bounds it above by 10 , but the possibility remains that a better bound is available.

Even without an exact value for h 0 , we may discuss the significance of its existence. On a cylinder less than a certain height, the horizontal vector field described by ϕ ( θ , t ) = 0 uniquely minimizes energy; on any tall cylinder, a slight turn up or down will show an improvement. An energy-minimizing function on the tall cylinder, if it exists, must still satisfy the differential equation Δ ϕ + sin ( 2 ϕ ) 2 = 0 , but ϕ = 0 is a trivial solution to this. We are left with two possible situations.

Perhaps solutions ϕ : [ 0 , 2 π ] × [ 0 , h ] R , ϕ ( θ , 0 ) = ϕ ( θ , h ) = 0 to the differential equation are unique for 0 < h < h 0 and non-unique for h 0 < h . This conclusion is plausible, if slightly uncomfortable. Alternatively, it is not obvious that on a cylinder of height greater than h 0 , there exists a vector field that attains the minimal energy. The compactness properties of our space have not been adequately explored to say if such a situation makes sense.

Computer approximations

We have seen that the partial differential equation Δ ϕ + sin ( 2 ϕ ) 2 = 0 does not yield to any of the standard analytical solving techniques. This motivates us to seek out numerical methods to use a computer to approximate the solution. MATLAB is an excellent environment in which to pursue this goal, as it has a powerful fsolve command which rapidly and accurately solves partial differential equations.

Polynomal interpolation and runge's phenomenon

When a computer “solves" a differential equation, it actually only assures that the differential equation is satisfied at a finite number of points. Between these points the computer uses polynomial interpolation to create a smooth solution. The question then arises as to how well we can trust the interpolation between these points. There is a classic example, called Runge's Phenomenon, of how interpolating using an evenly spaced grid leads to disastrous results. Suppose we would like to approximate the function

Questions & Answers

are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
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Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
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Renato
Berger describes sociologists as concerned with
Mueller Reply
what is hormones?
Wellington
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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