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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

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Professor
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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
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Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
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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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