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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

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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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