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Re: Solve this differential equation with periodic boundary conditions:

• To: mathgroup at smc.vnet.net
• Subject: [mg93882] Re: Solve this differential equation with periodic boundary conditions:
• From: dh <dh at metrohm.com>
• Date: Thu, 27 Nov 2008 05:35:21 -0500 (EST)
• References: <ggj7hn$j75$1@smc.vnet.net>

Hi Charlie,

I doubt if there are any non-trivial solutions (non constant) to your 

problem.

Consider z=u', then z''=z^3. The solution will be a curve that everwhere 

  bends away from the x axis. This is incompatible with periodicity if 

the function needs to be continuous and differentiable.

hope this helps, Daniel



Charlie Brummitt wrote:

> Hello,

> I am trying to solve the nonlinear differential equation

> 

> (u'[x])^3 - u'''[x] = 0

> 

> with periodic boundary conditions

> 

> u[0] = u[L]

> 

> u'[0] = u'[L]

> 

> 

> (Note: The equation is (du/dx)^3 - (third derivative of u) = 0.)

> 

> 

> I am trying the following ansatz (which clearly satisfies the boundary

> conditions)

> 

> u[x] = Sum(n=1 to infinity)  a_n Sin[2 pi n x / L] + b_n Cos[2 pi x / L].

> 

> 

> When you plug this into the differential equation, it reduces to

> 

> Sum(n=1 to infinity) n^3 (-a_n Cos[2 pi n x / L] + b_n Sin[2 pi n x / L] -

> (Sum(n=1 to infinity) n (a_n Cos[2 pi n x / L] - b_n Sin[2 pi n x / L]))^3 =

> 0     (*)

> 

> By equation the coefficients of each of the "modes," we get nonlinear

> algebraic equations for the a_i's and b_i's. The question becomes: can we

> solve for finitely many a_i, b_i by truncating the solution? If so, is this

> solution reasonable, or do the a_i's and b_i's change appreciably if we

> include more and more a_i's and b_i's?

> 

> I have tried entering the left-hand side of equation (*) into Mathematica as

> a function of k, where the sums run from n = 1 to k (rather than n = 1 to

> infinity). I am having trouble equation coefficients of the various modes

> and solving for the a_i and b_i.

> 

> Can anyone help?

> 

> Much thanks,

> 

> Charlie Brummitt

> 

>