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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
>
>
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