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 > >