Solve this differential equation with periodic boundary conditions: (u'[x])^3 - u'''[x] = 0
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- Subject: [mg93808] Solve this differential equation with periodic boundary conditions: (u'[x])^3 - u'''[x] = 0
- From: "Charlie Brummitt" <cbrummitt at wisc.edu>
- Date: Wed, 26 Nov 2008 05:14:22 -0500 (EST)
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