19-22 October 2011
Hotel Roanoke, Roanoke VA
US/Eastern timezone
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An Approximation to the Periodic Solution of a Differential Equation of Abel

22 Oct 2011, 11:33
Crystal Ballroom B (Hotel Roanoke, Roanoke VA)

Crystal Ballroom B

Hotel Roanoke, Roanoke VA


Ronald Mickens (Clark Atlanta University)


The Abel equation, in canonical form, is y' = sin t - y^3 (*) and corresponds to the singular (epsilon -> 0) limit of the nonlinear, forced oscillator epsilon y" + y' + y^3 = sin t, epsilon -> 0. (**) Equation (*) has the property that it has a unique periodic solution defined on (-infty, infty). Further, as t increases, all solutions are attracted into the strip |y| < 1 and any two different solutions y_1(t) and y_2(t) satisfy the condition Lim [y_1(t) - y_2(t)] = 0, (***) t -> infty and for t negatively decreasing, each solution, except for the periodic solution, becomes unbounded.[1] Our purpose is to calculate an approximation to the unique periodic solution of Eq. (*) using the method of harmonic balance. We also determine an estimation for the blow-up time of the non-periodic solutions. [1] U. Elias, American Mathematical Monthly, vol.115, (Feb. 2008), pps. 147-149.

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