Construction of An Approximate Periodic Solution to a Modified Lewis Equation

There have been many papers published on the construction of approximate periodic solutions of various types of non-linear differential equations. Many techniques have also been developed for obtaining approximate solutions. Among them are the method of Krylov-Bogoliubov-Mitropolsky, the harmonic balance, and the averaging method. We investigate the periodic solution of a modified Lewis equation \"{x}+$x^{3}$=?($1-|x|$)\'{x}, * where ? is a small and positive parameter, because of its strong cubic nonlinearity. Using an extension of the Mickens-Oyedeji method [1] developed by Cveticanin [2], we calculate the exact angular frequency for the equation \"{x}+$x^{3}$=0, ** and using the result of a first-order averaging method to calculate the approximate periodic solution of Eq. (*). Our result is compared with numerical calculations and there is a good agreement between our result and numerical calculations.