A modified Lyapunov method and its applications to ODE |
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Authors: | Manuel Gadella Luis Pedro Lara |
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Affiliation: | 1. Departamento de Física Teórica, Atómica y Optica and IMUVA, Facultad de Ciencias, Universidad de Valladolid, Paseo Belén 7, Valladolid, 47011 Spain;2. Instituto de Física Rosario, CONICET-UNR, Bv. 27 de Febrero, Rosario, S2000EKF Santa Fe, Argentina Departamento de Sistemas, Universidad del Centro Educativo Latinoamericano, Av. Pellegrini 1332, Rosario, S2000 Argentina |
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Abstract: | Here, we propose a method to obtain local analytic approximate solutions of ordinary differential equations with variable coefficients, or even some nonlinear equations, inspired in the Lyapunov method, where instead of polynomial approximations, we use truncated Fourier series with variable coefficients as approximate solutions. In the case of equations admitting periodic solutions, an averaging over the coefficients gives global solutions. We show that, under some restrictive condition, the method is equivalent to the Picard-Lindelöf method. After some numerical experiments showing the efficiency of the method, we apply it to equations of interest in physics, in which we show that our method possesses an excellent precision even with low iterations. |
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