Periodic analytic approximate solutions for the Mathieu equation

Física Teórica, Atómica y Óptic

A modified Lyapunov method and its applications to ODE

Producción CientíficaHere, 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.Ministerio de Ciencia e Innovación with funding from the European Union NextGenerationEU (PRTRC17.I1)Ministerio de Ciencia e Innovación project (PID2020-113406GB-I00

A variational modification of the Harmonic Balance method to obtain approximate Floquet exponents

Producción CientíficaWe propose a modification of a method based on Fourier analysis to obtainthe Floquet characteristic exponents for periodic homogeneous linear systems,which shows a high precision. This modification uses a variational principle tofind the correct Floquet exponents among the solutions of an algebraic equation.Once we have these Floquet exponents, we determine explicit approximatedsolutions. We test our results on systems for which exact solutions are knownto verify the accuracy of our method including one-dimensional periodicpotentials of interest in quantum physics. Using the equivalent linear system,we also study approximate solutions for homogeneous linear equations withperiodic coefficients.Ministerio de Ciencia e Innovación (proyect PID2020-113406GB-I0), with funding from the European Union NextGenerationEU and Consejería de Educación through QCAYLE (PRTRC17.I1)National University of Rosario (Argentina) grant ING19/i40