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IFOHAM – a generalization of the Picard-Lindelöf iteration method. (IFOHAM – a generalization of the Picard-Lindelöff iteration method.) (English) Zbl 1454.65054

Pinelas, Sandra (ed.) et al., Differential and difference equations with applications. Selected papers based on the presentations at the fourth international conference, ICDDEA 2019, Lisbon, Portugal, July 1–5, 2019. Cham: Springer. Springer Proc. Math. Stat. 333, 497-516 (2020).
Summary: IFOHAM (Iterative First order HAM) is an iterative technique based on the first order equation of the Homotopy Analysis Method (HAM). It can be shown that IFOHAM generalizes Picard-Lindeloff’s iteration algorithm and can be used to solve nonlinear differential equations. In this work IFOHAM will be implemented in an symbolic computer environment and we will analyze and test its applicability to find series solutions of second order nonlinear differential equations with periodic solutions. In particular, we will show that the IFOHAM method is able to identify the fundamental frequencies as well as the amplitudes of such periodic solutions. Knowledge of these parameters is of particular importance in design and maintenance activities as it characterizes the oscillatory behavior of many real systems with nonlinear responses. The results of tests performed using the IFOHAM method will be compared with results available in the literature using the HAM as well as with results obtained using classical numerical techniques to solve differential equations.
For the entire collection see [Zbl 1445.34003].

MSC:

65L99 Numerical methods for ordinary differential equations
34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations
65L05 Numerical methods for initial value problems involving ordinary differential equations

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