A relationship between three analytical approaches to nonlinear problems

0. Introduction A number of mathematical techniques have been applied for solving nonlinear problems. Energy Balance [1–3], homotopy analysis [4], variational iteration [5], homotopy perturbation [6,7], the max–min approach [8,9], frequency amplitude formulation [10] and perturbation methods [11] are the most prominent approaches that those have been employed for solving a large number of nonlinear equations. Researchers have used these methods for solving nonlinear problems in wave propagation [12], biomathematics [13] and structural dynamics [14]. It should be noted that both variational and Hamiltonian approaches were proposed by He over the past years. The variational approach is a powerful method for solving nonlinear equations, and a lot of researchers have employed it as a tool for solving nonlinear problems [15–18]. Similarly, theHamiltonian approachhas beenused for solving conservative oscillator problems, andmany authors have applied it to obtain the amplitude–frequency relationship of a number of conservative systems [19–29]. More recently, Beléndez et al. [30] have shown a relationship between the harmonic balance method (HBM) and the Hamiltonian approach (HA) in a first approximation with the cosine solution. Beléndez et al. [30] demonstrated HBM and HA to have the same solutions in a first approximation, and then they examined this worthwhile and valuable conclusion for generalized conservative oscillators and obtained a highly valuable result which is the resemblance of the Hamiltonian approach and harmonic balance method for a large number of oscillators. The aforementioned conclusion motivated us to investigate a relationship between other approaches in first-order and higher approximations. In the present study, the relationship between the variational approach (VA) and the Hamiltonian approach (HA) is exhibited and then, according to Ref. [30], the direct relationship between VA and HBM is concluded. Additionally, the relationship between HA and VA is analytically demonstrated for higher order approximations. ∗ Corresponding author. E-mail addresses: [email protected] (A. Yildirim), [email protected] (H. Askari), [email protected] (M.K. Yazdi), [email protected] (Y. Khan). 0893-9659/$ – see front matter© 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.aml.2012.02.001 1730 A. Yildirim et al. / Applied Mathematics Letters 25 (2012) 1729–1733 1. The relationship between VA, HA and HBM in the first approximation Consider conservative oscillators of a general kind with a single degree of freedom: ẍ + f (x) = 0 x(0) = A, ẋ(0) = 0. (1) Their variational principle can be easily established using the semi-inverse method: