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Exploring a piece-wise-nonlinear method Hector Vazquez-Leal

Received: 28 June 2013 / Accepted: 8 August 2013 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2013

Abstract This work introduces the piece-wise-nonlinear method as a novel tool with high potential to find approximate representations of nonlinear dynamic problems. We present three cases study showing the strength of the method to generate highly accurate approximations for three nonlinear oscillators. Keywords

Curve fitting · Piece-wise linear

Mathematics Subject Classification (2000)

34L30

1 Introduction The curves fitting for experimental data and approximate solution of nonlinear differential equations are important problems in sciences because many physical phenomena require analytic representations for simulation purposes. Nowadays, the piece-wise-linear (PWL) method (Trejo-Guerra et al. 2012, 2013) is a widely used tool in engineering to solve a variety of problems. The main advantage of this method over other techniques is that it can represent virtually any curve using straight-line segments. Therefore, we propose a generalization of the aforementioned method denominated piece-wise-nonlinear (PWN) method. PWN represents curves by polynomials segments. A well-know drawback of PWL method is that its derivative is a discontinue curve. Nonetheless, PWN possesses a continuous derivative, converting this method into an attractive tool when high-resolution approximations, with smooth derivatives, are required. We will show the use of PWN method by approximating

Communicated by Cristina Turner. H. Vazquez-Leal (B) Electronic Instrumentation and Atmospheric Sciences School, University of Veracruz, Cto. Gonzalo Aguirre Beltrán S/N, Xalapa, VER 91000, México e-mail: [email protected]; [email protected]

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three nonlinear oscillators and comparing the results to other semi-analytic (Marinca and Herisanu 2011) or numerical methods (Enright et al. 1986; Fehlberg 1970; Tlelo-Cuautle et al. 2007; Tlelo-Cuautle and Munoz-Pacheco 2007). This paper is organized as follows. In Sect. 2, we introduce the basic concept of the PWN method. The solutions of three highly nonlinear problems are presented in Sect. 3. In Sect. 4, numerical simulations and a discussion about the results are provided. Finally, a brief conclusion is given in Sect. 5. 2 Basic concept of PWN We propose that a given curve or collection of points (t, y(t)) can be fitted by the following expression k v0 + i=1 vi |t − ti |n y(t) = , ti+1 > ti , (1) q w0 + i=1 wi |t − ti |n where v0 , v1 , v2 , . . . , vk and w0 , w1 , w2 , . . . , wk are unknowns to be determined by the PWN method, ti are the joint points, k is the number of segments, and n is the polynomial order. The basic process of PWN method can be described as: • First, we choose a set of sample points for the joint of polynomial segments t j = j,

j = 0, 1, 2, . . . , 2k + 1,

where t0 is the starting point and  represents the separation of sample points. • Second, sample points (2) are substi