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Introduction

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The Optimal Homotopy Asymptotic Method

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Abstract

A nonlinear dynamical system is not a tidy subject, but is vital in engineering practice. “Nonlinear systems are surely the rule, not the exception, not only in research, but also in the engineering world” [1]. The world around us is inherently nonlinear and the most nonlinear phenomena are models of our real-life problems. The most successful class of rules for describing practice phenomena are differential equations. All major theories of engineering are stated in terms of differential equations. Differential equations lie at the basis of scientific mathematical philosophy [2] of our scientific world. This scientific philosophy began with the discovery of the calculus by Newton and Leibnitz and continues to the present days. The first two hundred years of this scientific philosophy, from Newton and Euler, through to Hamilton and Maxwell, produced many stunning successes in formulating the “rules of the world”. Nonlinear differential equations are widely used as models to describe complex physical phenomena in various fields of science as fluid dynamics, solid state physics, heat transfer, vibrations, electrical machines, chemical kinetics and so on. The evolution of some systems can be studied by means of the linear differential equations, but majority of the physics systems do not lead to linear differential equations but to nonlinear differential equations. A system of linear differential equations is one for which the dependent quantities or variables only appear to the first power. If terms are present which involve products of the dependent variables, or other powers or other mathematical forms, the system is said to be nonlinear. A linear dynamical system is one which the dynamic rule is linearly proportional to the system variables. Linear systems can be analyzed by breaking the problem into pieces and then adding these pieces together to build a complete solution. This property of linear differential equations is called the principle of superposition. It is the cornerstone from which all linear theory is built. But, unfortunately, the solutions of a nonlinear equation cannot usually be added together to build a larger solution. The principle of superposition fails to hold for nonlinear systems. A general feature of all nonlinear dynamical equations including nonlinear differential equations is the “breakdown” of linear additivity or superposition. Because of this breakdown, many mathematical techniques (among which Laplace transform, Fourier analysis etc) for solving linear differential equations no longer work or are useful for attempting to solve nonlinear differential equations [3].

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Authors and Affiliations

  1. Politehnica University of Timisoara, Timisoara, Romania

    Vasile Marinca & Nicolae Herisanu

  2. Center for Advanced and Fundamental Technical Research, Romanian Academy, Timisoara Branch, Timisoara, Romania

    Vasile Marinca & Nicolae Herisanu

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  1. Vasile Marinca
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Marinca, V., Herisanu, N. (2015). Introduction. In: The Optimal Homotopy Asymptotic Method. Springer, Cham. https://doi.org/10.1007/978-3-319-15374-2_1

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  • DOI: https://doi.org/10.1007/978-3-319-15374-2_1

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-15373-5

  • Online ISBN: 978-3-319-15374-2

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