Vasile Marinca - Optimal Auxiliary Functions Method for Nonlinear Dynamical Systems

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The analytical investigation of nonlinear dynamical systems is one of the most important but difficult tasks, and the present monograph consists of numerous examples from various domains of engineering and applied sciences. Any problem of motion in nonlinear dynamical systems can be assimilated only by working with differential equations which are applied in concrete examples. All examples presented here and treated from an analytical point of view are accompanied by comparisons with numerical results and sometimes with exact solutions, or with other known results in the literature.

The analytical technique presented in this book is an analytical approximation method applicable for highly nonlinear systems independent of the presence of small or large parameters into the governing equations or in the initial/boundary conditions.

A good knowledge of different approximate methods, especially the method of harmonic balance, the method of multiple scales, the optimal homotopy asymptotic method, or KrylovBogoliubov method, led to a better choice of the so-called auxiliary functions, which are decisive for the technique proposed here. In contrast to the other known techniques, our approach provides us with a simple way to control and adjust the convergence regions of solutions corresponding to nonlinear dynamical systems. The methodology proposed in this book is totally different from all other known analytical techniques, especially regarding the choice of the linear operators and optimal auxiliary functions, as well as the determination of the optimal convergence-control parameters.

The success of the present method is an important milestone in any field of exact sciences and techniques. Besides a wide field of applications, the proposed procedure can often be used to provide comparisons with the results obtained by other procedures.

The intended readers of this book include undergraduate students and graduate students doing projects on doctoral research in the field of nonlinear dynamical systems; researchers, engineers and university teachers will also find this book useful.

The work is based on the results obtained by the authors in the last years of research in the field of nonlinear dynamical systems. New results are illustrated by numerical examples. It is assumed that the reader already has minimal knowledge on how to differentiate and integrate elementary functions. Also, computer skills would be essential because computer simulation is a powerful tool for examination, confirmation and sometimes for refutation of the obtained results.

The book is divided into 31 chapters. The Chap. , the optimal auxiliary functions method is applied piecewise.

Here are treated models from various fields of engineering such as mechanical vibration, thermodynamics, fluid mechanics, astronomy, electrical machine and so on. Most models will demand some independent thinking and are selected in order to illustrate the main ideas of our procedure, which allowed the readers to understand the present material.