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## IntechOpen Book Series Nonlinear Systems Volume 2

Walter E. Legnani has a degree in Physics and a PhD in the area of Physics from the University of Buenos Aires (Argentina), and a thesis developed at the University of Colorado and the National Centre for Atmospheric Research (USA). He also has a postdoc from the Department of Applied Mathematics and Theoretical Physics, Cambridge University and Reading University (UK). He is a full professor in signal and system analysis and has run several post-

graduate courses. He is also Director of the Signals and Image Processing Centre (Universidad Tecnologica Nacional-Facultad Regional Buenos Aires), a director of several postgraduate theses, a board scientific member of the Binational Centre Argentina Italia (CAIMAR), a former full professor at Favaloro University, a former secretary of science, technology and postgraduate studies (UTN), and a member of several international scientific societies and evaluation committees. He is a peer reviewer and guest editor of prestigious journals of high impact. www.researchgate.net/profile/Walter\_Legnani

Dr. Terry E. Moschandreou is a professor in applied mathematics at the University of Western Ontario in the School of Mathematical and Statistical Sciences where he has taught for several years. He received his PhD degree in Applied Mathematics from the University of Western Ontario in 1996. The greater part of his professional life has been spent at the University of Western Ontario and Fanshawe College in London, Ontario, Canada. Dr. Moschandreou is also

currently working for Goode Educational Services where he teaches students advanced calculus and linear algebra. For a short period, he worked at the National Technical University of Athens, Greece. Dr. Moschandreou is the author of several research articles in blood flow and oxygen transport in the microcirculation, general fluid dynamics, and theory of differential equations. Also, he has contributed to the field of finite element modeling of the upper airways in sleep apnea as well as surgical brain deformation modeling. More recently, he has been working with the partial differential equations of multiphase flow and level set methods as used in fluid dynamics.

#### **Editors of Volume 2: Walter E. Legnani**

Signals and Images Processing Centre, Universidad Tecnologica Nacional Facultad Regional Buenos Aires, Argentina

#### **Terry E. Moschandreou**

School of Mathematical and Statistical Sciences, University of Western Ontario, Canada

**Book Series Editor: Mahmut Reyhanoglu, PhD** University of North Carolina Asheville, Department of Engineering Asheville, North Carolina, USA

## Scope of the Series

Contents

**Section 1**

of Equations

*and Umar Audu Omesa*

Dynamic Visualization

*and Osvaldo A. Rosso*

Finite Delay

**Preface III**

Theoretical Aspects of Nonlinear Systems **1**

**Chapter 1 3**

**Chapter 2 23**

**Chapter 3 45**

**Chapter 4 57**

**Chapter 5 85**

**Chapter 6 105**

**Chapter 7 125**

A Review on Fractional Differential Equations and a Numerical

*by María I. Troparevsky, Silvia A. Seminara and Marcela A. Fabio*

Numerical Solutions to Some Families of Fractional Order

A Shamanskii-Like Accelerated Scheme for Nonlinear Systems

Modified Moving Least Squares Method for Two-Dimensional

*by Massoumeh Poura'bd Rokn Saraei and Mashaallah Matinfar*

Informational Time Causal Planes: A Tool for Chaotic Map

On the Stabilization of Infinite Dimensional Semilinear Systems

Existence, Regularity, and Compactness Properties in the *α*-Norm for Some Partial Functional Integrodifferential Equations with

Method to Solve Some Boundary Value Problems

Differential Equations by Laguerre Polynomials *by Adnan Khan, Kamal Shah and Danfeng Luo*

*by Ibrahim Mohammed Sulaiman, Mustafa Mamat*

Linear and Nonlinear Systems of Integral Equations

*by Felipe Olivares, Lindiane Souza, Walter Legnani*

*by El Hassan Zerrik and Abderrahman Ait Aadi*

*by Boubacar Diao, Khalil Ezzinbi and Mamadou Sy*

The series will be both on classical materials, such as nonlinear dynamics, stability and optimality, and more modern topics such as differential geometry, nonlinear control theory and applications in robotics. The books can be used as a reference and guide in the active literature in these fields.

Topics will broadly include, but are not limited to:


## Contents



Preface

Nonlinear dynamical systems have been used in the most diverse areas of scientific knowledge. Along with this, differential equations of fractional order, whose theoretical formulation continues to grow and whose applications are increasingly diverse, have attracted outstanding interest. This book gathers clear examples of these fields along with the most recent knowledge. The contributions are from diverse authors from a remarkable variety of countries and show a diversity in fields of applications. This fact confirms the abundant current interest in these topics in

Fractional calculus (FC) has roots that are deep in the theory of differential calculus. FC occurs in applications such as chaos and dynamical systems, modeling of memory-dependent theory, and complex media, for example in the study of porous media. Further applications are seen in the fields of digital circuits, heat diffusion,

The development of FC is due to contributions from mathematicians like Euler, Liouville, Riemann, and Letnikov. Due to limitations in classical methods as applied to dynamical systems, FC has proven to be an efficient tool for this stream of study. Existence theory and hyperbolic differential equations are ery important parts of the study of FC. One of the most important contributions of FC is the Caputo fractional derivative. FC involves both derivatives and integrals up to an arbitrary

The Grunwald–Letnikov definition of fractional derivatives and the Riemann– Liouville definition are also important and use the gamma function. These operators associated with fractional derivatives are global operators defining memory events. The part of FC used in this book is new and has many of the features of FC that are

In addition to fractional theory, *Nonlinear Systems*, which is divided into theoretical

In the theoretical section of the book, in the context of FC methods, Chapter 1, "A Review on Fractional Differential Equations and a Numerical Method to Solve Some

"Numerical Solutions to Some Families of Fractional Order Differential Equations."

sequence of iterates, which is a Newton iterate followed by several "cord" iterates.

Chapter 4 looks at the topic of "Modified Moving Least Squares Method for Two-Dimensional Linear and Nonlinear Systems of Integral Equations." In the moving

In addition, related to FC, numerical methods is the content of Chapter 2,

Chapter 3, "A Shamanskii-Like Accelerated Method for Systems of Nonlinear Equation," starts with an initial iterate and moves through an intermediate

the scientific and academic community.

robotic theory, and controller tuning.

order, which can be real or complex.

and applied sections, has the following contributions.

It is a generalization that encapsulates Newton's method.

Boundary Value Problems," is proposed.

important in the literature.
