Preface

**Section 2**

Disease Prediction

*and Ricardo L. Armentano*

Recent Applications in Nonlinear Systems **147**

**Chapter 8 149**

**Chapter 9 167**

**Chapter 10 181**

**Chapter 11 205**

**Chapter 12 219**

**Chapter 13 241**

**Chapter 14 255**

Three-Dimensional Shell in Disorders of Human Vascular System

*by Vladimir Stepanovich Korolev, Elena Nikolaevna Polyakhova*

Nonlinear Friction Model for Passive Suspension System

Electrostatically Driven MEMS Resonator: Pull-in Behavior

Nonlinear Oxygen Transport with Poiseuille Hemodynamic

*by Terry E. Moschandreou and Keith C. Afas*

Nonlinear Resonances in 3D Printed Structures

*by Parag Chatterjee, Leandro J. Cymberknop*

Mathematical Modeling and Well-Posedness of

Problems of Control Motion of Solar Sail Spacecraft

*by Vishakha Jadaun and Nitin Raja Singh*

in the Photogravitational Fields

*and Irina Yurievna Pototskaya*

Identification and Effectiveness

*by Ali I. H. Al-Zughaibi*

and Non-linear Phenomena

Flow in a Micro-Channel

*by Barun Pratiher*

**II**

Nonlinear Systems in Healthcare towards Intelligent

*by Astitva Tripathi and Anil K. Bajaj*

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 the scientific and academic community.

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, robotic theory, and controller tuning.

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 order, which can be real or complex.

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 important in the literature.

In addition to fractional theory, *Nonlinear Systems*, which is divided into theoretical and applied sections, has the following contributions.

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 Boundary Value Problems," is proposed.

In addition, related to FC, numerical methods is the content of Chapter 2, "Numerical Solutions to Some Families of Fractional Order Differential Equations."

Chapter 3, "A Shamanskii-Like Accelerated Method for Systems of Nonlinear Equation," starts with an initial iterate and moves through an intermediate sequence of iterates, which is a Newton iterate followed by several "cord" iterates. It is a generalization that encapsulates Newton's method.

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

least squares method an approximation value can be expressed as a linear combination of shape functions and known function values. The moving least squares method reconstructs continuous functions from a set of non-organized point samples of a biased weighted least squares indicator.

It is not a point of arrival, only a starting point to be enriched with future developments that show the advances in the production of knowledge in these

**Walter E. Legnani**

**Terry E. Moschandreou**

University of Western, Ontario, Canada

Argentina

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

School of Mathematical and Statistical Sciences,

exciting fields of scientific work.

**V**

In the light of chaotic systems, a bi-dimensional and a causal plane are defined, in which different dynamical regimes appear very clear and give information on the process involved. This is the subject of Chapter 5, "Informational Time Causal Planes: A Tool for Chaotic Maps Dynamics Visualization."

Stability is an important area of nonlinear systems and is considered in Chapter 6, "On the Stabilization for Infinite Dimensional Semi-Linear Systems."

For Chapter 7, "Existence, Regularity and Compactness Properties in the Alpha-Norm for Some Partial Functional Integro-Differential Equations with Delay" is considered. The objective of this chapter is to study the alpha-norm, existence, continuity dependence in initial data, regularity, and compactness of solutions of mild solution for some semi-linear partial functional integro-differential equations in abstract Banach space.

In the applied section, "Recent Applications in Nonlinear Systems," Chapter 1 covers "Nonlinear Resonances in 3D Printed Structures." Here, nonlinear resonators are studied and the nonlinear behavior of such structures is analyzed. Computational methods are employed for structural design and the case of one to two internal resonances of hyperelastic materials.

The importance of health care cannot be understated and Chapter 2 considers the "Nonlinear Systems in Healthcare Towards Intelligent Disease Prediction." Here predictive analytics are considered with examples of intelligent systems toward disease prediction.

Shell structures are examined in Chapter 3, "Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human Vascular System" where a shell structure is a general three-dimensional structure that is elongated in two directions and thinned out in the other direction. In the human vascular system, the human anatomy develops cyst-related diseases with progressive severity. These cysts can be modeled as shells, albeit in higher dimensions.

Further applications are found in Chapter 4, "Features of Optimal Control in Photo-Gravitational Fields," and Chapter 5, "Nonlinear Friction Model Identification and Effectiveness." Chapter 6 of this section, "Electrostatically Driven MEMS Resonator: Pull-In Behavior and Nonlinear Phenomena," presents an interesting application to the stability and bifurcation analysis of highly nonlinear, electrically driven micro-electro-mechanical systems (MEMS).

Finally, in Chapter 7, "An Approximate Analytical Solution for Non-Linear Oxygen with Poiseuille Hemodynamic Flow in a Micro-Channel," an important nonlinear model of hemodynamic flow with oxygen transport is considered.

We hope that this book will be useful material for both graduate students and researchers in general.

It is not a point of arrival, only a starting point to be enriched with future developments that show the advances in the production of knowledge in these exciting fields of scientific work.

> **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

least squares method an approximation value can be expressed as a linear combination of shape functions and known function values. The moving least squares method reconstructs continuous functions from a set of non-organized point sam-

In the light of chaotic systems, a bi-dimensional and a causal plane are defined, in which different dynamical regimes appear very clear and give information on the process involved. This is the subject of Chapter 5, "Informational Time Causal

Stability is an important area of nonlinear systems and is considered in Chapter 6,

For Chapter 7, "Existence, Regularity and Compactness Properties in the Alpha-Norm for Some Partial Functional Integro-Differential Equations with Delay" is considered. The objective of this chapter is to study the alpha-norm, existence, continuity dependence in initial data, regularity, and compactness of solutions of mild solution for some semi-linear partial functional integro-differential equations

In the applied section, "Recent Applications in Nonlinear Systems," Chapter 1 covers "Nonlinear Resonances in 3D Printed Structures." Here, nonlinear resonators are studied and the nonlinear behavior of such structures is analyzed. Computational methods are employed for structural design and the case of one to two

The importance of health care cannot be understated and Chapter 2 considers the "Nonlinear Systems in Healthcare Towards Intelligent Disease Prediction." Here predictive analytics are considered with examples of intelligent systems toward

Shell structures are examined in Chapter 3, "Mathematical Modeling and Well-Posedness of Three-Dimensional Shell in Disorders of Human Vascular System" where a shell structure is a general three-dimensional structure that is elongated in two directions and thinned out in the other direction. In the human vascular system, the human anatomy develops cyst-related diseases with progressive severity.

Further applications are found in Chapter 4, "Features of Optimal Control in Photo-Gravitational Fields," and Chapter 5, "Nonlinear Friction Model Identification and Effectiveness." Chapter 6 of this section, "Electrostatically Driven MEMS Resonator: Pull-In Behavior and Nonlinear Phenomena," presents an interesting application to the stability and bifurcation analysis of highly nonlinear, electrically driven

Finally, in Chapter 7, "An Approximate Analytical Solution for Non-Linear Oxygen with Poiseuille Hemodynamic Flow in a Micro-Channel," an important nonlinear

We hope that this book will be useful material for both graduate students and

These cysts can be modeled as shells, albeit in higher dimensions.

model of hemodynamic flow with oxygen transport is considered.

"On the Stabilization for Infinite Dimensional Semi-Linear Systems."

ples of a biased weighted least squares indicator.

in abstract Banach space.

disease prediction.

researchers in general.

**IV**

internal resonances of hyperelastic materials.

micro-electro-mechanical systems (MEMS).

Planes: A Tool for Chaotic Maps Dynamics Visualization."

Section 1

Theoretical Aspects

of Nonlinear Systems

**1**

Section 1
