Ronald E. Mickens's Books

Truly Nonlinear Oscillations

Harmonic Balance, Parameter Expansions, Iteration, and Averaging Methods

By: Ronald E. Mickens

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Truly Nonlinear Oscillations

Harmonic Balance, Parameter Expansions, Iteration, and Averaging Methods

By: Ronald E. Mickens

This unique book provides a concise presentation of many of the fundamental strategies for calculating approximations to the oscillatory solutions of "truly nonlinear" (TNL) oscillator equations. The volume gives a general overview of the author's work on harmonic balance, iteration and combined linearization-averaging methods. However, full discussions are also presented on parameter expansion procedures and a first-order averaging technique for TNL oscillators. The calculational basis of each method is clarified by applying them to a set of standard TNL oscillator equations. This allows a direct comparison to be made among the various methods. The book is self-contained and therefore suitable for both classroom use and self-study by students and professionals who desire to learn, understand, and apply this technique to the field of nonlinear oscillations.

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Page Count

261

Publisher

World Scientific

Published Date

ISBN 10

981429165X

ISBN 13

9789814291651

Advances in the Applications of Nonstandard Finite Diffference Schemes

By: Ronald E. Mickens

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Advances in the Applications of Nonstandard Finite Diffference Schemes

By: Ronald E. Mickens

This volume provides a concise introduction to the methodology of nonstandard finite difference (NSFD) schemes construction and shows how they can be applied to the numerical integration of differential equations occurring in the natural, biomedical, and engineering sciences. These methods had their genesis in the work of Mickens in the 1990's and are now beginning to be widely studied and applied by other researchers. The importance of the book derives from its clear and direct explanation of NSFD in the introductory chapter along with a broad discussion of the future directions needed to advance the topic.

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668

Publisher

World Scientific

Published Date

ISBN 10

9812564047

ISBN 13

9789812564047

Applications of Nonstandard Finite Difference Schemes

By: Ronald E. Mickens

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Applications of Nonstandard Finite Difference Schemes

By: Ronald E. Mickens

The main purpose of this book is to provide a concise introduction to the methods and philosophy of constructing nonstandard finite difference schemes and illustrate how such techniques can be applied to several important problems. Chapter I gives an overview of the subject and summarizes previous work. Chapters 2 and 3 consider in detail the construction and numerical implementation of schemes for physical problems involving convection-diffusion-reaction equations, that arise in groundwater pollution and scattering of electromagnetic waves using Maxwell's equations. Chapter 4 examines certain mathematical issues related to the nonstandard discretization of competitive and cooperative models for ecology. The application chapters illustrate well the power of nonstandard methods. In particular, for the same accuracy as obtained by standard techniques, larger step sizes can be used. This volume will satisfy the needs of scientists, engineers, and mathematicians who wish to know how to construct nonstandard schemes and see how these are applied to obtain numerical solutions of the differential equations which arise in the study of nonlinear dynamical systems modeling important physical phenomena.

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Page Count

268

Publisher

World Scientific

Published Date

ISBN 10

981024133X

ISBN 13

9789810241339

Advances In The Applications Of Nonstandard Finite Difference Schemes

By: Ronald E Mickens

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Advances In The Applications Of Nonstandard Finite Difference Schemes

By: Ronald E Mickens

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665

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World Scientific

Published Date

ISBN 10

9814479861

ISBN 13

9789814479868

Nonstandard Finite Difference Models of Differential Equations

By: Ronald E. Mickens

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Nonstandard Finite Difference Models of Differential Equations

By: Ronald E. Mickens

This book provides a clear summary of the work of the author on the construction of nonstandard finite difference schemes for the numerical integration of differential equations. The major thrust of the book is to show that discrete models of differential equations exist such that the elementary types of numerical instabilities do not occur. A consequence of this result is that in general bigger step-sizes can often be used in actual calculations and/or finite difference schemes can be constructed that are conditionally stable in many instances whereas in using standard techniques no such schemes exist. The theoretical basis of this work is centered on the concepts of ?exact? and ?best? finite difference schemes. In addition, a set of rules is given for the discrete modeling of derivatives and nonlinear expressions that occur in differential equations. These rules often lead to a unique nonstandard finite difference model for a given differential equation.

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264

Publisher

World Scientific

Published Date

ISBN 10

9810214588

ISBN 13

9789810214586

Mathematical Methods For The Natural And Engineering Sciences (Second Edition)

By: Ronald E Mickens

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Mathematical Methods For The Natural And Engineering Sciences (Second Edition)

By: Ronald E Mickens

This second edition provides a broad range of methods and concepts required for the analysis and solution of equations which arise in the modeling of phenomena in the natural, engineering, and applied mathematical sciences. It may be used productively by both undergraduate and graduate students, as well as others who wish to learn, understand, and apply these techniques. Detailed discussions are also given for several topics that are not usually included in standard textbooks at this level of presentation: qualitative methods for differential equations, dimensionalization and scaling, elements of asymptotics, difference equations and several perturbation procedures. Further, this second edition includes several new topics covering functional equations, the Lambert-W function, nonstandard sets of periodic functions, and the method of dominant balance. Each chapter contains a large number of worked examples and provides references to the appropriate books and literature.

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Page Count

640

Publisher

World Scientific Publishing Company

Published Date

ISBN 10

9813202726

ISBN 13

9789813202726

Generalized Trigonometric and Hyperbolic Functions

By: Ronald E. Mickens

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Generalized Trigonometric and Hyperbolic Functions

By: Ronald E. Mickens

Generalized Trigonometric and Hyperbolic Functions highlights, to those in the area of generalized trigonometric functions, an alternative path to the creation and analysis of these classes of functions. Previous efforts have started with integral representations for the inverse generalized sine functions, followed by the construction of the associated cosine functions, and from this, various properties of the generalized trigonometric functions are derived. However, the results contained in this book are based on the application of both geometrical phase space and dynamical systems methodologies. Features Clear, direct construction of a new set of generalized trigonometric and hyperbolic functions Presentation of why x2+y2 = 1, and related expressions, may be interpreted in three distinct ways All the constructions, proofs, and derivations can be readily followed and understood by students, researchers, and professionals in the natural and mathematical sciences

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Page Count

171

Publisher

CRC Press

Published Date

ISBN 10

0429821085

ISBN 13

9780429821080

Mathematical Methods for the Natural and Engineering Sciences

By: Ronald E. Mickens

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Mathematical Methods for the Natural and Engineering Sciences

By: Ronald E. Mickens

This book provides a variety of methods required for the analysis and solution of equations which arise in the modeling of phenomena from the natural and engineering sciences. It can be used productively by both undergraduate and graduate students, as well as others who need to learn and understand these techniques. A detailed discussion is also presented for several topics that are usually not included in standard textbooks at this level: qualitative methods for differential equations, dimensionalization and scaling, elements of asymptotics, difference equations, and various perturbation methods. Each chapter contains a large number of worked examples and provides references to the appropriate literature.

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544

Publisher

World Scientific

Published Date

ISBN 10

9812387501

ISBN 13

9789812387509

Difference Equations, Second Edition

By: Ronald E. Mickens

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Difference Equations, Second Edition

By: Ronald E. Mickens

In recent years, the study of difference equations has acquired a new significance, due in large part to their use in the formulation and analysis of discrete-time systems, the numerical integration of differential equations by finite-difference schemes, and the study of deterministic chaos. The second edition of Difference Equations: Theory and Applications provides a thorough listing of all major theorems along with proofs. The text treats the case of first-order difference equations in detail, using both analytical and geometrical methods. Both ordinary and partial difference equations are considered, along with a variety of special nonlinear forms for which exact solutions can be determined. Numerous worked examples and problems allow readers to fully understand the material in the text. They also give possible generalization of the theorems and application models. The text's expanded coverage of application helps readers appreciate the benefits of using difference equations in the modeling and analysis of "realistic" problems from a broad range of fields. The second edition presents, analyzes, and discusses a large number of applications from the mathematical, biological, physical, and social sciences. Discussions on perturbation methods and difference equation models of differential equation models of differential equations represent contributions by the author to the research literature. Reference to original literature show how the elementary models of the book can be extended to more realistic situations. Difference Equations, Second Edition gives readers a background in discrete mathematics that many workers in science-oriented industries need as part of their general scientific knowledge. With its minimal mathematical background requirements of general algebra and calculus, this unique volume will be used extensively by students and professional in science and technology, in areas such as applied mathematics, control theory, population science, economics, and electronic circuits, especially discrete signal processing.

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461

Publisher

CRC Press

Published Date

ISBN 10

1000152898

ISBN 13

9781000152890

Difference Equations

Theory, Applications and Advanced Topics, Third Edition

By: Ronald E. Mickens

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Difference Equations

Theory, Applications and Advanced Topics, Third Edition

By: Ronald E. Mickens

Difference Equations: Theory, Applications and Advanced Topics, Third Edition provides a broad introduction to the mathematics of difference equations and some of their applications. Many worked examples illustrate how to calculate both exact and approximate solutions to special classes of difference equations. Along with adding several advanced to

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551

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CRC Press

Published Date

ISBN 10

1482230798

ISBN 13

9781482230796

Mathematical Modelling with Differential Equations

By: Ronald E. Mickens

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Mathematical Modelling with Differential Equations

By: Ronald E. Mickens

Mathematical Modelling with Differential Equations aims to introduce various strategies for modelling systems using differential equations. Some of these methodologies are elementary and quite direct to comprehend and apply while others are complex in nature and require thoughtful, deep contemplation. Many topics discussed in the chapter do not appear in any of the standard textbooks and this provides users an opportunity to consider a more general set of interesting systems that can be modelled. For example, the book investigates the evolution of a "toy universe," discusses why "alternate futures" exists in classical physics, constructs approximate solutions to the famous Thomas—Fermi equation using only algebra and elementary calculus, and examines the importance of "truly nonlinear" and oscillating systems. Features Introduces, defines, and illustrates the concept of "dynamic consistency" as the foundation of modelling. Can be used as the basis of an upper-level undergraduate course on general procedures for mathematical modelling using differential equations. Discusses the issue of dimensional analysis and continually demonstrates its value for both the construction and analysis of mathematical modelling.

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284

Publisher

CRC Press

Published Date

ISBN 10

1000585557

ISBN 13

9781000585551

Oscillations in Planar Dynamic Systems

By: Ronald E. Mickens

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Oscillations in Planar Dynamic Systems

By: Ronald E. Mickens

This book provides a concise presentation of the major techniques for determining analytic approximations to the solutions of planar oscillatory dynamic systems. These systems model many important phenomena in the sciences and engineering. In addition to the usual perturbation procedures, the book gives the details of when and how to correctly apply the method of harmonic balance for both first-order and higher-order calculations. This procedure is rarely given or discussed fully in standard textbooks. The basic philosophy of the book stresses how to initiate and complete the calculation of approximate solutions. This is done by a clear presentation of necessary background materials and by the working out of many examples.

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340

Publisher

World Scientific

Published Date

ISBN 10

9810222920

ISBN 13

9789810222925

An Introduction to Nonlinear Oscillations

By: Ronald E. Mickens

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An Introduction to Nonlinear Oscillations

By: Ronald E. Mickens

An introductory account of the equations describing nonlinear oscillations & the methods for solving them.

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234

Publisher

CUP Archive

Published Date

ISBN 10

0521222087

ISBN 13

9780521222082

Edward Bouchet

The First African-American Doctorate

By: Ronald E. Mickens

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Edward Bouchet

The First African-American Doctorate

By: Ronald E. Mickens

Edward A. Bouchet was the first African-American to receive the doctorate in any field of knowledge in the United States and that area was physics. He was granted the degree in 1876 from Yale University making him at that time one of the few persons to hold the physics doctorate from an American univeristy. Bouchet played a significant role in the education of African-Americans during the last quarter of the 19th century through his teaching and mentoring activities at the Institute for Colored Youth in Philadelphia, Pennsylvania. He was one among a small number of African-Americans who achieved advanced training and education within decades of the American civil war. These people provided direction, leadership, and role models for what eventually became the civil/human rights movements. The year 2001 marks the 125th celebration of his receiving the doctorate degree. This book gives a summary of his life and career.

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156

Publisher

World Scientific

Published Date

ISBN 10

9810249098

ISBN 13

9789810249090

Nonstandard Finite Difference Schemes: Methodology And Applications

By: Ronald E Mickens

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Nonstandard Finite Difference Schemes: Methodology And Applications

By: Ronald E Mickens

This second edition of Nonstandard Finite Difference Models of Differential Equations provides an update on the progress made in both the theory and application of the NSFD methodology during the past two and a half decades. In addition to discussing details related to the determination of the denominator functions and the nonlocal discrete representations of functions of dependent variables, we include many examples illustrating just how this should be done.Of real value to the reader is the inclusion of a chapter listing many exact difference schemes, and a chapter giving NSFD schemes from the research literature. The book emphasizes the critical roles played by the 'principle of dynamic consistency' and the use of sub-equations for the construction of valid NSFD discretizations of differential equations.

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Page Count

332

Publisher

World Scientific

Published Date

ISBN 10

981122255X

ISBN 13

9789811222559

Generalized Trigonometric and Hyperbolic Functions

By: Ronald E. Mickens

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Generalized Trigonometric and Hyperbolic Functions

By: Ronald E. Mickens

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Page Count

212

Publisher

CRC Press

Published Date

ISBN 10

0429821093

ISBN 13

9780429821097

Book's by Ronald E. Mickens

Truly Nonlinear Oscillations

Truly Nonlinear Oscillations

Advances in the Applications of Nonstandard Finite Diffference Schemes

Advances in the Applications of Nonstandard Finite Diffference Schemes

Applications of Nonstandard Finite Difference Schemes

Applications of Nonstandard Finite Difference Schemes

Advances In The Applications Of Nonstandard Finite Difference Schemes

Advances In The Applications Of Nonstandard Finite Difference Schemes

Nonstandard Finite Difference Models of Differential Equations

Nonstandard Finite Difference Models of Differential Equations

Mathematical Methods For The Natural And Engineering Sciences (Second Edition)

Mathematical Methods For The Natural And Engineering Sciences (Second Edition)

Generalized Trigonometric and Hyperbolic Functions

Generalized Trigonometric and Hyperbolic Functions

Mathematical Methods for the Natural and Engineering Sciences

Mathematical Methods for the Natural and Engineering Sciences

Difference Equations, Second Edition

Difference Equations, Second Edition

Difference Equations

Difference Equations

Mathematical Modelling with Differential Equations

Mathematical Modelling with Differential Equations

Oscillations in Planar Dynamic Systems

Oscillations in Planar Dynamic Systems

An Introduction to Nonlinear Oscillations

An Introduction to Nonlinear Oscillations

Edward Bouchet

Edward Bouchet

Nonstandard Finite Difference Schemes: Methodology And Applications

Nonstandard Finite Difference Schemes: Methodology And Applications

Generalized Trigonometric and Hyperbolic Functions

Generalized Trigonometric and Hyperbolic Functions

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