Numerical Analysis with Applications in Mechanics and Engineering

The books Fractional Calculus with Applications in Mechanics: Vibrations and Diffusion Processes and Fractional Calculus with Applications in Mechanics: Wave Propagation, Impact and Variational Principles contain various applications of fractional calculus to the fields of classical mechanics. Namely, the books study problems in fields such as viscoelasticity of fractional order, lateral vibrations of a rod of fractional order type, lateral vibrations of a rod positioned on fractional order viscoelastic foundations, diffusion-wave phenomena, heat conduction, wave propagation, forced oscillations of a body attached to a rod, impact and variational principles of a Hamiltonian type. The books will be useful for graduate students in mechanics and applied mathematics, as well as for researchers in these fields. Part 1 of this book presents an introduction to fractional calculus. Chapter 1 briefly gives definitions and notions that are needed later in the book and Chapter 2 presents definitions and some of the properties of fractional integrals and derivatives. Part 2 is the central part of the book. Chapter 3 presents the analysis of waves in fractional viscoelastic materials in infinite and finite spatial domains. In Chapter 4, the problem of oscillations of a translatory moving rigid body, attached to a heavy, or light viscoelastic rod of fractional order type, is studied in detail. In Chapter 5, the authors analyze a specific engineering problem of the impact of a viscoelastic rod against a rigid wall. Finally, in Chapter 6, some results for the optimization of a functional containing fractional derivatives of constant and variable order are presented.

Features a solid foundation of mathematical and computational tools to formulate and solve real-world ODE problems across various fields With a step-by-step approach to solving ordinary differential equations (ODEs), Differential Equation Analysis in Biomedical Science and Engineering: Ordinary Differential Equation Applications with R successfully applies computational techniques for solving real-world ODE problems that are found in a variety of fields, including chemistry, physics, biology, and physiology. The book provides readers with the necessary knowledge to reproduce and extend the computed numerical solutions and is a valuable resource for dealing with a broad class of linear and nonlinear ordinary differential equations. The author’s primary focus is on models expressed as systems of ODEs, which generally result by neglecting spatial effects so that the ODE dependent variables are uniform in space. Therefore, time is the independent variable in most applications of ODE systems. As such, the book emphasizes details of the numerical algorithms and how the solutions were computed. Featuring computer-based mathematical models for solving real-world problems in the biological and biomedical sciences and engineering, the book also includes: R routines to facilitate the immediate use of computation for solving differential equation problems without having to first learn the basic concepts of numerical analysis and programming for ODEs Models as systems of ODEs with explanations of the associated chemistry, physics, biology, and physiology as well as the algebraic equations used to calculate intermediate variables Numerical solutions of the presented model equations with a discussion of the important features of the solutions Aspects of general ODE computation through various biomolecular science and engineering applications Differential Equation Analysis in Biomedical Science and Engineering: Ordinary Differential Equation Applications with R is an excellent reference for researchers, scientists, clinicians, medical researchers, engineers, statisticians, epidemiologists, and pharmacokineticists who are interested in both clinical applications and interpretation of experimental data with mathematical models in order to efficiently solve the associated differential equations. The book is also useful as a textbook for graduate-level courses in mathematics, biomedical science and engineering, biology, biophysics, biochemistry, medicine, and engineering.

Features a solid foundation of mathematical and computational tools to formulate and solve real-world ODE problems across various fields With a step-by-step approach to solving ordinary differential equations (ODEs), Differential Equation Analysis in Biomedical Science and Engineering: Ordinary Differential Equation Applications with R successfully applies computational techniques for solving real-world ODE problems that are found in a variety of fields, including chemistry, physics, biology, and physiology. The book provides readers with the necessary knowledge to reproduce and extend the computed numerical solutions and is a valuable resource for dealing with a broad class of linear and nonlinear ordinary differential equations. The author’s primary focus is on models expressed as systems of ODEs, which generally result by neglecting spatial effects so that the ODE dependent variables are uniform in space. Therefore, time is the independent variable in most applications of ODE systems. As such, the book emphasizes details of the numerical algorithms and how the solutions were computed. Featuring computer-based mathematical models for solving real-world problems in the biological and biomedical sciences and engineering, the book also includes: R routines to facilitate the immediate use of computation for solving differential equation problems without having to first learn the basic concepts of numerical analysis and programming for ODEs Models as systems of ODEs with explanations of the associated chemistry, physics, biology, and physiology as well as the algebraic equations used to calculate intermediate variables Numerical solutions of the presented model equations with a discussion of the important features of the solutions Aspects of general ODE computation through various biomolecular science and engineering applications Differential Equation Analysis in Biomedical Science and Engineering: Ordinary Differential Equation Applications with R is an excellent reference for researchers, scientists, clinicians, medical researchers, engineers, statisticians, epidemiologists, and pharmacokineticists who are interested in both clinical applications and interpretation of experimental data with mathematical models in order to efficiently solve the associated differential equations. The book is also useful as a textbook for graduate-level courses in mathematics, biomedical science and engineering, biology, biophysics, biochemistry, medicine, and engineering.

This book contains mathematical preliminaries in which basic definitions of fractional derivatives and spaces are presented. The central part of the book contains various applications in classical mechanics including fields such as: viscoelasticity, heat conduction, wave propagation and variational Hamilton–type principles. Mathematical rigor will be observed in the applications. The authors provide some problems formulated in the classical setting and some in the distributional setting. The solutions to these problems are presented in analytical form and these solutions are then analyzed numerically. Theorems on the existence of solutions will be presented for all examples discussed. In using various constitutive equations the restrictions following from the second law of thermodynamics will be implemented. Finally, the physical implications of obtained solutions will be discussed in detail.

This book develops a new vision in geomechanics which will be of interest to researchers and engineers. It begins with the key theoretical features of dissipative structures induced by elementary contact friction within geomaterials in slow motion, their multi-scale expression in key tensor relations and associated features including strain localization and shear banding.

Computational contact mechanics is a broad topic which brings together algorithmic, geometrical, optimization and numerical aspects for a robust, fast and accurate treatment of contact problems. This book covers all the basic ingredients of contact and computational contact mechanics: from efficient contact detection algorithms and classical optimization methods to new developments in contact kinematics and resolution schemes for both sequential and parallel computer architectures. The book is self-contained and intended for people working on the implementation and improvement of contact algorithms in a finite element software. Using a new tensor algebra, the authors introduce some original notions in contact kinematics and extend the classical formulation of contact elements. Some classical and new resolution methods for contact problems and associated ready-to-implement expressions are provided. Contents: 1. Introduction to Computational Contact. 2. Geometry in Contact Mechanics. 3. Contact Detection. 4. Formulation of Contact Problems. 5. Numerical Procedures. 6. Numerical Examples. About the Authors Vladislav A. Yastrebov is a postdoctoral-fellow in Computational Solid Mechanics at MINES ParisTech in France. His work in computational contact mechanics was recognized by the CSMA award and by the Prix Paul Caseau of the French Academy of Technology and Electricit? de France.

Driven by advances in computer technology, engineering analysis has developed rapidly and extensively in recent times; Visualization of Fields and Applications in Engineering presents the basic techniques for tensor field visualization and mapping of engineering data. Focusing on the fundamental aspects of post processing databases and applications outputs, the author explores existing theories and their integration in tensor field visualization and analysis. The subject covers fundamental theories through to integrated, multi-disciplinary technologies with practical applications in engineering, computer /general sciences. Visualization of Fields and Applications in Engineering is suitable for academic use and to serve as a source of reference. It will appeal to those who work in the engineering and science professions or in pursuit of academic training/ research. Offers a unique engineering approach to basic techniques for tensor field visualization and mapping Collates together material currently disseminated throughout the literature into one accessible point of reference Presents examples with applications beyond and across many disciplines.

An authoritative text that presents the current problems, theories, and applications of mathematical analysis research Mathematical Analysis and Applications: Selected Topics offers the theories, methods, and applications of a variety of targeted topics including: operator theory, approximation theory, fixed point theory, stability theory, minimization problems, many-body wave scattering problems, Basel problem, Corona problem, inequalities, generalized normed spaces, variations of functions and sequences, analytic generalizations of the Catalan, Fuss, and Fuss–Catalan Numbers, asymptotically developable functions, convex functions, Gaussian processes, image analysis, and spectral analysis and spectral synthesis. The authors—a noted team of international researchers in the field— highlight the basic developments for each topic presented and explore the most recent advances made in their area of study. The text is presented in such a way that enables the reader to follow subsequent studies in a burgeoning field of research. This important text: Presents a wide-range of important topics having current research importance and interdisciplinary applications such as game theory, image processing, creation of materials with a desired refraction coefficient, etc. Contains chapters written by a group of esteemed researchers in mathematical analysis Includes problems and research questions in order to enhance understanding of the information provided Offers references that help readers advance to further study Written for researchers, graduate students, educators, and practitioners with an interest in mathematical analysis, Mathematical Analysis and Applications: Selected Topics includes the most recent research from a range of mathematical fields. Michael Ruzhansky, Ph.D., is Professor in the Department of Mathematics at Imperial College London, UK. Dr. Ruzhansky was awarded the Ferran Sunyer I Balaguer Prize in 2014. Hemen Dutta, Ph.D., is Senior Assistant Professor of Mathematics at Gauhati University, India. Ravi P. Agarwal, Ph.D., is Professor and Chair of the Department of Mathematics at Texas A&M University-Kingsville, Kingsville, USA.

H andbook and reference guide for students and practitioners of statistical regression-based analyses in R   Handbook of Regression Analysis  with Applications in R, Second Edition  is a comprehensive and up-to-date guide to conducting complex regressions in the R statistical programming language. The authors’ thorough treatment of “classical” regression analysis in the first edition is complemented here by their discussion of more advanced topics including time-to-event survival data and longitudinal and clustered data.  The book further pays particular attention to methods that have become prominent in the last few decades as increasingly large data sets have made new techniques and applications possible. These include:  Regularization methods Smoothing methods Tree-based methods In the new edition of the  Handbook , the data analyst’s toolkit is explored and expanded. Examples are drawn from a wide variety of real-life applications and data sets. All the utilized R code and data are available via an author-maintained website.  Of interest to undergraduate and graduate students taking courses in statistics and regression, the  Handbook of Regression Analysis  will also be invaluable to practicing data scientists and statisticians.

A comprehensive introduction to social network analysis that hones in on basic centrality measures, social links, subgroup analysis, data sources, and more Written by military, industry, and business professionals, this book introduces readers to social network analysis, the new and emerging topic that has recently become of significant use for industry, management, law enforcement, and military practitioners for identifying both vulnerabilities and opportunities in collaborative networked organizations. Focusing on models and methods for the analysis of organizational risk, Social Network Analysis with Applications provides easily accessible, yet comprehensive coverage of network basics, centrality measures, social link theory, subgroup analysis, relational algebra, data sources, and more. Examples of mathematical calculations and formulas for social network measures are also included. Along with practice problems and exercises, this easily accessible book covers: The basic concepts of networks, nodes, links, adjacency matrices, and graphs Mathematical calculations and exercises for centrality, the basic measures of degree, betweenness, closeness, and eigenvector centralities Graph-level measures, with a special focus on both the visual and numerical analysis of networks Matrix algebra, outlining basic concepts such as matrix addition, subtraction, multiplication, and transpose and inverse calculations in linear algebra that are useful for developing networks from relational data Meta-networks and relational algebra, social links, diffusion through networks, subgroup analysis, and more An excellent resource for practitioners in industry, management, law enforcement, and military intelligence who wish to learn and apply social network analysis to their respective fields, Social Network Analysis with Applications is also an ideal text for upper-level undergraduate and graduate level courses and workshops on the subject.