Chapman & Hall/CRC Applied Mathematics & Nonlinear Science – serie

More than ever before, complicated mathematical procedures are integral to the success and advancement of technology, engineering, and even industrial production. Knowledge of and experience with these procedures is therefore vital to present and future scientists, engineers and technologists.Mathematical Methods in Physics and Engineering with Mathematica clearly demonstrates how to solve difficult practical problems involving ordinary and partial differential equations and boundary value problems using the software package Mathematica (4.x). Avoiding mathematical theorems and numerical methods-and requiring no prior experience with the software-the author helps readers learn by doing with step-by-step recipes useful in both new and classical applications.Mathematica and FORTRAN codes used in the book's examples and exercises are available for download from the Internet. The author's clear explanation of each Mathematica command along with a wealth of examples and exercises make Mathematical Methods in Physics and Engineering with Mathematica an outstanding choice both as a reference for practical problem solving and as a quick-start guide to using a leading mathematics software package.

Nonlinear Optimal Control Theory presents a deep, wide-ranging introduction to the mathematical theory of the optimal control of processes governed by ordinary differential equations and certain types of differential equations with memory. Many examples illustrate the mathematical issues that need to be addressed when using optimal control techniques in diverse areas.Drawing on classroom-tested material from Purdue University and North Carolina State University, the book gives a unified account of bounded state problems governed by ordinary, integrodifferential, and delay systems. It also discusses Hamilton-Jacobi theory. By providing a sufficient and rigorous treatment of finite dimensional control problems, the book equips readers with the foundation to deal with other types of control problems, such as those governed by stochastic differential equations, partial differential equations, and differential games.

Green’s Functions and Linear Differential Equations: Theory, Applications, and Computation presents a variety of methods to solve linear ordinary differential equations (ODEs) and partial differential equations (PDEs). The text provides a sufficient theoretical basis to understand Green’s function method, which is used to solve initial and boundary value problems involving linear ODEs and PDEs. It also contains a large number of examples and exercises from diverse areas of mathematics, applied science, and engineering.Taking a direct approach, the book first unravels the mystery of the Dirac delta function and then explains its relationship to Green’s functions. The remainder of the text explores the development of Green’s functions and their use in solving linear ODEs and PDEs. The author discusses how to apply various approaches to solve initial and boundary value problems, including classical and general variations of parameters, Wronskian method, Bernoulli’s separation method, integral transform method, method of images, conformal mapping method, and interpolation method. He also covers applications of Green’s functions, including spherical and surface harmonics.Filled with worked examples and exercises, this robust, self-contained text fully explains the differential equation problems, includes graphical representations where necessary, and provides relevant background material. It is mathematically rigorous yet accessible enough for readers to grasp the beauty and power of the subject.

Most existing books on evolution equations tend either to cover a particular class of equations in too much depth for beginners or focus on a very specific research direction. Thus, the field can be daunting for newcomers to the field who need access to preliminary material and behind-the-scenes detail. Taking an applications-oriented, conversational approach, Discovering Evolution Equations with Applications: Volume 2-Stochastic Equations provides an introductory understanding of stochastic evolution equations. The text begins with hands-on introductions to the essentials of real and stochastic analysis. It then develops the theory for homogenous one-dimensional stochastic ordinary differential equations (ODEs) and extends the theory to systems of homogenous linear stochastic ODEs. The next several chapters focus on abstract homogenous linear, nonhomogenous linear, and semi-linear stochastic evolution equations. The author also addresses the case in which the forcing term is a functional before explaining Sobolev-type stochastic evolution equations. The last chapter discusses several topics of active research.Each chapter starts with examples of various models. The author points out the similarities of the models, develops the theory involved, and then revisits the examples to reinforce the theoretical ideas in a concrete setting. He incorporates a substantial collection of questions and exercises throughout the text and provides two layers of hints for selected exercises at the end of each chapter.Suitable for readers unfamiliar with analysis even at the undergraduate level, this book offers an engaging and accessible account of core theoretical results of stochastic evolution equations in a way that gradually builds readers’ intuition.

A defining feature of nonlinear hyperbolic equations is the occurrence of shock waves. While the popular shock-capturing methods are easy to implement, shock-fitting techniques provide the most accurate results. A Shock-Fitting Primer presents the proper numerical treatment of shock waves and other discontinuities. The book begins by recounting the events that lead to our understanding of the theory of shock waves and the early developments related to their computation. After presenting the main shock-fitting ideas in the context of a simple scalar equation, the author applies Colombeau’s theory of generalized functions to the Euler equations to demonstrate how the theory recovers well-known results and to provide an in-depth understanding of the nature of jump conditions. He then extends the shock-fitting concepts previously discussed to the one-dimensional and quasi-one-dimensional Euler equations as well as two-dimensional flows. The final chapter explores existing and future developments in shock-fitting methods within the framework of unstructured grid methods.Throughout the text, the techniques developed are illustrated with numerous examples of varying complexity. On the accompanying downloadable resources, MATLAB® codes serve as concrete examples of how to implement the ideas discussed in the book.

Modeling and Control in Vibrational and Structural Dynamics: A Differential Geometric Approach describes the control behavior of mechanical objects, such as wave equations, plates, and shells. It shows how the differential geometric approach is used when the coefficients of partial differential equations (PDEs) are variable in space (waves/plates), when the PDEs themselves are defined on curved surfaces (shells), and when the systems have quasilinear principal parts.To make the book self-contained, the author starts with the necessary background on Riemannian geometry. He then describes differential geometric energy methods that are generalizations of the classical energy methods of the 1980s. He illustrates how a basic computational technique can enable multiplier schemes for controls and provide mathematical models for shells in the form of free coordinates. The author also examines the quasilinearity of models for nonlinear materials, the dependence of controllability/stabilization on variable coefficients and equilibria, and the use of curvature theory to check assumptions.With numerous examples and exercises throughout, this book presents a complete and up-to-date account of many important advances in the modeling and control of vibrational and structural dynamics.

Discovering Evolution Equations with Applications: Volume 1-Deterministic Equations provides an engaging, accessible account of core theoretical results of evolution equations in a way that gradually builds intuition and culminates in exploring active research. It gives nonspecialists, even those with minimal prior exposure to analysis, the foundation to understand what evolution equations are and how to work with them in various areas of practice.After presenting the essentials of analysis, the book discusses homogenous finite-dimensional ordinary differential equations. Subsequent chapters then focus on linear homogenous abstract, nonhomogenous linear, semi-linear, functional, Sobolev-type, neutral, delay, and nonlinear evolution equations. The final two chapters explore research topics, including nonlocal evolution equations. For each class of equations, the author develops a core of theoretical results concerning the existence and uniqueness of solutions under various growth and compactness assumptions, continuous dependence upon initial data and parameters, convergence results regarding the initial data, and elementary stability results.By taking an applications-oriented approach, this self-contained, conversational-style book motivates readers to fully grasp the mathematical details of studying evolution equations. It prepares newcomers to successfully navigate further research in the field.

Modern Tools to Perform Numerical DifferentiationThe original direct differential quadrature (DQ) method has been known to fail for problems with strong nonlinearity and material discontinuity as well as for problems involving singularity, irregularity, and multiple scales. But now researchers in applied mathematics, computational mechanics, and engineering have developed a range of innovative DQ-based methods to overcome these shortcomings. Advanced Differential Quadrature Methods explores new DQ methods and uses these methods to solve problems beyond the capabilities of the direct DQ method.After a basic introduction to the direct DQ method, the book presents a number of DQ methods, including complex DQ, triangular DQ, multi-scale DQ, variable order DQ, multi-domain DQ, and localized DQ. It also provides a mathematical compendium that summarizes Gauss elimination, the Runge–Kutta method, complex analysis, and more. The final chapter contains three codes written in the FORTRAN language, enabling readers to quickly acquire hands-on experience with DQ methods.Focusing on leading-edge DQ methods, this book helps readers understand the majority of journal papers on the subject. In addition to gaining insight into the dynamic changes that have recently occurred in the field, readers will quickly master the use of DQ methods to solve complex problems.

Modern Tools to Perform Numerical DifferentiationThe original direct differential quadrature (DQ) method has been known to fail for problems with strong nonlinearity and material discontinuity as well as for problems involving singularity, irregularity, and multiple scales. But now researchers in applied mathematics, computational mechanics, and engineering have developed a range of innovative DQ-based methods to overcome these shortcomings. Advanced Differential Quadrature Methods explores new DQ methods and uses these methods to solve problems beyond the capabilities of the direct DQ method.After a basic introduction to the direct DQ method, the book presents a number of DQ methods, including complex DQ, triangular DQ, multi-scale DQ, variable order DQ, multi-domain DQ, and localized DQ. It also provides a mathematical compendium that summarizes Gauss elimination, the Runge–Kutta method, complex analysis, and more. The final chapter contains three codes written in the FORTRAN language, enabling readers to quickly acquire hands-on experience with DQ methods.Focusing on leading-edge DQ methods, this book helps readers understand the majority of journal papers on the subject. In addition to gaining insight into the dynamic changes that have recently occurred in the field, readers will quickly master the use of DQ methods to solve complex problems.

Discovering Evolution Equations with Applications: Volume 1-Deterministic Equations provides an engaging, accessible account of core theoretical results of evolution equations in a way that gradually builds intuition and culminates in exploring active research. It gives nonspecialists, even those with minimal prior exposure to analysis, the foundation to understand what evolution equations are and how to work with them in various areas of practice.After presenting the essentials of analysis, the book discusses homogenous finite-dimensional ordinary differential equations. Subsequent chapters then focus on linear homogenous abstract, nonhomogenous linear, semi-linear, functional, Sobolev-type, neutral, delay, and nonlinear evolution equations. The final two chapters explore research topics, including nonlocal evolution equations. For each class of equations, the author develops a core of theoretical results concerning the existence and uniqueness of solutions under various growth and compactness assumptions, continuous dependence upon initial data and parameters, convergence results regarding the initial data, and elementary stability results.By taking an applications-oriented approach, this self-contained, conversational-style book motivates readers to fully grasp the mathematical details of studying evolution equations. It prepares newcomers to successfully navigate further research in the field.

Most existing books on evolution equations tend either to cover a particular class of equations in too much depth for beginners or focus on a very specific research direction. Thus, the field can be daunting for newcomers to the field who need access to preliminary material and behind-the-scenes detail. Taking an applications-oriented, conversational approach, Discovering Evolution Equations with Applications: Volume 2-Stochastic Equations provides an introductory understanding of stochastic evolution equations. The text begins with hands-on introductions to the essentials of real and stochastic analysis. It then develops the theory for homogenous one-dimensional stochastic ordinary differential equations (ODEs) and extends the theory to systems of homogenous linear stochastic ODEs. The next several chapters focus on abstract homogenous linear, nonhomogenous linear, and semi-linear stochastic evolution equations. The author also addresses the case in which the forcing term is a functional before explaining Sobolev-type stochastic evolution equations. The last chapter discusses several topics of active research.Each chapter starts with examples of various models. The author points out the similarities of the models, develops the theory involved, and then revisits the examples to reinforce the theoretical ideas in a concrete setting. He incorporates a substantial collection of questions and exercises throughout the text and provides two layers of hints for selected exercises at the end of each chapter.Suitable for readers unfamiliar with analysis even at the undergraduate level, this book offers an engaging and accessible account of core theoretical results of stochastic evolution equations in a way that gradually builds readers’ intuition.

A defining feature of nonlinear hyperbolic equations is the occurrence of shock waves. While the popular shock-capturing methods are easy to implement, shock-fitting techniques provide the most accurate results. A Shock-Fitting Primer presents the proper numerical treatment of shock waves and other discontinuities. The book begins by recounting the events that lead to our understanding of the theory of shock waves and the early developments related to their computation. After presenting the main shock-fitting ideas in the context of a simple scalar equation, the author applies Colombeau’s theory of generalized functions to the Euler equations to demonstrate how the theory recovers well-known results and to provide an in-depth understanding of the nature of jump conditions. He then extends the shock-fitting concepts previously discussed to the one-dimensional and quasi-one-dimensional Euler equations as well as two-dimensional flows. The final chapter explores existing and future developments in shock-fitting methods within the framework of unstructured grid methods.Throughout the text, the techniques developed are illustrated with numerous examples of varying complexity. On the accompanying downloadable resources, MATLAB® codes serve as concrete examples of how to implement the ideas discussed in the book.

Modeling and Control in Vibrational and Structural Dynamics: A Differential Geometric Approach describes the control behavior of mechanical objects, such as wave equations, plates, and shells. It shows how the differential geometric approach is used when the coefficients of partial differential equations (PDEs) are variable in space (waves/plates), when the PDEs themselves are defined on curved surfaces (shells), and when the systems have quasilinear principal parts.To make the book self-contained, the author starts with the necessary background on Riemannian geometry. He then describes differential geometric energy methods that are generalizations of the classical energy methods of the 1980s. He illustrates how a basic computational technique can enable multiplier schemes for controls and provide mathematical models for shells in the form of free coordinates. The author also examines the quasilinearity of models for nonlinear materials, the dependence of controllability/stabilization on variable coefficients and equilibria, and the use of curvature theory to check assumptions.With numerous examples and exercises throughout, this book presents a complete and up-to-date account of many important advances in the modeling and control of vibrational and structural dynamics.

Green’s Functions and Linear Differential Equations: Theory, Applications, and Computation presents a variety of methods to solve linear ordinary differential equations (ODEs) and partial differential equations (PDEs). The text provides a sufficient theoretical basis to understand Green’s function method, which is used to solve initial and boundary value problems involving linear ODEs and PDEs. It also contains a large number of examples and exercises from diverse areas of mathematics, applied science, and engineering.Taking a direct approach, the book first unravels the mystery of the Dirac delta function and then explains its relationship to Green’s functions. The remainder of the text explores the development of Green’s functions and their use in solving linear ODEs and PDEs. The author discusses how to apply various approaches to solve initial and boundary value problems, including classical and general variations of parameters, Wronskian method, Bernoulli’s separation method, integral transform method, method of images, conformal mapping method, and interpolation method. He also covers applications of Green’s functions, including spherical and surface harmonics.Filled with worked examples and exercises, this robust, self-contained text fully explains the differential equation problems, includes graphical representations where necessary, and provides relevant background material. It is mathematically rigorous yet accessible enough for readers to grasp the beauty and power of the subject.

Nonlinear Optimal Control Theory presents a deep, wide-ranging introduction to the mathematical theory of the optimal control of processes governed by ordinary differential equations and certain types of differential equations with memory. Many examples illustrate the mathematical issues that need to be addressed when using optimal control techniques in diverse areas.Drawing on classroom-tested material from Purdue University and North Carolina State University, the book gives a unified account of bounded state problems governed by ordinary, integrodifferential, and delay systems. It also discusses Hamilton-Jacobi theory. By providing a sufficient and rigorous treatment of finite dimensional control problems, the book equips readers with the foundation to deal with other types of control problems, such as those governed by stochastic differential equations, partial differential equations, and differential games.

More than ever before, complicated mathematical procedures are integral to the success and advancement of technology, engineering, and even industrial production. Knowledge of and experience with these procedures is therefore vital to present and future scientists, engineers and technologists.Mathematical Methods in Physics and Engineering with Mathematica clearly demonstrates how to solve difficult practical problems involving ordinary and partial differential equations and boundary value problems using the software package Mathematica (4.x). Avoiding mathematical theorems and numerical methods-and requiring no prior experience with the software-the author helps readers learn by doing with step-by-step recipes useful in both new and classical applications.Mathematica and FORTRAN codes used in the book's examples and exercises are available for download from the Internet. The author's clear explanation of each Mathematica command along with a wealth of examples and exercises make Mathematical Methods in Physics and Engineering with Mathematica an outstanding choice both as a reference for practical problem solving and as a quick-start guide to using a leading mathematics software package.

Unlike the classical Sturm theorems on the zeros of solutions of second-order ODEs, Sturm's evolution zero set analysis for parabolic PDEs did not attract much attention in the 19th century, and, in fact, it was lost or forgotten for almost a century. Briefly revived by Pólya in the 1930's and rediscovered in part several times since, it was not until the 1980's that the Sturmian argument for PDEs began to penetrate into the theory of parabolic equations and was found to have several fundamental applications.Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications focuses on geometric aspects of the intersection comparison for nonlinear models creating finite-time singularities. After introducing the original Sturm zero set results for linear parabolic equations and the basic concepts of geometric analysis, the author presents the main concepts and regularity results of the geometric intersection theory (G-theory). Here he considers the general singular equation and presents the geometric notions related to the regularity and interface propagation of solutions. In the general setting, the author describes the main aspects of the ODE-PDE duality, proves existence and nonexistence theorems, establishes uniqueness and optimal Bernstein-type estimates, and derives interface equations, including higher-order equations. The final two chapters explore some special aspects of discontinuous and continuous limit semigroups generated by singular parabolic equations.Much of the information presented here has never before been published in book form. Readable and self-contained, this book forms a unique and outstanding reference on second-order parabolic PDEs used as models for a wide range of physical problems.

Introduction to Fuzzy Systems provides students with a self-contained introduction that requires no preliminary knowledge of fuzzy mathematics and fuzzy control systems theory. Simplified and readily accessible, it encourages both classroom and self-directed learners to build a solid foundation in fuzzy systems.After introducing the subject, the authors move directly into presenting real-world applications of fuzzy logic, revealing its practical flavor. This practicality is then followed by basic fuzzy systems theory. The book also offers a tutorial on fuzzy control theory, based mainly on the well-known classical Proportional-Integral-Derivative (PID) controllers theory and design methods. In particular, the text discusses fuzzy PID controllers in detail, including a description of the new notion of generalized verb-based fuzzy-logic control theory.Introduction to Fuzzy Systems is primarily designed to provide training for systems and control majors, both senior undergraduate and first year graduate students, to acquaint them with the fundamental mathematical theory and design methodology required to understand and utilize fuzzy control systems.

Offering the only existing finite element (FE) codes for Maxwell equations that support hp refinements on irregular meshes, Computing with hp-ADAPTIVE FINITE ELEMENTS: Volume 1. One- and Two-Dimensional Elliptic and Maxwell Problems presents 1D and 2D codes and automatic hp adaptivity. This self-contained source discusses the theory and implementation of hp-adaptive FE methods, focusing on projection-based interpolation and the corresponding hp-adaptive strategy.The book is split into three parts, progressing from simple to more advanced problems. Part I examines the hp elements for the standard 1D model elliptic problem. The author develops the variational formulation and explains the construction of FE basis functions. The book then introduces the 1D code (1Dhp) and automatic hp adaptivity. This first part ends with a study of a 1D wave propagation problem. In Part II, the book proceeds to 2D elliptic problems, discussing two model problems that are slightly beyond standard-level examples: 3D axisymmetric antenna problem for Maxwell equations (example of a complex-valued, indefinite problem) and 2D elasticity (example of an elliptic system). The author concludes with a presentation on infinite elements - one of the possible tools to solve exterior boundary-value problems. Part III focuses on 2D time-harmonic Maxwell equations. The book explains the construction of the hp edge elements and the fundamental de Rham diagram for the whole family of hp discretizations. Next, it explores the differences between the elliptic and Maxwell versions of the 2D code, including automatic hp adaptivity. Finally, the book presents 2D exterior (radiation and scattering) problems and sample solutions using coupled hp finite/infinite elements.In Computing with hp-ADAPTIVE FINITE ELEMENTS, the information provided, including many unpublished details, aids in solving elliptic and Maxwell problems.

With a focus on 1D and 2D problems, the first volume of Computing with hp-ADAPTIVE FINITE ELEMENTS prepared readers for the concepts and logic governing 3D code and implementation. Taking the next step in hp technology, Volume II Frontiers: Three-Dimensional Elliptic and Maxwell Problems with Applications presents the theoretical foundations of the 3D hp algorithm and provides numerical results using the 3Dhp code developed by the authors and their colleagues.The first part of the book focuses on fundamentals of the 3D theory of hp methods as well as issues that arise when the code is implemented. After a review of boundary-value problems, the book examines exact hp sequences, projection-based interpolation, and De Rham diagrams. It also presents the 3D version of the automatic hp-adaptivity package, a two-grid solver for highly anisotropic hp meshes and goal-oriented Krylov iterations, and a parallel implementation of the 3D code.The second part explores several recent projects in which the 3Dhp code was used and illustrates how these applications have greatly driven the development of 3D hp technology. It encompasses acoustic and electromagnetic (EM) scattering problems, an analysis of complex structures with thin-walled components, and challenging simulations of logging tools. The book concludes with a look at the future of hp methods.Spearheaded by a key developer of this technology with more than 20 years of research in the field, this self-contained, comprehensive resource will help readers overcome the difficulties in coding hp-adaptive elements.