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  • Cover image of <span class="hlFld-Title">Calculus for the Natural Sciences</span>

    Calculus for the Natural Sciences

    Published: 2023

    This textbook is intended for a two-semester course on calculus of one variable. The target audience is comprised of students in biology, chemistry, mathematics, physics, and related disciplines, as well as professionals in these areas. It grew out of the Symbiosis Project at East Tennessee State University.

  • Cover image of <span class="hlFld-Title">Nonlinear Time Scale Systems in Standard and Nonstandard Forms</span>

    Nonlinear Time Scale Systems in Standard and Nonstandard Forms

    Published: 2014

    Singular perturbation theory plays a significant role in analysis and design of control laws. It presents control engineers with the advantage of model reduction that simplifies both the software and hardware implementation of control algorithms. Through this book, our goal is to present the reader with control design techniques that extend these benefits to a larger class of systems, specifically to those that are not traditionally controlled through this methodology. Our aim in doing so is to show the reader that some classes of nonminimum phase control problems can be actively controlled in real-time through use of singular perturbation methods.

  • Cover image of <span class="hlFld-Title">A Primer on Mathematical Models in Biology</span>

    A Primer on Mathematical Models in Biology


    It was on a placid canoe trip at a Gordon Conference in 2002 that Lee Segel told me that he was writing a new book in mathematical biology. In the prime of his health and at the peak of his mathematical career, Lee asked me to agree to act as shepherd to this book “in case” anything happened to prevent his completion of the project. The request was purely “academic” at that time, and I agreed to this formal arrangement with the certainty that it would require no actual work. It came as a great shock that Lee Segel passed away on January 31, 2005, after a sudden and devastating illness. This was a great loss to his many friends, students, coworkers, and admirers in the applied mathematics and mathematical biology communities.

  • Cover image of <span class="hlFld-Title">Nonlinear Programming</span>

    Nonlinear Programming

    Published: 2010

    Chemical engineering applications have been a source of challenging optimization problems for over 50 years. For many chemical process systems, detailed steady state and dynamic behavior can now be described by a rich set of detailed nonlinear models, and relatively small changes in process design and operation can lead to significant improvements in efficiency, product quality, environmental impact, and profitability.With these characteristics, it is not surprising that systematic optimization strategies have played an important role in chemical engineering practice. In particular, over the past 35 years, nonlinear programming (NLP) has become an indispensable tool for the optimization of chemical processes. These tools are now applied at research and process development stages, in the design stage, and in the online operation of these processes. More recently, the scope of these applications is being extended to cover more challenging, large-scale tasks including process control based on the optimization of nonlinear dynamic models, as well as the incorporation of nonlinear models into strategic planning functions.

    Moreover, the ability to solve large-scale process optimization models cheaply, even online, is aided by recent breakthroughs in nonlinear programming, including the development of modern barrier methods, deeper understanding of line search and trust region strategies to aid global convergence, efficient exploitation of second derivatives in algorithmic development, and the availability of recently developed and widely used NLP codes, including those for barrier methods [81, 391, 404], sequential quadratic programming (SQP) [161, 159], and reduced gradient methods [119, 245, 285]. Finally, the availability of optimization modeling environments, such as AIMMS, AMPL, and GAMS, as well as the NEOS server, has made the formulation and solution of optimization accessible to a much wider user base. All of these advances have a huge impact in addressing and solving process engineering problems previously thought intractable. In addition to developments in mathematical programming, research in process systems engineering has led to optimization modeling formulations that leverage these algorithmic advances, with specific model structure and characteristics that lead to more efficient solutions.

    This text attempts to make these recent optimization advances accessible to engineers and practitioners. Optimization texts for engineers usually fall into two categories. First, excellent mathematical programming texts (e.g., [134, 162, 294, 100, 227]) emphasize fundamental properties and numerical analysis, but have few specific examples with relevance to real-world applications, and are less accessible to practitioners. On the other hand, equally good engineering texts (e.g., [122, 305, 332, 53]) emphasize applications with well-known methods and codes, but often without their underlying fundamental properties. While their approach is accessible and quite useful for engineers, these texts do not aid in a deeper understanding of the methods or provide extensions to tackle large-scale problems efficiently.

  • Cover image of <span class="hlFld-Title">Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems</span>

    Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems

    Published: 2009

    This is an introduction to “polynomial continuation,” which is used to compute the solutions to systems of polynomial equations. The book shows how to solve practical problems but maintains an elementary mathematical perspective. The first two chapters illustrate most of the important concepts and numerical processes, using only high-school mathematics and some simple computer programs. Since 1987, when the book was first published, the field has advanced through many developments, noted below. Still, I have been gratified that students continue to take the trouble to let me know that they have found this book a useful starting point.

    The concrete, empirical, and conversational style of this book arose from my experiences as a mathematician at General Motors working with engineers. They sometimes were not convinced by proofs, but their mechanical intuition responded well to numerical experiments that demonstrated the “feel” of the concepts. I learned at this time the paradox that a verbal explanation can defeat its purpose, if made to a tactile person, no matter how correct the explanation, no matter how skilled the person. Consequently, the language in this book stays as basic as possible for as long as possible: subscripts are not used until the end of Chapter 2, the only “spaces” referenced are Euclidean, and no concept or computer code is more general than needed. Yet, the mathematical facts are complete and proven, not necessarily when they are introduced, but eventually. For this, I have relied on a few results from differential topology, avoiding more abstract mathematics.

    Carrying out the numerical exercises using the Fortran code provided is important to the learning experience of the book.

  • Cover image of <span class="hlFld-Title">Singular Perturbations and Hysteresis</span>

    Singular Perturbations and Hysteresis


    Time relaxation and hysteresis are common strongly nonlinear phenomena which occur in many industrial, physical and economic systems. The wording ‘strongly nonlinear’ means that linearization will not encapsulate the observed phenomena. Often these two types of phenomena manifest different stages of the same or similar processes. A number of fundamental hysteresis models can be considered as limit cases of time relaxation processes, or admit an approximation by a differential equation which is singular with respect to a particular parameter.

    However, the amount of interaction between practitioners of theories of systems with time relaxation and systems with hysteresis (and between the ‘relaxation’ and ‘hysteresis’ research communities) is quite low. Thus, the International Workshop on Relaxation Oscillations & Hysteresis was held at University College Cork, Ireland on April 1–6, 2002 as an initial attempt to address this situation. Among the aims of the workshop were

    • to bring together leading experts in time relaxation and hysteresis phenomena in applied problems;

    • to discuss important problems in areas such as reacting systems, semiconductor lasers, shock phenomena in economic modelling, fluid mechanics, etc. with the emphasis on hysteresis and singular perturbations;

    • to learn and to share modern techniques in areas of common interest.

    Further details concerning the workshop can found at

    This book is based on results of the workshop. It is hoped that it will facilitate the cross-fertilization process between these two important topics, and the emergence of promising new areas of research.

  • Cover image of <span class="hlFld-Title">Optimal Control of Viscous Flow</span>

    Optimal Control of Viscous Flow

    Published: 1998

    Control of fluid flow has numerous applications spanning many branches of engineering science. We list below some of the most evident applications.

    Mechanical Engineering Applications

    1. Internal combustion engines: control of efficiency and stability of combustion dynamics.

    2. Internal combustion engines: emission and NOx-control.

    3. Air breathing engines: control of combustion and control of jet noise.

    In these cases, control of fluid dynamics would be the central issue since it is the driving process for combustion and jet noise.

    4. Airbreathing engines: control of turbine/compressor aerodynamics.

    5. Heat exchangers: control of heat transfer.

    Electrical/Computer Engineering Applications

    6. Control of heat rejection devices in computers: control of heat transfer.

    7. Flow control problems in ink jet printers.

    8. Manufacturing problems in microelectronic chips: control of heat transfer.

    Nuclear Fusion Engineering Applications

    9. MHD-control and magnetic fusion, Tokamak stability and control.

    Civil Engineering Applications

    10. Control of pollutant transport.

    11. Oil recovery/transport problems

    Manufacturing Engineering Applications

    12. Control of manufacturing processes and heat transfer issues.

    13. Control of metal forming

    14. Control of coating processes.

    Chemical Engineering Applications

    15. Control of chemical reactors.

    16. Electromagnetic control of conducting fluids such as liquid metals and salt water: control of Lorenz force distributions.

    17. Control of nuclear reactors.

    Aerospace Engineering Applications

    18. Control of aerodynamic surfaces: boundary layer control.

    19. Control of jets and wakes.

    Atmospheric Sciences

    20. Weather prediction: variational methods in data assimilation, adjoint methods.

    Naval Engineering

    21. Control of underwater vehicles, control of UUVs and AUVs.

  • Cover image of <span class="hlFld-Title">The Boundary Function Method for Singular Perturbation Problems</span>

    The Boundary Function Method for Singular Perturbation Problems


    This book was written as a textbook on one of the effective asymptotic methods in the theory of singular perturbations, the boundary function method. Singularly perturbed equations are often used as mathematical models describing processes in physics, chemical kinetics, and mathematical biology, and they often arise during investigation of applied problems of technology and engineering. This explains the interest of applied scientists in asymptotic methods (both from the mathematical and practical viewpoints). Historical aspects of development of different asymptotic techniques can be found, e.g., in O'Malley [114]; a short review of the history of research on singularly perturbed problems in Russia (and other parts of the former Soviet Union) is given in Vasil'eva [147].

    There are a number of introductory as well as advanced books on asymptotic methods that discuss some aspects of different perturbation techniques and their applications. Among them, let us mention, e.g., the monographs and textbooks by Chang and Howes [41], Cole [43], Bush [14], Erdélyi [48], Hinch [62], Hoppensteadt [64], Kevorkian [76], Murdock [97], Murray [98], Nayfeh [102]–[104], O'Malley [113], [114], Smith [128], Van Dyke [139], Vishik [162], and Wasow [166]. Many books are devoted to a description of some specific asymptotic method and present material starting from the basic underlying ideas and finishing with sophisticated calculations for concrete examples. This book can be considered both as an introduction to the basics of one of the asymptotic methods, the boundary function method, also known as the method of boundary layer corrections (cf. O'Malley [113], [114], Smith [128]), and as the survey of results, most of which (especially on the problems for partial differential equations) were presented only in the journal literature. The core of the book reflects the development of this asymptotic technique during the last forty years in the group of small parameter specialists in the Physics Department of Moscow State University (starting with the seminal results by Tikhonov [129]–[131], which were extended by Vasil'eva, Butuzov, and several generations of their students). Two monographs on the application of the boundary function method to singularly perturbed ordinary differential equations, integro-differential equations and equations with small delay as well as to singularly perturbed problems in the critical cases (singular singularly perturbed problems) were written by Vasil'eva and Butuzov [148], [149].

  • Cover image of <span class="hlFld-Title">Contemporary Problems in Statistical Physics</span>

    Contemporary Problems in Statistical Physics

    Published: 1994

    The founders of physics as we know it, Kepler, Galileo, and Newton, recognized only a deterministic world in which each phenomenon had an associated cause. The philosophy of determinism is an integral part of Newton's mechanics, which consists of a very specific set of rules for the analysis of any mechanical system. One of the first successes of Newton's theory was the complete analysis of the motion of two point particles under mutual gravitational attraction. This was immediately and successfully applied to the solution of many problems in celestial dynamics. However, the extension of this analysis to systems comprised of three or more bodies poses not inconsequential mathematical problems which have not been entirely overcome at this time, although a considerable amount is known about such systems following three centuries of research. Since classical mechanics cannot furnish an exact solution to the three-body problem it is hardly imaginable that dynamical properties of mechanical systems consisting of larger numbers of particles can be studied in any exact way without introducing artificial restrictions on the nature of the system. Because of this consideration the notion of statistical methodology as being applied to the analysis of physical systems became a candidate for consideration, particularly for the analysis of gases, for which numbers of the order of 1023 are the important ones. Early work along these lines was initiated by an under-appreciated physicist named Waterhouse in the 1840s, and later and more comprehensively by the better-known Maxwell and Boltzmann.

  • Cover image of <span class="hlFld-Title">Mathematical Problems in Linear Viscoelasticity</span>

    Mathematical Problems in Linear Viscoelasticity

    Published: 1992

    The aim of this book is to investigate the connection between thermodynamic restrictions and the well posedness of initial and boundary value problems. On the one hand, we elaborate a thorough thermodynamic analysis of linear viscoelasticity; new results are established and previous ones are shown to follow as particular cases from the general scheme. On the other hand, we show that significant improvements can be obtained in existence, uniqueness, and asymptotic stability theorems by starting from the thermodynamic restrictions as mathematical hypotheses for the initial-boundary value problems. On the whole, various subjects are examined: well posedness of Cauchy's problems, variational and extremum principles, wave propagation, thermodynamics of continuous media, modelling of viscoelastic materials. The central role played by thermodynamic restrictions on the properties of the solutions makes these subjects deeply interrelated.

    All along, mathematical statements and detailed proofs of results are framed in the pertinent physical context. The material developed is essentially self-contained. The reader is required to have general familiarity with standard techniques of modern analysis and basic concepts in continuum thermodynamics. This book is then of interest to mathematicians, physicists, and engineers.

  • Cover image of <span class="hlFld-Title">Dynamics of Internal Layers and Diffusive Interfaces</span>

    Dynamics of Internal Layers and Diffusive Interfaces

    Published: 1988

    Interfacial phenomena are commonplace in physics, chemistry, biology, and in various disciplines bridging these fields. They occur whenever a continuum is present that can exist in at least two different chemical or physical “states,” and there is some mechanism that generates or enforces a spatial separation between these states. The separation boundary is then called an interface. In the examples studied here, the separation boundary and its internal structure result from the balance between two opposing tendencies: a diffusive effect that attempts to mix and smooth the properties of the material and a physical or chemical mechanism that works to drive it to one or the other pure state.

    This latter is an “unmixing” tendency. In our cases, it is one of the following: (1) a chemical kinetic mechanism with two stable steady states or two attracting slow manifolds in concentration space; (2) a double-well potential that drives a substance into one of two possible phases, such as solid or liquid; (3) an imposed electric field that affects different kinds of ions in different ways; (4) a chemical reaction rate that is so sensitive to temperature that a temperature isocline can serve as an interface separating (a) the region in space where the reaction (and other reactions it triggers) has gone to completion, from (b) the other region, where the reaction is so slow due to the lower temperature that it can be neglected; or finally (5) a very complex biophysical process responsible for the triggering of physical change in biological tissue, followed by its recovery to its original state.

    This fifth mechanism, together with the diffusive-type process alluded to above, is responsible for the propagation of signals along a nerve axon or cardiac tissue, and is quite commonly modeled by systems of the type studied in Chap. 4. The Belousov—Zhabotinskii and other excitable chemical reagents subject to mechanism (1) above are appropriately modeled the same way, and are also treated in Chap. 4.

  • Cover image of <span class="hlFld-Title">Multigrid Methods</span>

    Multigrid Methods

    Published: 1987

    As all manuscripts of this type must be, this volume of the SIAM series Frontiers in Applied Mathematics was designed as much by compromise as by anything else. Decisions were carefully made about such things as notation, structure, content, style, purpose, and targeted audience, but, almost always, tradeoffs were involved.

    We first decided on our basic purpose, the underlying formal structure we wanted, and what we expected of our audience. Our intent was a fairly cohesive book that developed, organized, and surveyed selected fields in the multigrid discipline. New material would be included, but only to fill gaps. The book would be directed toward readers who were well versed in related fields, especially the numerical solution of partial differential equations, but who were otherwise unfamiliar with multigrid methods.

    Our biggest problem was designing a common system of notation and conventions. The system had to be simple, clean, versatile, unambiguous, and general enough to meet our diverse needs. At the end of this Preface, we summarize the design that emerged from our compromises.

    It is crucial that the reader pay close attention to our notation. Most significantly, in lieu of iteration subscripts, we use dynamic variables, much as variables are used in computer languages. This puts the responsibility on the reader to always have in mind the sense of these variables, especially those that represent coarse grid quantities. It is a serious warning that significant subtleties in our notation must not be overlooked. We point out some of these subtleties in Chapter 1.

    Since this volume is but one of several in the SIAM series Frontiers in Applied Mathematics, we have conformed to certain common objectives of the series including development of a basic theme and topic, incorporation of a survey of tutorial character, treatment of new developments in the field, and solicitation of individual contributions from experts in each area. However, because of the flexibility that this series allows, our volume departs a little from the others in character, primarily by tending toward a more unified text. From a pedagogical perspective, we have written the first chapter so that it may be used in the classroom as a basis for a module on multigrid methods; other chapters may provide resources for further topics of study.

  • Cover image of <span class="hlFld-Title">Lectures on Mathematical Combustion</span>

    Lectures on Mathematical Combustion

    Published: 1983

    The material contained herein is the written version of ten lectures on mathematical combustion given during a CBMS-NSF Regional Conference held at Colorado State University in June 1982. A few changes have been made, largely in the form of amplifications, but the result is faithful to what was delivered there. Even the lecture style is retained, which explains much and itself needs explanation.

    We were, and are, primarily concerned with conveying the excitement of this new mathematical science. Full treatments of the material and pedantry over original sources (even though very recent) would not have contributed to such a goal. Most missing information can be found in our Cambridge monograph Theory of Laminar Flames (TLF), which appeared just after the conference. The remainder, corresponding to more recent topics, is referenced here. To the same end we limited ourselves to work that appeared to have unusual significance, studiously avoiding a review. If someone with a finger in the mathematical combustion pie is thereby offended, (s)he should know that we did it for the greater good.

    The lectures were a five-day commercial for TLF; its preface should be read as a pre-preface here, giving the spirit of the enterprise. In no sense is the present text a reader's digest, however; if some phrases are the same (though we admit nothing), the reason is undoubtedly the uniqueness of perfection. Where the same topic is discussed, a fresh light has been thrown on it. In many instances, that was done automatically by adopting the so-called constant-density approximation from the start (and only abandoning it when absolutely necessary). But several topics here do not appear in TLF; examples are concentrated in the stability theory, which has surged forwards in the '80s. The theory was allocated one chapter of twelve in TLF; here it has commandeered thirty per cent of the lectures.