This book presents methods for the computational solution of differential equations, both ordinary and partial, time-dependent and steady-state. Finite difference methods are introduced and analyzed in the first four chapters, and finite element methods are studied in chapter five. A very general-purpose and widely-used finite element program, PDE2D, which implements many of the methods studied in the earlier chapters, is presented and documented in Appendix A.The book contains the relevant theory and error analysis for most of the methods studied, but also emphasizes the practical aspects involved in implementing the methods. Students using this book will actually see and write programs (FORTRAN or MATLAB) for solving ordinary and partial differential equations, using both finite differences and finite elements. In addition, they will be able to solve very difficult partial differential equations using the software PDE2D, presented in Appendix A. PDE2D solves very general steady-state, time-dependent and eigenvalue PDE systems, in 1D intervals, general 2D regions, and a wide range of simple 3D regions.The Windows version of PDE2D comes free with every purchase of this book. More information at www.pde2d.com/contact.

This book maps Christopher Isherwood's intellectual and aesthetic reflections from the late 1930s through the late 1970s. Drawing on the queer theory of Eve Sedgwick and the ethical theory of Michel Foucault, Carr illuminates Isherwood's post-war development of a queer ethos through his focus on the aesthetic, social, and historical politics of the 1930s in his novels Prater Violet (1945), The World in the Evening (1954), and Down There on a Visit (1962), and in his memoir, Christopher and His Kind: 1929 1939 (1976)."

The "Hyperboloidal Foliation Method" introduced in this monograph is based on a (3 + 1) foliation of Minkowski spacetime by hyperboloidal hypersurfaces. This method allows the authors to establish global-in-time existence results for systems of nonlinear wave equations posed on a curved spacetime. It also allows to encompass the wave equation and the Klein-Gordon equation in a unified framework and, consequently, to establish a well-posedness theory for a broad class of systems of nonlinear wave-Klein-Gordon equations. This book requires certain natural (null) conditions on nonlinear interactions, which are much less restrictive that the ones assumed in the existing literature. This theory applies to systems arising in mathematical physics involving a massive scalar field, such as the Dirac-Klein-Gordon systems.

This monographs presents new spherical mean value relations for classical boundary value problems of mathematical physics. The derived spherical mean value relations provide equivalent integral formulations of original boundary value problems. Direct and converse mean value theorems are proved for scalar elliptic equations (the Laplace, Helmholtz and diffusion equations), parabolic equations, high-order elliptic equations (biharmonic and metaharmonic equations), and systems of elliptic equations (the Lami equation, systems of diffusion and elasticity equations). In addition, applications to the random walk on spheres method are given.

Vladimir Maz'ya: Friend and mathematician. Recollections.- On Maz'ya's work in potential theory and the theory of function spaces.- 1. Introduction.- 2. Embeddings and isoperimetric inequalities.- 3. Regularity of solutions.- 4. Boundary regularity.- 5. Nonlinear potential theory.- Maz'ya's works in the linear theory of water waves.- 1. Introduction.- 2. The unique solvability of the water wave problem.- 3. The Neumann-Kelvin problem.- 4. Asymptotic expansions for transient water waves due to brief and high-frequency disturbances.- Maz'ya's work on integral and pseudodifferential operators.- 1. Non-elliptic operators.- 2. Oblique derivative problem: breakthrough in the generic case of degeneration.- 3. Estimates for differential operators in the half-space.- 4. The characteristic Cauchy problem for hyperbolic equations.- 5. New methods for solving ill-posed boundary value problems.- 6. Applications of multiplier theory to integral operators.- 7. Integral equations of harmonic potential theory on general non-regular surfaces.- 8. Boundary integral equations on piecewise smooth surfaces.- Contributions of V. Maz'ya to the theory of boundary value problems in nonsmooth domains.- 1. Maz'ya's early work on boundary value problems in nonsmooth domains.- 2. General elliptic boundary value problems in domains with point singularities.- 3. Boundary value problems in domains with edges.- 4. Spectral properties of operator pencils generated by elliptic boundary value problems in a cone.- 5. Applications to elastostatics and hydrodynamics.- 6. Singularities of solutions to nonlinear elliptic equations at a cone vertex.- On some potential theoretic themes in function theory.- 1. Approximation theory.- 2. Uniqueness properties of analytic functions.- 3. The Cauchy problem for the Laplace equation.- Approximate approximations and their applications.- 1. Introduction.- 2. Quasi-interpolation.- 3. Generating functions for quasi-interpolation of high order.- 4. Semi-analytic cubature formulas.- 5. Cubature of integral operators over bounded domains.- 6. Approximate wavelets.- 7. Numerical algorithms based upon approximate approximations.- Maz'ya's work on the biography of Hadamard.- Isoperimetric inequalities and capacities on Riemannian manifolds.- 1. Introduction.- 2. Capacity of balls.- 3. Parabolicity of manifolds.- 4. Isoperimetric inequality and Sobolev inequality.- 5. Capacity and the principal frequency.- 6. Cheeger's inequality.- 7. Eigenvalues of balls on spherically symmetric manifolds.- 8. Heat kernel on spherically symmetric manifolds.- Multipliers of differentiable functions and their traces.- 1. Introduction.- 2. Description and properties of multipliers.- 3. Multipliers in the space of Bessel potentials as traces of multipliers.- An asymptotic theory of nonlinear abstract higher order ordinary differential equations.- Sobolev spaces for domains with cusps.- 1. Introduction.- 2. Extension theorems.- 3. Embedding theorems.- 4. Boundary values of Sobolev functions.- Extension theorems for Sobolev spaces.- 1. Introduction.- 2. Extensions with preservation of class.- 3. Estimates for the minimal norm of an extension operator.- 4. Extensions with deterioration of class.- Contributions of V.G. Maz'ya to analysis of singularly perturbed boundary value problems.- 1. Introduction.- 2. Domain with a small hole.- 3. General asymptotic theory by Maz'ya, Nazarov and Plamenevskii.- 4. Asymptotics of solutions of boundary integral equations under a small perturbation of a corner.- 5. Compound asymptotics for homogenization problems.- 6. Boundary value problems in 3D-1D multi-structures.- Asymptotic analysis of a mixed boundary value problem in a singularly degenerating domain.- 1. Introduction.- 2. Formulation of the problem.- 3. The leading order approximation.- A history of the Cosserat spectrum.- 1. Introduction.- 2. The first boundary value problem of elastostatics.- 3. The second and other boundary-value problems.- 4. Applications and o...

This book covers a highly relevant and timely topic that is of wide interest, especially in finance, engineering and computational biology. The introductory material on simulation and stochastic differential equation is very accessible and will prove popular with many readers. While there are several recent texts available that cover stochastic differential equations, the concentration here on inference makes this book stand out. No other direct competitors are known to date. With an emphasis on the practical implementation of the simulation and estimation methods presented, the text will be useful to practitioners and students with minimal mathematical background. What's more, because of the many R programs, the information here is appropriate for many mathematically well educated practitioners, too.

This unique book on ordinary differential equations addresses practical issues of composing and solving such equations by large number of examples and homework problems with solutions. These problems originate in engineering, finance, as well as science at appropriate levels that readers with the basic knowledge of calculus, physics or economics are assumed able to follow.

This unique book on ordinary differential equations addresses practical issues of composing and solving such equations by large number of examples and homework problems with solutions. These problems originate in engineering, finance, as well as science at appropriate levels that readers with the basic knowledge of calculus, physics or economics are assumed able to follow.

This volume is on initial-boundary value problems for parabolic partial differential equations of second order. It rewrites the problems as abstract Cauchy problems or evolution equations, and then solves them by the technique of elementary difference equations. Because of this, the volume assumes less background and provides an easy approach for readers to understand.

This volume provides a systematic introduction to the theory of the multidimensional Mellin transformation in a distributional setting. In contrast to the classical texts on the Mellin and Laplace transformations, this work concentrates on the "local" properties of the Mellin transformations, ie on those properties of the Mellin transforms of distributions "u" which are preserved under multiplication of "u" by cut-off functions (of various types). The main part of the book is devoted to the local study of regularity of solutions to linear Fuchsian partial differential operators on a corner, which demonstrates the appearance of "non-discrete" asymptotic expansions (at the vertex) and of resurgence effects in the spirit of J. Ecalle. The book constitutes a part of a program to use the Mellin transformation as a link between the theory of second micro-localization, resurgence theory and the theory of the generalized Borel transformation. Chapter 1 contains the basic theorems and definitions of the theory of distributions and Fourier transformations which are used in the succeeding chapters. This material includes proofs which are partially transformed into exercises with hints. Chapter 2 presents a systematic treatment of the Mellin transform in several dimensions. Chapter 3 is devoted to Fuchsian-type singular differential equations. While aimed at researchers and graduate students interested in differential equations and integral transforms, this book can also be recommended as a graduate text for students of mathematics and engineering.

Transmutation operators in differential equations and spectral theory can be used to reveal the relations between different problems, and often make it possible to transform difficult problems into easier ones. Accordingly, they represent an important mathematical tool in the theory of inverse and scattering problems, of ordinary and partial differential equations, integral transforms and equations, special functions, harmonic analysis, potential theory, and generalized analytic functions. This volume explores recent advances in the construction and applications of transmutation operators, while also sharing some interesting historical notes on the subject.

In this book several connections between probability theory and wave propagation are explored. The connection comes via the probabilistic (or path integral) representation of both the (fixed frequency) Green functions and of the propagators -operators mapping initial into present time data. The formalism includes both waves in continuous space and in discrete structures.
One of the main applications of the formalism developed is to inverse problems in wave propagation. Using the probabilistic formalism, the parameters of the medium and the surfaces determining the region of propagation appear explicitly in the path integral representation of the Green functions and propagators. This fact is what provides a useful starting point for inverse problem formulation.

Audience: The book is suitable for advanced graduate students in the mathematical, physical or in the engineering sciences. The presentation is quite self-contained, and not extremely rigorous.

This monograph presents a synopsis of fluid dynamics based on the personal scientific experience of the author who has contributed immensely to the field. The interested reader will also benefit from the general historical context in which the material is presented in the book. The book covers a wide range of relevant topics of the field, and the main tool being rational asymptotic modelling (RAM) approach. The target audience primarily comprises experts in the field of fluid dynamics, but the book may also be beneficial for graduate students.

The Institute for Mathematics and its Applications (IMA) devoted its 1997-1998 program to Emerging Applications of Dynamical Systems. Dynamical systems theory and related numerical algorithms provide powerful tools for studying the solution behavior of differential equations and mappings. In the past 25 years computational methods have been developed for calculating fixed points, limit cycles, and bifurcation points. A remaining challenge is to develop robust methods for calculating more complicated objects, such as higher- codimension bifurcations of fixed points, periodic orbits, and connecting orbits, as well as the calcuation of invariant manifolds. Another challenge is to extend the applicability of algorithms to the very large systems that result from discretizing partial differential equations. Even the calculation of steady states and their linear stability can be prohibitively expensive for large systems (e.g. 10_3- -10_6 equations) if attempted by simple direct methods. Several of the papers in this volume treat computational methods for low and high dimensional systems and, in some cases, their incorporation into software packages. A few papers treat fundamental theoretical problems, including smooth factorization of matrices, self -organized criticality, and unfolding of singular heteroclinic cycles. Other papers treat applications of dynamical systems computations in various scientific fields, such as biology, chemical engineering, fluid mechanics, and mechanical engineering.

This book is a collection of papers in memory of Gu Chaohao on the subjects of Differential Geometry, Partial Differential Equations and Mathematical Physics that Gu Chaohao made great contributions to with all his intelligence during his lifetime.All contributors to this book are close friends, colleagues and students of Gu Chaohao. They are all excellent experts among whom there are 9 members of the Chinese Academy of Sciences. Therefore this book will provide some important information on the frontiers of the related subjects.

This two-volume monograph is a comprehensive and up-to-date presentation of the theory and applications of kinetic equations. The first volume covers many-particle dynamics, Maxwell models of the Boltzmann equation (including their exact and self-similar solutions), and hydrodynamic limits beyond the Navier-Stokes level.

This book is a comprehensive collection of known results about the Lozi map, a piecewise-affine version of the Henon map. Henon map is one of the most studied examples in dynamical systems and it attracts a lot of attention from researchers, however it is difficult to analyze analytically. Simpler structure of the Lozi map makes it more suitable for such analysis. The book is not only a good introduction to the Lozi map and its generalizations, it also summarizes of important concepts in dynamical systems theory such as hyperbolicity, SRB measures, attractor types, and more.

Ever since China and Vietnam resumed diplomatic contacts and reopened the border in 1991, the borderland region has become part of the vibrant growing economies of both countries and drawn many from the interior provinces to the borderland for new economic adventures. This book examines Chinese-Vietnamese relationships at the borderland through every day cross-border interaction in trade and tourism activities. It looks into the historical underlining of bilateral relations of the two countries which often shape people's perceptions of the 'other' and interpretation of intentions of acts in their daily interaction. Albeit Chinese and Vietnamese have lived side by side for centuries, their interaction in the space of trade and modern tourism in post-war and post-reform China and Vietnam is something novel to both people. The book provides a 'bottom-up' approach to examine the localized experiences of inter-state relations. It illustrates the changes the vibrant economic process has brought to the borderland communities, and how the revived contacts and interaction have generated a contested space for examining Vietnamese-Chinese relationships and demonstrating trans-border cultural politics. A novel study of the strategic development of the borderland within the new political economy at China-Southeast Asia border region, this book is of interest to academics in the field of Anthropology, Border Studies, Social and Cultural Studies and Asian Studies.

This open access book presents a comprehensive survey of modern operator techniques for boundary value problems and spectral theory, employing abstract boundary mappings and Weyl functions. It includes self-contained treatments of the extension theory of symmetric operators and relations, spectral characterizations of selfadjoint operators in terms of the analytic properties of Weyl functions, form methods for semibounded operators, and functional analytic models for reproducing kernel Hilbert spaces. Further, it illustrates these abstract methods for various applications, including Sturm-Liouville operators, canonical systems of differential equations, and multidimensional Schroedinger operators, where the abstract Weyl function appears as either the classical Titchmarsh-Weyl coefficient or the Dirichlet-to-Neumann map. The book is a valuable reference text for researchers in the areas of differential equations, functional analysis, mathematical physics, and system theory. Moreover, thanks to its detailed exposition of the theory, it is also accessible and useful for advanced students and researchers in other branches of natural sciences and engineering.

The aim of this third edition is to give an accessible and essentially self-contained account of pseudo-differential operators based on the previous edition. New chapters notwithstanding, the elementary and detailed style of earlier editions is maintained in order to appeal to the largest possible group of readers. The focus of this book is on the global theory of elliptic pseudo-differential operators on Lp(Rn).The main prerequisite for a complete understanding of the book is a basic course in functional analysis up to the level of compact operators. It is an ideal introduction for graduate students in mathematics and mathematicians who aspire to do research in pseudo-differential operators and related topics.

The theory of one-dimensional ergodic operators involves a beautiful synthesis of ideas from dynamical systems, topology, and analysis. Additionally, this setting includes many models of physical interest, including those operators that model crystals, disordered media, or quasicrystals. This field has seen substantial progress in recent decades, much of which has yet to be discussed in textbooks. Beginning with a refresher on key topics in spectral theory, this volume presents the basic theory of discrete one-dimensional Schrodinger operators with dynamically defined potentials. It also includes a self-contained introduction to the relevant aspects of ergodic theory and topological dynamics. This text is accessible to graduate students who have completed one-semester courses in measure theory and complex analysis. It is intended to serve as an introduction to the field for junior researchers and beginning graduate students as well as a reference text for people already working in this area. It is well suited for self-study and contains numerous exercises (many with hints).

Iterative Optimization in Inverse Problems brings together a number of important iterative algorithms for medical imaging, optimization, and statistical estimation. It incorporates recent work that has not appeared in other books and draws on the author's considerable research in the field, including his recently developed class of SUMMA algorithms. Related to sequential unconstrained minimization methods, the SUMMA class includes a wide range of iterative algorithms well known to researchers in various areas, such as statistics and image processing. Organizing the topics from general to more specific, the book first gives an overview of sequential optimization, the subclasses of auxiliary-function methods, and the SUMMA algorithms. The next three chapters present particular examples in more detail, including barrier- and penalty-function methods, proximal minimization, and forward-backward splitting. The author also focuses on fixed-point algorithms for operators on Euclidean space and then extends the discussion to include distance measures other than the usual Euclidean distance. In the final chapters, specific problems illustrate the use of iterative methods previously discussed. Most chapters contain exercises that introduce new ideas and make the book suitable for self-study. Unifying a variety of seemingly disparate algorithms, the book shows how to derive new properties of algorithms by comparing known properties of other algorithms. This unifying approach also helps researchers-from statisticians working on parameter estimation to image scientists processing scanning data to mathematicians involved in theoretical and applied optimization-discover useful related algorithms in areas outside of their expertise.

Until now, no book addressed convexity, monotonicity, and variational inequalities together. Generalized Convexity, Nonsmooth Variational Inequalities, and Nonsmooth Optimization covers all three topics, including new variational inequality problems defined by a bifunction.

The first part of the book focuses on generalized convexity and generalized monotonicity. The authors investigate convexity and generalized convexity for both the differentiable and nondifferentiable case. For the nondifferentiable case, they introduce the concepts in terms of a bifunction and the Clarke subdifferential.

The second part offers insight into variational inequalities and optimization problems in smooth as well as nonsmooth settings. The book discusses existence and uniqueness criteria for a variational inequality, the gap function associated with it, and numerical methods to solve it. It also examines characterizations of a solution set of an optimization problem and explores variational inequalities defined by a bifunction and set-valued version given in terms of the Clarke subdifferential.

Integrating results on convexity, monotonicity, and variational inequalities into one unified source, this book deepens your understanding of various classes of problems, such as systems of nonlinear equations, optimization problems, complementarity problems, and fixed-point problems. The book shows how variational inequality theory not only serves as a tool for formulating a variety of equilibrium problems, but also provides algorithms for computational purposes.

The present volume develops the theory of integration in Banach spaces, martingales and UMD spaces, and culminates in a treatment of the Hilbert transform, Littlewood-Paley theory and the vector-valued Mihlin multiplier theorem. Over the past fifteen years, motivated by regularity problems in evolution equations, there has been tremendous progress in the analysis of Banach space-valued functions and processes. The contents of this extensive and powerful toolbox have been mostly scattered around in research papers and lecture notes. Collecting this diverse body of material into a unified and accessible presentation fills a gap in the existing literature. The principal audience that we have in mind consists of researchers who need and use Analysis in Banach Spaces as a tool for studying problems in partial differential equations, harmonic analysis, and stochastic analysis. Self-contained and offering complete proofs, this work is accessible to graduate students and researchers with a background in functional analysis or related areas.