Differential calculus and equations Books

Book SynopsisThis book offers a modern introduction to Nevanlinna theory and its intricate relation to the theory of normal families, algebraic functions, asymptotic series, and algebraic differential equations.Following a comprehensive treatment of Nevanlinna’s theory of value distribution, the author presents advances made since Hayman’s work on the value distribution of differential polynomials and illustrates how value- and pair-sharing problems are linked to algebraic curves and Briot–Bouquet differential equations. In addition to discussing classical applications of Nevanlinna theory, the book outlines state-of-the-art research, such as the effect of the Yosida and Zalcman–Pang method of re-scaling to algebraic differential equations, and presents the Painlevé–Yosida theorem, which relates Painlevé transcendents and solutions to selected 2D Hamiltonian systems to certain Yosida classes of meromorphic functions.Aimed at graduate students interested in recent developments in the field and researchers working on related problems, Nevanlinna Theory, Normal Families, and Algebraic Differential Equations will also be of interest to complex analysts looking for an introduction to various topics in the subject area. With examples, exercises and proofs seamlessly intertwined with the body of the text, this book is particularly suitable for the more advanced reader.Trade Review“The book by Steinmetz is clearly written, including a substantial number of exercises related to and complementing the actual text.” (Ilpo Laine, Mathematical Reviews, June, 2018)“The list of references contains more than 200 items including very recent results of the author and other. … I recommend this book to any person who is interested in complex analysis, in particular, in value distribution theory and complex differential equations.” (Igor Chyzhykov, zbMATH, 2018)Table of ContentsIntroduction and preface.- Selected Topics in Complex Analysis.- Nevanlinna Theory.- Selected Applications of Nevanlinna Theory.- Normal Families.- Algebraic Differential Equations.- Higher-Order Algebraic Differential Equations.- Index.

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Book SynopsisThis book collects together lectures by some of the leaders in the field of partial differential equations and geometric measure theory. It features a wide variety of research topics in which a crucial role is played by the interaction of fine analytic techniques and deep geometric observations, combining the intuitive and geometric aspects of mathematics with analytical ideas and variational methods. The problems addressed are challenging and complex, and often require the use of several refined techniques to overcome the major difficulties encountered. The lectures, given during the course "Partial Differential Equations and Geometric Measure Theory'' in Cetraro, June 2–7, 2014, should help to encourage further research in the area. The enthusiasm of the speakers and the participants of this CIME course is reflected in the text.Table of ContentsAlberto Farina and Enrico Valdinoci:Introduction.-Alessio Figalli:Global Existence for the Semi-Geostrophic Equations via Sobolev Estimates for Monge-Ampère.-Ireneo Peral Alonso: On Some Elliptic and Parabolic Equations Related to Growth Models.- Enrico Valdinoci: All Functions are (locally) S-harmonic (up to a small error) – and Applications

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Book SynopsisThis book presents four lectures on recent research in commutative algebra and its applications to algebraic geometry. Aimed at researchers and graduate students with an advanced background in algebra, these lectures were given during the Commutative Algebra program held at the Vietnam Institute of Advanced Study in Mathematics in the winter semester 2013 -2014. The first lecture is on Weyl algebras (certain rings of differential operators) and their D-modules, relating non-commutative and commutative algebra to algebraic geometry and analysis in a very appealing way. The second lecture concerns local systems, their homological origin, and applications to the classification of Artinian Gorenstein rings and the computation of their invariants. The third lecture is on the representation type of projective varieties and the classification of arithmetically Cohen -Macaulay bundles and Ulrich bundles. Related topics such as moduli spaces of sheaves, liaison theory, minimal resolutions, and Hilbert schemes of points are also covered. The last lecture addresses a classical problem: how many equations are needed to define an algebraic variety set-theoretically? It systematically covers (and improves) recent results for the case of toric varieties.Table of Contents1. Notes on Weyl Algebras and D-modules.- 2. Inverse Systems of Local Rings.- 3. Lectures on the Representation Type of a Projective Variety.- 4. Simplicial Toric Varieties which are set-theoretic Complete Intersections.

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Book SynopsisThis book takes readers on a multi-perspective tour through state-of-the-art mathematical developments related to the numerical treatment of PDEs based on splines, and in particular isogeometric methods. A wide variety of research topics are covered, ranging from approximation theory to structured numerical linear algebra. More precisely, the book provides (i) a self-contained introduction to B-splines, with special focus on approximation and hierarchical refinement, (ii) a broad survey of numerical schemes for control problems based on B-splines and B-spline-type wavelets, (iii) an exhaustive description of methods for computing and analyzing the spectral distribution of discretization matrices, and (iv) a detailed overview of the mathematical and implementational aspects of isogeometric analysis. The text is the outcome of a C.I.M.E. summer school held in Cetraro (Italy), July 2017, featuring four prominent lecturers with different theoretical and application perspectives. The book may serve both as a reference and an entry point into further research.Table of ContentsFoundations of Spline Theory: B-Splines, Spline Approximation, and Hierarchical Refinement.- Adaptive Multiscale Methods for the Numerical Treatment of Systems of PDEs.- Generalized Locally Toeplitz Sequences: A Spectral Analysis Tool for Discretized Differential Equations.- Isogeometric Analysis: Mathematical and Implementational Aspects, with Applications.

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Book SynopsisDifferenzialgleichungen sind Ihnen ein Buch mit sieben Siegeln? Kein Problem! Im ersten Teil liefert Ihnen dieses Buch wirklich alles, was Sie an Handwerkszeug zum Lösen von Differenzialgleichungen benötigen. Anschließend erfahren Sie, was Differenzialgleichungen überhaupt sind und mit welchen Methoden man sie lösen kann. Im dritten Teil wird es ernst: Sie werden einfache Differenzialgleichungen rechnerisch lösen. Aber keine Sorge: Vielfältige Beispiele geben Ihnen die Gelegenheit, die Verfahren gründlich zu üben. Und damit Sie wissen, warum Sie sich all diesen Mühen unterziehen, werden Sie zuletzt auf berühmte Differenzialgleichungen aus Biologie, Chemie, Physik und Ökonomie treffen.

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Book SynopsisSince the early 70's, mixed finite elements have been the object of a wide and deep study by the mathematical and engineering communities. The fundamental role of this method for many application fields has been worldwide recognized and its use has been introduced in several commercial codes. An important feature of mixed finite elements is the interplay between theory and application. Discretization spaces for mixed schemes require suitable compatibilities, so that simple minded approximations generally do not work and the design of appropriate stabilizations gives rise to challenging mathematical problems. This volume collects the lecture notes of a C.I.M.E. course held in Summer 2006, when some of the most world recognized experts in the field reviewed the rigorous setting of mixed finite elements and revisited it after more than 30 years of practice. Applications, in this volume, range from traditional ones, like fluid-dynamics or elasticity, to more recent and active fields, like electromagnetism.Table of ContentsMixed Finite Element Methods.- Finite Elements for the Stokes Problem.- Polynomial Exact Sequences and Projection-Based Interpolation with Application to Maxwell Equations.- Finite Element Methods for Linear Elasticity.- Finite Elements for the Reissner–Mindlin Plate.

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Book SynopsisThe main theme is the integration of the theory of linear PDE and the theory of finite difference and finite element methods. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. The chapters on elliptic equations are preceded by a chapter on the two-point boundary value problem for ordinary differential equations. Similarly, the chapters on time-dependent problems are preceded by a chapter on the initial-value problem for ordinary differential equations. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. The required background on linear functional analysis and Sobolev spaces is reviewed in an appendix. The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering.Trade ReviewFrom the reviews:"The book under review is an introduction to the field of linear partial differential equations and to standard methods for their numerical solution. … The balanced combination of mathematical theory with numerical analysis is an essential feature of the book. … The book is easily accessible and concentrates on the main ideas while avoiding unnecessary technicalities. It is therefore well suited as a textbook for a beginning graduate course in applied mathematics." (A. Ostermann, IMN - Internationale Mathematische Nachrichten, Vol. 59 (198), 2005)"This book, which is aimed at beginning graduate students of applied mathematics and engineering, provides an up to date synthesis of mathematical analysis, and the corresponding numerical analysis, for elliptic, parabolic and hyperbolic partial differential equations. … This widely applicable material is attractively presented in this impeccably well-organised text. … Partial differential equations with numerical methods covers a lot of ground authoritatively and without ostentation and with a constant focus on the needs of practitioners." (Nick Lord, The Mathematical Gazette, March, 2005)"Larsson and Thomée … discuss numerical solution methods of linear partial differential equations. They explain finite difference and finite element methods and apply these concepts to elliptic, parabolic, and hyperbolic partial differential equations. … The text is enhanced by 13 figures and 150 problems. Also included are appendixes on mathematical analysis preliminaries and a connection to numerical linear algebra. Summing Up: Recommended. Upper-division undergraduates through faculty." (D. P. Turner, CHOICE, March, 2004)"This book presents a very well written and systematic introduction to the finite difference and finite element methods for the numerical solution of the basic types of linear partial differential equations (PDE). … the book is very well written, the exposition is clear, readable and very systematic." (Emil Minchev, Zentralblatt MATH, Vol. 1025, 2003)"The author’s purpose is to give an elementary, relatively short, and readable account of the basic types of linear partial differential equations, their properties, and the most commonly used methods for their numerical solution. … We warmly recommend it to advanced undergraduate and beginning graduate students of applied mathematics and/or engineering at every university of the world." (Ferenc Móricz, Acta Scientiarum Mathematicarum, Vol. 71, 2005)"The presentation of the book is smart and very classical; it is more a reference book for applied mathematicians … . The convergence results, error estimates, variation formulations, all the theorems proofs, are very clear and well presented, the annexes A and B summary the necessary background for the understanding, without redundant generalisation or forgotten matter. The bibliography is presented by theme, well targeted on the topic of the book." (Anne Lemaitre, Physicalia Magazine, Vol. 28 (1), 2006)“Offers basic theory of linear partial differential equations and discusses the most commonly used numerical methods to solve these equations. … There are two appendices providing some extra basic material, useful to help understanding some of the theoretical principles that might be unfamiliar to unexperienced readers and students. The text is elementary and meant for students in mathematics, physics, engineering. … The bibliography is well arranged according to the important issues, which makes it easy to get informed about possible references for further study.” (Paula Bruggen, Bulletin of the Belgian Mathematical Society, Vol. 15 (1), 2008)Table of ContentsA Two-Point Boundary Value Problem.- Elliptic Equations.- Finite Difference Methods for Elliptic Equations.- Finite Element Methods for Elliptic Equations.- The Elliptic Eigenvalue Problem.- Initial-Value Problems for Ordinary Differential Equations.- Parabolic Equations.- Finite Difference Methods for Parabolic Problems.- The Finite Element Method for a Parabolic Problem.- Hyperbolic Equations.- Finite Difference Methods for Hyperbolic Equations.- The Finite Element Method for Hyperbolic Equations.- Some Other Classes of Numerical Methods.

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Book SynopsisThis book gives an introduction to the finite element method as a general computational method for solving partial differential equations approximately. Our approach is mathematical in nature with a strong focus on the underlying mathematical principles, such as approximation properties of piecewise polynomial spaces, and variational formulations of partial differential equations, but with a minimum level of advanced mathematical machinery from functional analysis and partial differential equations. In principle, the material should be accessible to students with only knowledge of calculus of several variables, basic partial differential equations, and linear algebra, as the necessary concepts from more advanced analysis are introduced when needed. Throughout the text we emphasize implementation of the involved algorithms, and have therefore mixed mathematical theory with concrete computer code using the numerical software MATLAB is and its PDE-Toolbox. We have also had the ambition to cover some of the most important applications of finite elements and the basic finite element methods developed for those applications, including diffusion and transport phenomena, solid and fluid mechanics, and also electromagnetics.​ Trade ReviewFrom the reviews:“The authors give an introduction to the finite element method as a general computational method for solving partial differential equations (PDEs) approximately. … The book should be accessible to students with only knowledge of calculus of several variables, basic partial differential equations, and linear algebra, as the necessary concepts from more advanced analysis are introduced when needed.” (Răzvan Răducanu, Zentralblatt MATH, Vol. 1263, 2013)Table of Contents1. Piecewise Polynomial Approximation in 1D.- 2. The Finite Element Method in 1D.- 3. Piecewise Polynomial Approximation in 2D.- 4. The Finite Element Method in 2D.- 5. Time-dependent Problems.- 6. Solving Large Sparse Linear Systems.- 7. Abstract Finite Element Analysis.- 8. The Finite Element.- 9. Non-linear Problems.- 10. Transport Problems.- 11. Solid Mechanics.- 12. Fluid Mechanics.- 13. Electromagnetics.- 14. Discontinuous Galerkin Methods.- A. Some Additional Matlab Code.- References.

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Book SynopsisIn this book, for the first time, a hitherto unknown general solution of the reliably known stress conditions is presented. This general solution forms a reliable and new starting point to get further in stress calculations than before. In this way, approximately realistic solutions can be found despite a recurring problem: the information deficits that are unavoidable due to the difficulty of exploring glaciers. This issue is demonstrated by the example of stagnating glaciers. For horizontally isotropic homogeneous tabular iceberg models, even mathematically exact unambiguous solutions of all relevant conditions are presented. All calculations use only elementary arithmetic operations, differentiations and integrations. The mathematical fundamentals are presented in detail and explained in many application examples. The integral operators specific to calculations of stresses facilitate the mathematical considerations. The stand-alone text allows the reader to understand what is involved even without considering the formulas. The author Peter Halfar is a theoretical physicist. He also developed a model of the movement of large ice caps (1983), which is still in use today.Table of ContentsI Introduction and fundamentals. Introduction.- Balance and boundary conditions.- Integral operators.- Forces and torques on surfaces.- Special solutions of balance conditions.- Weightless stress tensor fields.- II The general solution of balance and boundary conditions. Weightless stress tensor fields with boundary conditions.- The general solution of balance and boundary conditions.- Models and model selection. III Applications and examples. Land glaciers.- Floating glaciers.- IV Appendix. Bibliography.- Explanation and list of symbols.

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Book SynopsisThese lecture notes are devoted to the analysis of a nonlocal equation in the whole of Euclidean space. In studying this equation, all the necessary material is introduced in the most self-contained way possible, giving precise references to the literature when necessary. The results presented are original, but no particular prerequisite or knowledge of the previous literature is needed to read this text. The work is accessible to a wide audience and can also serve as introductory research material on the topic of nonlocal nonlinear equations.Table of ContentsIntroduction.- The problem studied in this monograph.- Functional analytical setting.- Existence of a minimal solution and proof of Theorem 2.2.2.- Regularity and positivity of the solution.- Existence of a second solution and proof of Theorem 2.2.4.

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Book SynopsisThis book treats some basic topics in the spectral theory of dynamical systems. The treatment is at a general level, but two more advanced theorems, one by H. Helson and W. Parry and the other by B. Host, are presented. Moreover, Ornstein's family of mixing rank one automorphisms is described with construction and proof. Systems of imprimitivity and their relevance to ergodic theory are discussed, and Baire category theorems of ergodic theory, scattered in the literature, are derived in a unified way. Riesz products are considered and they are used to describe the spectral types and eigenvalues of rank one automorphisms.The major change in this edition is that a new chapter titled Calculus of Generalized Riesz Products has been added. This is based on some recent work of the author with El Houcein El Abdalaoui and supplements the material presented elsewhere in the book.

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Book SynopsisThis monograph is devoted to the existence and stability (Ulam-Hyers-Rassias stability and asymptotic stability) of solutions for various classes of functional differential equations or inclusions involving the Hadamard or Hilfer fractional derivative. Some equations present delay which may be finite, infinite, or state-dependent. Others are subject to impulsive effect which may be fixed or non-instantaneous.Readers will find the book self-contained and unified in presentation. It provides the necessary background material required to go further into the subject and explores the rich research literature in detail. Each chapter concludes with a section devoted to notes and bibliographical remarks and all abstract results are illustrated by examples. The tools used include many classical and modern nonlinear analysis methods such as fixed-point theorems, as well as some notions of Ulam stability, attractivity and the measure of non-compactness as well as the measure of weak noncompactness. It is useful for researchers and graduate students for research, seminars, and advanced graduate courses, in pure and applied mathematics, physics, mechanics, engineering, biology, and all other applied sciences.

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Book SynopsisThis book explains the notion of Brakke’s mean curvature flow and its existence and regularity theories without assuming familiarity with geometric measure theory. The focus of study is a time-parameterized family of k-dimensional surfaces in the n-dimensional Euclidean space (1 ≤ k < n). The family is the mean curvature flow if the velocity of motion of surfaces is given by the mean curvature at each point and time. It is one of the simplest and most important geometric evolution problems with a strong connection to minimal surface theory. In fact, equilibrium of mean curvature flow corresponds precisely to minimal surface. Brakke’s mean curvature flow was first introduced in 1978 as a mathematical model describing the motion of grain boundaries in an annealing pure metal. The grain boundaries move by the mean curvature flow while retaining singularities such as triple junction points. By using a notion of generalized surface called a varifold from geometric measure theory which allows the presence of singularities, Brakke successfully gave it a definition and presented its existence and regularity theories. Recently, the author provided a complete proof of Brakke’s existence and regularity theorems, which form the content of the latter half of the book. The regularity theorem is also a natural generalization of Allard’s regularity theorem, which is a fundamental regularity result for minimal surfaces and for surfaces with bounded mean curvature. By carefully presenting a minimal amount of mathematical tools, often only with intuitive explanation, this book serves as a good starting point for the study of this fascinating object as well as a comprehensive introduction to other important notions from geometric measure theory.

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Book SynopsisThis book aims to establish a foundation for fractional derivatives and fractional differential equations. The theory of fractional derivatives enables considering any positive order of differentiation. The history of research in this field is very long, with its origins dating back to Leibniz. Since then, many great mathematicians, such as Abel, have made contributions that cover not only theoretical aspects but also physical applications of fractional calculus. The fractional partial differential equations govern phenomena depending both on spatial and time variables and require more subtle treatments. Moreover, fractional partial differential equations are highly demanded model equations for solving real-world problems such as the anomalous diffusion in heterogeneous media. The studies of fractional partial differential equations have continued to expand explosively. However we observe that available mathematical theory for fractional partial differential equations is not still complete. In particular, operator-theoretical approaches are indispensable for some generalized categories of solutions such as weak solutions, but feasible operator-theoretic foundations for wide applications are not available in monographs.To make this monograph more readable, we are restricting it to a few fundamental types of time-fractional partial differential equations, forgoing many other important and exciting topics such as stability for nonlinear problems. However, we believe that this book works well as an introduction to mathematical research in such vast fields.Trade Review“The book is written nicely and useful as an introductory book on time fractional derivatives in abstract spaces.” (Syed Abbas, zbMATH 1485.34002, 2022)Table of Contents

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Book SynopsisThis monograph provides a comprehensive overview on a class of nonlinear evolution equations, such as nonlinear Schrödinger equations, nonlinear Klein-Gordon equations, KdV equations as well as Navier-Stokes equations and Boltzmann equations. The global wellposedness to the Cauchy problem for those equations is systematically studied by using the harmonic analysis methods.This book is self-contained and may also be used as an advanced textbook by graduate students in analysis and PDE subjects and even ambitious undergraduate students.Table of ContentsFourier Multiplier, Function Spaces; Navier-Stokes Equation; Strichartz Estimates for Linear Dispersive Equations; Local and Global Wellposedness for Nonlinear Dispersive Equations; The Low Regularity Theory for the Nonlinear Dispersive Equations; Frequency-Uniform Decomposition Method; Conservations, Morawetz' Inequalities of NLS; Boltzmann Equation without Angular Cutoff.

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Book SynopsisThis book in the BCAM SpringerBriefs series is a treatise on the p-Laplace equation. It is based on lectures by the author that were originally delivered at the Summer School in Jyväskylä, Finland, in August 2005 and have since been updated and extended to cover various new topics, including viscosity solutions and asymptotic mean values. The p-Laplace equation is a far-reaching generalization of the ordinary Laplace equation, but it is non-linear and degenerate (p>2) or singular (p<2). Thus it requires advanced methods. Many fascinating properties of the Laplace equation are, in some modified version, extended to the p-Laplace equation. Nowadays the theory is almost complete, although some challenging problems remain open.Trade Review“The book will provide the reader with an up-to-date overview on the p-Laplace equation and will uncover ideas behind some concepts and involved results in the field.” (Vladimir Bobkov, Mathematical Reviews, December, 2019)“The book is a very useful contribution to the growing literature on this circle of ideas. I wholeheartedly recommend this book both as a textbook, as well as for independent study.” (Vicenţiu D. Rădulescu, zbMATH 1421.35002, 2019)Table of Contents1 Introduction.- 2 The Dirichlet problem and weak solutions.- 3 Regularity theory.- 4 Differentiability.- 5 On p-superharmonic functions.- 6 Perron's method.- 7 Some remarks in the complex plane.- 8 The infinity Laplacian.- 9 Viscosity solutions.- 10 Asymptotic mean values.- 11 Some open problems.- 12 Inequalities for vectors.

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Book SynopsisThis volume 6 of the Collected Works comprises 27 papers by V.I.Arnold, one of the most outstanding mathematicians of all times, written in 1991 to 1995. During this period Arnold's interests covered Vassiliev’s theory of invariants and knots, invariants and bifurcations of plane curves, combinatorics of Bernoulli, Euler and Springer numbers, geometry of wave fronts, the Berry phase and quantum Hall effect. The articles include a list of problems in dynamical systems, a discussion of the problem of (in)solvability of equations, papers on symplectic geometry of caustics and contact geometry of wave fronts, comments on problems of A.D.Sakharov, as well as a rather unusual paper on projective topology. The interested reader will certainly enjoy Arnold’s 1994 paper on mathematical problems in physics with the opening by-now famous phrase “Mathematics is the name for those domains of theoretical physics that are temporarily unfashionable.” The book will be of interest to the wide audience from college students to professionals in mathematics or physics and in the history of science. The volume also includes translations of two interviews given by Arnold to the French and Spanish media. One can see how worried he was about the fate of Russian and world mathematics and science in general.Table of Contents1 Bernoulli–Euler updown numbers associated with function singularities, their combinatorics and arithmetics.- 2 Congruences for Euler, Bernoulli and Springer numbers of Coxeter groups.- 3 The calculus of snakes and the combinatorics of Bernoulli, Euler and Springer numbers of Coxeter groups.- 4 Springer numbers and Morsification spaces.- 5 Polyintegrable flows.- 6 Bounds for Milnor numbers of intersections in holomorphic dynamical systems.- 7 Some remarks on symplectic monodromy of Milnor fibrations.- 8 Topological properties of Legendre projections in contact geometry of wave fronts [On topological properties of Legendre projections in contact geometry of wave fronts].- 9 Sur les propriétés topologiques des projections lagrangiennes en géométrie symplectique des caustiques [On topological properties of Lagrangian projections in symplectic geometry of caustics].- 10 Plane curves, their invariants, perestroikas and classifications (with an appendix by F. Aicardi).- 11 Invariants and perestroikas of plane fronts.- 12 The Vassiliev theory of discriminants and knots.- 13 The geometry of spherical curves and the algebra of quaternions.- 14 Remarks on eigenvalues and eigenvectors of Hermitian matrices, Berry phase, adiabatic connections and quantum Hall effect.- 15 Problems on singularities and dynamical systems.- 16 Sur quelques problèmes de la théorie des systèmes dynamiques [On some problems in the theory of dynamical systems].- 17 Mathematical problems in classical physics.- 18 Problèmes résolubles et problèmes irrésolubles analytiques et géométriques [Solvable and unsolvable analytic and geometric problems].- 19 Projective topology.- 20 Questions à V.I. Arnold (an interview with M. Audin and P. Iglésias) [Questions to V.I. Arnold].- 21 En todo matemático hay un ángel y un demonio (an interview with Marimar Jiménez) [In every mathematician, there is an angel and a devil].- 22 Will Russian mathematics survive?.- 23 Will mathematics survive? Report on the Zurich Congress.- 24 Why study mathematics? What mathematicians think about it.- 25 Preface to the Russian translation of the book by M.F. Atiyah “The Geometry and Physics of Knots”.- 26 A comment on one of A.D. Sakharov’s “Amateur Problems”.- 27 Comments on two of A.D. Sakharov’s “Amateur Problems”.- Acknowledgements.

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Book SynopsisTrade Review"Praise for the previous edition: "Linear Systems Theory gives a good presentation of the main topics on linear systems as well as more advanced topics related to controller design. The scholarship is sound and the book is very well written and readable.""---Ian Petersen, University of New South Wales"Praise for the previous edition: "This book provides a sound basis for an excellent course on linear systems theory. It covers a breadth of material in a fast-paced and mathematically focused way. It can be used by students wishing to specialize in this subject, as well as by those interested in this topic generally.""---Geir E. Dullerud, University of Illinois, Urbana-Champaign

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Book SynopsisCompletely revised and greatly expanded, the new edition of this text takes readers who have been exposed to only basic courses in analysis through the modern general theory of random processes and stochastic integrals as used by systems theorists, electronic engineers and, more recently, those working in quantitative and mathematical finance.Trade Review“As supplementary reading for a second course or as s comprehensive (!) resource for the general theory of processes aimed at Ph. D. students and scholars, this second edition will stay a valuable resource.” (René L. Schilling, Mathematical Reviews, October, 2016)“This is a fundamental book in modern stochastic calculus and its applications: rich contents, well structured material, comprehensive coverage of all significant results given with complete proofs and well illustrated by examples, carefully written text. Hence, there are more than enough reasons to strongly recommend the book to a wide audience. Among them, there are good and motivated graduate university students. … Also, the book is an excellent reference book.” (Jordan M. Stoyanov, zbMATH 1338.60001, 2016)Table of ContentsPart I: Measure Theoretic Probability.- Measure Integral.- Probabilities and Expectation.- Part II: Stochastic Processes.- Filtrations, Stopping Times and Stochastic Processes.- Martingales in Discrete Time.- Martingales in Continuous Time.- The Classification of Stopping Times.- The Progressive, Optional and Predicable -Algebras.- Part III: Stochastic Integration.- Processes of Finite Variation.- The Doob-Meyer Decomposition.- The Structure of Square Integrable Martingales.- Quadratic Variation and Semimartingales.- The Stochastic Integral.- Random Measures.- Part IV: Stochastic Differential Equations.- Ito's Differential Rule.- The Exponential Formula and Girsanov's Theorem.- Lipschitz Stochastic Differential Equations.- Markov Properties of SDEs.- Weak Solutions of SDEs.- Backward Stochastic Differential Equations.- Part V: Applications.- Control of a Single Jump.- Optimal Control of Drifts and Jump Rates.- Filtering. Part VI: Appendices.

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Book SynopsisProvides a quick, but very readable introduction to stochastic differential equations—that is, to differential equations subject to additive “white noise" and related random disturbances. The exposition is strongly focused upon the interplay between probabilistic intuition and mathematical rigour.Trade Review... [A]n interesting and unusual introduction to stochastic differential equations...topical and appealing to a wide audience. ... This is interesting stuff and, because of Evans' always clear explanations, it is fun too." - MAA ReviewsTable of Contents Preface Introduction A crash course in probability theory Brownian motion and “white noise” Stochastical integrals Stochastic differential equations Applications Appendix Exercises Notes and suggested reading Bibliography Index

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Book SynopsisThis undergraduate textbook is a rigorous mathematical introduction to dynamical systems and an accessible guide for students transitioning from calculus to advanced mathematics. It has many student-friendly features, such as graded exercises that range from straightforward to more difficult with hints, and includes concrete applications of real analysis and metric space theory to dynamical problems. Proofs are complete and carefully explained, and there is opportunity to practice manipulating algebraic expressions in an applied context of dynamical problems. After presenting a foundation in one-dimensional dynamical systems, the text introduces students to advanced subjects in the latter chapters, such as topological and symbolic dynamics. It includes two-dimensional dynamics, Sharkovsky''s theorem, and the theory of substitutions, and takes special care in covering Newton''s method. Mathematica code is available online, so that students can see implementation of many of the dynamicalTrade Review'This remarkable book provides a thoroughly field-tested way of teaching analysis while introducing dynamical systems. Combining lightness with rigor, it motivates and applies a wide range of subjects in the theory of metric spaces as it explores a broad variety of topics in dynamics.' Boris Hasselblatt, Tufts University, Massachusetts'This is a most impressive book. The author presents a range of topics which are not usually included in a book at this level (for example Sharkovsky's theorem, fractals, substitutions). The writing is clear and there are exercises of varying difficulty. A fine undergraduate text, which will also be of interest to graduate students and researchers in dynamics.' Joseph Auslander, Professor Emeritus of Mathematics, University of Maryland'This carefully written book introduces the student to a wealth of examples in dynamical systems, including several modern topics such as complex dynamics, topological dynamics and substitutions.' Cesar E. Silva, Williams College, Massachusetts'More rigorous than other undergraduate texts but less daunting than graduate books, this book is perfect for a core course on chaotic dynamic systems for undergraduates in their junior or senior year. Thoughtful, clear, and written with just the right amount of detail, Goodson develops the necessary tools required for an in-depth study of dynamical systems.' Alisa DeStefano, College of the Holy Cross, Massachusetts'… readers familiar with the basics of calculus, linear algebra, topology, and some real analysis will find that the topics are presented in an interesting manner, making this a good treatment of discrete dynamical systems … Summing Up: Recommended. Upper-division undergraduates and above; faculty and professionals.' M. D. Sanford, CHOICE'I think that this attractive textbook would be a welcome addition to the bookshelf of just about anyone with an interest in fractals, chaos, or dynamical systems. It presents most of the basic concepts in these fields at a level appropriate for senior math majors. Additional[ly], it has an extended treatment of substitution dynamical systems - the only undergraduate textbook I'm aware of that does so.' Christopher P. Grant, Mathematical Reviews'This book is a good example of what is possible as an introduction to this broad material of chaos, dynamical systems, fractals, tilings, substitutions, and many other related aspects. To bring all this in one volume and at a moderate mathematical level is an ambitious plan but these notes are the result of many years of teaching experience … The extraordinary combination of abstraction linked to simple yet appealing examples is the secret ingredient that is mastered wonderfully in this text.' Adhemar Bultheel, European Mathematical SocietyTable of Contents1. The orbits of one-dimensional maps; 2. Bifurcations and the logistic family; 3. Sharkovsky's theorem; 4. Dynamics on metric spaces; 5. Countability, sets of measure zero, and the Cantor set; 6. Devaney's definition of chaos; 7. Conjugacy of dynamical systems; 8. Singer's theorem; 9. Conjugacy, fundamental domains, and the tent family; 10. Fractals; 11. Newton's method for real quadratics and cubics; 12. Coppel's theorem and a proof of Sharkovsky's theorem; 13. Real linear transformations, the Hénon Map, and hyperbolic toral automorphisms; 14. Elementary complex dynamics; 15. Examples of substitutions; 16. Fractals arising from substitutions; 17. Compactness in metric spaces and an introduction to topological dynamics; 18. Substitution dynamical systems; 19. Sturmian sequences and irrational rotations; 20. The multiple recurrence theorem of Furstenberg and Weiss; Appendix A: theorems from calculus; Appendix B: the Baire category theorem; Appendix C: the complex numbers; Appendix D: Weyl's equidistribution theorem.

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a huge range and FREE tracked UK delivery on ALL orders.

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Book SynopsisAimed at graduates and researchers, and requiring only a basic knowledge of multi-variable calculus, this introduction to computer-based partial differential equation (PDE) modeling provides readers with the practical methods necessary to develop and use PDE mathematical models in biomedical engineering. Taking an applied approach, rather than using abstract mathematics, the reader is instructed through six biomedical example applications, each example characterized by step-by-step discussions of established numerical methods and implemented in reliable computer routines. Adopting this technique, the reader will understand how PDE models are formulated, implemented and tested. Supported by a set of rigorously tested general purpose PDE routines online, and with enhanced understanding through animations, this book will be ideal for anyone faced with interpreting large experimental data sets that need to be analyzed with PDE models in biomedical engineering.Table of Contents1. Introduction to partial differential equation integration in space and time; 2. Antibody binding kinetics; 3. Acid-mediated tumour growth; 4. Retinal oxygen transport; 5. Hemodialyzer dynamics; 6. Epidermal wound healing; 7. Drug distribution from a polymer matrix.

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Book SynopsisA modern introduction to synchronization phenomena, this text presents recent discoveries and the current state of research in the field, from low-dimensional systems to complex networks. The book describes some of the main mechanisms of collective behaviour in dynamical systems, including simple coupled systems, chaotic systems, and systems of infinite-dimension. After introducing the reader to the basic concepts of nonlinear dynamics, the book explores the main synchronized states of coupled systems and describes the influence of noise and the occurrence of synchronous motion in multistable and spatially-extended systems. Finally, the authors discuss the underlying principles of collective dynamics on complex networks, providing an understanding of how networked systems are able to function as a whole in order to process information, perform coordinated tasks, and respond collectively to external perturbations. The demonstrations, numerous illustrations and application examples will Table of ContentsPreface; 1. Introduction and main concepts; 2. Low-dimensional systems; 3. Multistable systems, coupled neurons and applications; 4. High-dimensional systems; 5. Complex networks; References; Index.

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Book SynopsisThis first introductory text to discrete integrable systems introduces key notions of integrability from the vantage point of discrete systems, also making connections with the continuous theory where relevant. While treating the material at an elementary level, the book also highlights many recent developments. Topics include: Darboux and BÃcklund transformations; difference equations and special functions; multidimensional consistency of integrable lattice equations; associated linear problems (Lax pairs); connections with Padà approximants and convergence algorithms; singularities and geometry; Hirota's bilinear formalism for lattices; intriguing properties of discrete Painlevà equations; and the novel theory of Lagrangian multiforms. The book builds the material in an organic way, emphasizing interconnections between the various approaches, while the exposition is mostly done through explicit computations on key examples. Written by respected experts in the field, the numerous exerTable of ContentsPreface; 1. Introduction to difference equations; 2. Discrete equations from transformations of continuous equations; 3. Integrability of P∆Es; 4. Interlude: lattice equations and numerical algorithms; 5. Continuum limits of lattice P∆Es; 6. One-dimensional lattices and maps; 7. Identifying integrable difference equations; 8. Hirota's bilinear method; 9. Multi-soliton solutions and the Cauchy matrix scheme; 10. Similarity reductions of integrable P∆Es; 11. Discrete Painlevé equations; 12. Lagrangian multiform theory; Appendix A. Elementary difference calculus and difference equations; Appendix B. Theta functions and elliptic functions; Appendix C. The continuous Painlevé equations and the Garnier system; Appendix D. Some determinantal identities; References; Index.

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Book SynopsisThe stability of stochastic differential equations in abstract, mainly Hilbert, spaces receives a unified treatment in this self-contained book. It covers basic theory as well as computational techniques for handling the stochastic stability of systems from mathematical, physical and biological problems. Its core material is divided into three parts devoted respectively to the stochastic stability of linear systems, non-linear systems, and time-delay systems. The focus is on stability of stochastic dynamical processes affected by white noise, which are described by partial differential equations such as the NavierStokes equations. A range of mathematicians and scientists, including those involved in numerical computation, will find this book useful. It is also ideal for engineers working on stochastic systems and their control, and researchers in mathematical physics or biology.Trade Review'The text itself is rather detailed, and therefore can be understood by graduate students and young researchers who have taken a solid course in stochastic analysis. Many examples are provided throughout the text to explain the finer points in the results.' Mar´ıa J. Garrido-Atienza, MathSciNetTable of ContentsPreface; 1. Preliminaries; 2. Stability of linear stochastic differential equations; 3. Stability of non linear stochastic differential equations; 4. Stability of stochastic functional differential equations; 5. Some applications related to stochastic stability; Appendix; References; Index.

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Book SynopsisThis introduction to finite difference and finite element methods is aimed at graduate students who need to solve differential equations. The prerequisites are few (basic calculus, linear algebra, and ODEs) and so the book will be accessible and useful to readers from a range of disciplines across science and engineering. Part I begins with finite difference methods. Finite element methods are then introduced in Part II. In each part, the authors begin with a comprehensive discussion of one-dimensional problems, before proceeding to consider two or higher dimensions. An emphasis is placed on numerical algorithms, related mathematical theory, and essential details in the implementation, while some useful packages are also introduced. The authors also provide well-tested MATLAB codes, all available online.Trade Review'The authors of this volume on finite difference and finite element methods provide a sound and complete exposition of these two numerical techniques for solving differential equations. The text is divided into two independent parts, tackling the finite difference and finite element methods separately. The parts offer a balanced mix of theory, application, and examples to offer readers a thorough introduction to the material. They utilize MATLAB programming to provide various codes illustrating the applications and examples. … Overall, the textbook offers a solid introduction to finite difference methods and finite element methods that should be useful to graduate students in mathematics as well as to students in applied and interdisciplinary fields, such as engineering and economics, who need to solve differential equations numerically.' S. L. Sullivan, ChoiceTable of Contents1. Introduction; Part I. Finite Difference Methods: 2. Finite difference methods for 1D boundary value problems; 3. Finite difference methods for 2D elliptic PDEs; 4. FD methods for parabolic PDEs; 5. Finite difference methods for hyperbolic PDEs; Part II. Finite Element Methods: 6. Finite element methods for 1D boundary value problems; 7. Theoretical foundations of the finite element method; 8. Issues of the FE method in one space dimension; 9. The finite element method for 2D elliptic PDEs; Appendix. Numerical solutions of initial value problems; References; Index.

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Book SynopsisMany problems encountered in applied mathematics or mathematical physics can be modelled by using differential equations under different boundary conditions. In this regard, linear and nonlinear partial differential equations are often used because of their strong capacity to describe and formulate many real-world problems governed by dynamical phenomena. There are many different methods to solve linear and nonlinear problems arising from different studies in various disciplines. However, due to lack of general existence theorems for establishing solutions, scientists have to seek alternative approaches and methods. In this context, the present work demonstrates different methods and approaches to obtain solutions to some class of differential equations given under different boundary conditions. The present book, where contemporary developments in the area of boundary value problems is shared, can be beneficial to advanced undergraduates, graduate students and researchers who are interested in the area of differential equations.Table of ContentsPreface; Boundary Value Problem of CO2 Production and Transport in Forest Sandy Soil; Boundary Value Problem of Hydrogen Thermal Desorption: Reduction to Fractional Differential Equation; Exact Absorbing Conditions for Initial Boundary Value Problems of Computational Electrodynamics: A Review; Diffraction Boundary Value Problems for Electromagnetic Theory of Inhomogeneous Multilayered Media: Riccati Equation Method; The Linearization Methods as a Basis to Derive the Relaxation and the Shooting Methods; The Existence of Boundary Value Problems of Fifth Painlevè Equation in a Complex Domain; Existence of Solutions for a Steklov Problem with Variable Exponent; On a Class of Kirchhoff Type Problems Involving Critical Exponents and Caffarelli-Kohn-Nirenberg Inequalities; Index.

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Book SynopsisThis book presents translations of selected Russian papers on the theoretical aspects of differential equations and applications of mathematical methods to modelling. These papers have been selected for their high scientific standards, innovative approaches, and topical interests.

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Book SynopsisThe aim of this volume is to introduce and exchange recent new topics on the areas of difference, differential, integrodifferential and integral equations, evolution equations, control and optimisation theory, dynamic system theory, queuing theory and electromagnetism and their applications.

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Book SynopsisThe aim of this volume is to introduce new topics on the areas of difference, differential, integrodifferential and integral equations, evolution equations, control and optimisation theory, dynamic system theory, queuing theory and electromagnetism and their applications.

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Book SynopsisThis book presents high-quality research on the theory and methods of linear or non-linear evolution equations as well as their further applications. Equations dealing with the asymptotic behaviour of solutions to evolution equations are included. This book also covers degenerate parabolic equations, abstract differential equations, comments on the Schrodinger equation, solutions in banach spaces, periodic and quasi-periodic solutions, concave Lagragian systems and integral equations.

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Book SynopsisThis book includes research on the Lax-Milgram theorem, which can be used to prove existence and uniqueness of weak solutions to partial differential equations and several examples of its application to relevant boundary value problems are presented. The authors also investigate nonlinear control problems for couple partial differential equations arising from climate and circulation dynamics in the equatorial zone; the integration of partial differential equations (PDE) with the help of non-commutative analysis over octonions and Cayley-Dickson algebras; and the existence and properties of solutions, applications in sequential optimal control with pointwise in time state constraints.

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Book SynopsisThe laws of nature are written in the language of partial differential equations. Therefore, these equations arise as models in virtually all branches of science and technology. Our goal in this book is to help you to understand what this vast subject is about. The book is an introduction to the field suitable for senior undergraduate and junior graduate students. Introductory courses in partial differential equations (PDEs) are given all over the world in various forms. The traditional approach to the subject is to introduce a number of analytical techniques, enabling the student to derive exact solutions of some simplified problems. Students who learn about computational techniques in other courses subsequently realize the scope of partial differential equations beyond paper and pencil. Our book is significantly different from the existing ones. We introduce both analytical theory, including the theory of classical solutions and that of weak solutions, and introductory techniques of ill-posed problems with reference to weak solutions. Besides, since computational techniques are commonly available and are currently used in all practical applications of partial differential equations, we incorporate classical finite difference methods and finite element methods in our book.

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Book SynopsisThis title is an introduction to the mathematical analysis of p- and hp-finite elements applied to elliptic problems in solid and fluid mechanics, and is suitable for graduate students and researchers who have had some prior exposure to finite element methods (FEM).Trade Review'Summarizing the book is the first theoretical book addressing the hp-version of the finite element method which is used today in practical computations. It is very well written and gives a very good review of the techniques and results in this relatively new direction in the FEM. It is highly recommended to anybody with mathematical interest for both learning and reference' ZAMMTable of ContentsVariational formulation of boundary value problems ; The Finite Element Method (FEM): definition, basic properties ; hp- Finite Elements in one dimension ; hp- Finite Elements in two dimensions ; Finite Element analysis of saddle point problems, mixed hp-FEM in incompressible fluid flow ; hp-FEM in the theory of elasticity

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Book SynopsisThis book provides a self-contained introduction to the mathematical theory of hyperbolic systems of conservation laws, with particular emphasis on the study of discontinuous solutions, characterized by the appearance of shock waves. This area has experienced substantial progress in very recent years thanks to the introduction of new techniques, in particular the front tracking algorithm and the semigroup approach. These techniques provide a solution to the long standing open problems of uniqueness and stability of entropy weak solutions. This monograph is the first to present a comprehensive account of these new, fundamental advances, mainly obtained by the author together with several collaborators. It also includes a detailed analysis of the stability and convergence of the front tracking algorithm. The book is addressed to graduate students as well as researchers. Both the elementary and the more advanced material are carefully explained, helping the reader''s visual intuition withTrade ReviewAn excellent and self-contained treatment of the mathematical theory of hyperbolic systems of conservation laws ... written in a clear and self-contained way and will be of great value for graduate students and specialists in the field. * EMS *

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Book SynopsisThis book provides a rapid introduction to the mathematical theory of compressible flow, giving a comprehensive account of the field and all important results up to the present day. The book is written in a clear, instructive and self-contained manner and will be accessible to a wide audience.Table of Contents1. Fundamental concepts and equations ; 2. Theoretical results for the Euler equations ; 3. Some mathematical tools for compressible flows ; 4. Weak solutions for steady compressible Navier-Stokes equations in barotropic regime ; 5. Strong solutions for steady compressible Navier-Stokes equations and small data ; 6. Some mathematical tools for non-steady equations ; 7. Weak solutions for non-stationary compressible Navier-Stokes equations ; 8. Global behavior of weak solutions ; 9. Strong solutions of non-steady compressible Navier-Stokes equations

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Book SynopsisThe theory of homogenization replaces a real composite material with an imaginary homogeneous one, to describe the macroscopic properties of the composite using the properties of the microscopic structure. This work illustrates the relevant mathematics, logic and methodology with examples.Trade Review'serve as good textbook for a post-graduate course' ZAMMTable of Contents1. Weak and weak - convergence in Banach spaces ; 2. Rapidly oscillating periodic functions ; 3. Some classes of Sobolev spaces ; 4. Some variational elliptic problems ; 5. Examples of periodic composite materials ; 6. Homogenization of elliptic equations: the convergence result ; 7. The multiple-scale method ; 8. Tartar's method of oscillating test functions ; 9. The two-scale convergence method ; 10. Homogenization in linearized elasticity ; 11. Homogenization of the heat equation ; 12. Homogenization of the wave equation ; 13. General Approaches to the non-periodic case ; References

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Book SynopsisHeun''s equation is a second-order differential equation which crops up in a variety of forms in a wide range of problems in applied mathematics. These include integral equations of potential theory, wave propogation, electrostatic oscillation, and Schrodinger''s equation. This volume brings together important research work for the first time, providing an important resource for all those interested in this mathematical topic. Both the current theory and the main areas of application are surveyed, and includes contributions from authoritative researchers such as Felix Arscott (Canada), P. Maroni (France), and Gerhard Wolf (Germany).Trade ReviewThere is a wealth of important results and open problems and the book is a welcome addition to the literature on these important special functions and their applications. * B D Sleeman, Zbl. Math. 847/96. *Table of ContentsA. HEUN'S EQUATION ; I: GENERAL AND POWER SERIES ; II: HYPERGEOMETRIC FUNCTION SERIES ; B. CONFLUENT HEUN EQUATION ; C. DOUBLE CONFLUENT HEUN EQUATION ; D. BICONFLUENT HEUN EQUATION ; E. TRICONFLUENT HEUN EQUATION

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Book SynopsisThis text is concerned with the quantitative aspects of the theory of nonlinear diffusion equations; equations which can be seen as nonlinear variations of the classical heat equation. They appear as mathematical models in different branches of Physics, Chemistry, Biology, and Engineering, and are also relevant in differential geometry and relativistic physics. Much of the modern theory of such equations is based on estimates and functional analysis.Concentrating on a class of equations with nonlinearities of power type that lead to degenerate or singular parabolicity (equations of porous medium type), the aim of this text is to obtain sharp a priori estimates and decay rates for general classes of solutions in terms of estimates of particular problems. These estimates are the building blocks in understanding the qualitative theory, and the decay rates pave the way to the fine study of asymptotics. Many technically relevant questions are presented and analyzed in detail. A systematic pTrade ReviewThis book is intended to introduce graduate students to the methods and results of nonlinear diffusion equations of porous medium type, as practised today. The present text, remarkable for generality and depth, is also notable for its author's concern, throughout, to keep the important issues about varieties clearly in the foreground ... [the book] succeeds admirably, in the reviewer's opinion, in introducing its difficult subject at a level appropriate for preparing future workers in the field. * Vicentiu Radulescu, Mathematical Reviews Issue 2007k *Table of ContentsPART I; PART II; PART III

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Book SynopsisSelf-adaptive discretization methods are now an indispensable tool for the numerical solution of partial differential equations that arise from physical and technical applications. The aim is to obtain a numerical solution within a prescribed tolerance using a minimal amount of work. The main tools in achieving this goal are a posteriori error estimates which give global and local information on the error of the numerical solution and which can easily be computed from the given numerical solution and the data of the differential equation. This book reviews the most frequently used a posteriori error estimation techniques and applies them to a broad class of linear and nonlinear elliptic and parabolic equations. Although there are various approaches to adaptivity and a posteriori error estimation, they are all based on a few common principles. The main aim of the book is to elaborate these basic principles and to give guidelines for developing adaptive schemes for new problems. ChaptersTrade ReviewError control and adaptive solution algorithms for finite element approximation are a key concern of every practitioner. The present text, written by a leading authority in the field who has made many important contributions, will be valuable for theoreticians and practitioners alike. * Mark Ainsworth, Professor of Applied Mathematics, Brown University *Table of Contents1. A Simple Model Problem ; 2. Implementation ; 3. Auxiliary Results ; 4. Linear Elliptic Equations ; 5. Nonlinear Elliptic Equations ; 6. Parabolic Equations

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