Differential calculus and equations Books

Book SynopsisThis volume covers the stability of nonautonomous differential equations in Banach spaces in the presence of nonuniform hyperbolicity. Topics under discussion include the Lyapunov stability of solutions, the existence and smoothness of invariant manifolds, and the construction and regularity of topological conjugacies. The exposition is directed to researchers as well as graduate students interested in differential equations and dynamical systems, particularly in stability theory.Trade ReviewFrom the reviews: “In this book, the authors give a unified presentation of a substantial body of work which they have carried out and which revolves around the concept of nonuniform exponential dichotomy. … This is a well-written book which contains many interesting results. The reader will find significant generalizations of the standard invariant manifold theories, of the Hartman-Grobman theorem … . Anyone interested in these topics will profit from reading this book.” (Russell A. Johnson, Mathematical Reviews, Issue 2010 b)Table of ContentsExponential dichotomies.- Exponential dichotomies and basic properties.- Robustness of nonuniform exponential dichotomies.- Stable manifolds and topological conjugacies.- Lipschitz stable manifolds.- Smooth stable manifolds in Rn.- Smooth stable manifolds in Banach spaces.- A nonautonomous Grobman–Hartman theorem.- Center manifolds, symmetry and reversibility.- Center manifolds in Banach spaces.- Reversibility and equivariance in center manifolds.- Lyapunov regularity and stability theory.- Lyapunov regularity and exponential dichotomies.- Lyapunov regularity in Hilbert spaces.- Stability of nonautonomous equations in Hilbert spaces.

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Book SynopsisIn recent years, the Fourier analysis methods have expereinced a growing interest in the study of partial differential equations. In particular, those techniques based on the Littlewood-Paley decomposition have proved to be very efficient for the study of evolution equations. The present book aims at presenting self-contained, state- of- the- art models of those techniques with applications to different classes of partial differential equations: transport, heat, wave and Schrödinger equations. It also offers more sophisticated models originating from fluid mechanics (in particular the incompressible and compressible Navier-Stokes equations) or general relativity. It is either directed to anyone with a good undergraduate level of knowledge in analysis or useful for experts who are eager to know the benefit that one might gain from Fourier analysis when dealing with nonlinear partial differential equations.Trade ReviewFrom the reviews:“The authors did make impressive contributions to a broad area of fluid dynamics. It is the first time that a coherent presentation of those research results is available, which will give easier access to the whole area to a broader audience. … It is a valuable contribution in the important area of the interest of the authors and will without question find its place in the mathematical libraries, and on the shelves of people working in those areas.” (Herbert Koch, Jahresbericht der Deutschen Mathematiker-Vereinigung, Vol. 115, 2014)“The aim of the present monograph is to introduce methods from Fourier analysis, and in particular techniques based on the Littlewood–Paley decomposition, for the solution of nonlinear partial differential equations. … The presentation is fairly self-contained and only requires a solid background in measure theory and functional analysis. It will be of value to both graduate students and researchers interested in application of Fourier analysis to partial differential equations.” (G. Teschl, Monatshefte für Mathematik, Vol. 165 (3-4), March, 2012)“This book is a well-written introduction to Fourier analysis, Littlewood-Paley theory and some of their applications to the theory of evolution equations. It is suitable for readers with a solid undergraduate background in analysis. A feature that distinguishes it from other books of this sort is its emphasis on using Littlewood-Paley decomposition to study nonlinear differential equations. … the references, historical background, and discussion of possible future developments at the end of each chapter are very convenient for its readers.” (Lijing Sun, Zentralblatt MATH, Vol. 1227, 2012)“This book intends to prepare the reader how to apply tools from Fourier analysis to directly solve problems arising in the theory of non linear partial differential equations. … The presentation is well structured and easy to follow. … This is a textbook for advanced undergraduate or beginning graduate students with a good background in real and functional analysis. … even active researchers or mathematicians interested in the application of Fourier-analytic tools will find this book very useful.” (Peter R. Massopust, Mathematical Reviews, Issue 2011 m)Table of ContentsPreface.- 1. Basic analysis.- 2. Littlewood-Paley theory.- 3. Transport and transport-diffusion equations.- 4. Quasilinear symmetric systems.- 5. Incompressible Navier-Stokes system.- 6. Anisotropic viscosity.- 7. Euler system for perfect incompressible fluids.- 8. Strichartz estimates and applications to semilinear dispersive equations.- 9. Smoothing effect in quasilinear wave equations.- 10.- The compressible Navier-Stokes system.- References. - List of notations.- Index.

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Book SynopsisIn this third volume of his modern introduction to quantum field theory, Eberhard Zeidler examines the mathematical and physical aspects of gauge theory as a principle tool for describing the four fundamental forces which act in the universe: gravitative, electromagnetic, weak interaction and strong interaction. Volume III concentrates on the classical aspects of gauge theory, describing the four fundamental forces by the curvature of appropriate fiber bundles. This must be supplemented by the crucial, but elusive quantization procedure. The book is arranged in four sections, devoted to realizing the universal principle force equals curvature: Part I: The Euclidean Manifold as a ParadigmPart II: Ariadne's Thread in Gauge TheoryPart III: Einstein's Theory of Special RelativityPart IV: Ariadne's Thread in Cohomology For students of mathematics the book is designed to demonstrate that detailed knowledge of the physical background helps to reveal interesting interrelationships among diverse mathematical topics. Physics students will be exposed to a fairly advanced mathematics, beyond the level covered in the typical physics curriculum. Quantum Field Theory builds a bridge between mathematicians and physicists, based on challenging questions about the fundamental forces in the universe (macrocosmos), and in the world of elementary particles (microcosmos). Trade ReviewFrom the reviews:“This book is the third volume of a complete exposition of the important mathematical methods used in modern quantum field theory. It presents the very basic formalism, important results, and the most recent advances emphasizing the applications to gauge theory. … the book’s greatest strength is Zeidler’s zeal to help students understand fundamental mathematics better. I thus find the book extremely useful since it signifies the role of mathematics for the road to reality … .” (Gert Roepstorff, Zentralblatt MATH, Vol. 1228, 2012)“The present book is a good companion to the literature on the subject of the volume title, especially for those already familiar with it. … the book touches upon a large number of subjects on the interface between mathematics and physics, providing a good overview of gauge theory in both fields. It contains lots of background material, many historical remarks, and an extensive bibliography that helps the interested reader to continue his or her more thorough studies elsewhere.” (Walter D. van Suijlekom, Mathematical Reviews, Issue 2012 m)Table of ContentsPrologue.- Part I. The Euclidean Manifold as a Paradigm: 1. The Euclidean Space E3 (Hilbert Space and Lie Algebra Structure).- 2. Algebras and Duality (Tensor Algebra, Grassmann Algebra, Cli_ord Algebra, Lie Algebra).- 3. Representations of Symmetries in Mathematics and Physics.- 4. The Euclidean Manifold E3.- 5. The Lie Group U(1) as a Paradigm in Harmonic Analysis and Geometry.- 6. Infinitesimal Rotations and Constraints in Physics.- 7. Rotations, Quaternions, the Universal Covering Group, and the Electron Spin.- 8. Changing Observers - A Glance at Invariant Theory Based on the Principle of the Correct Index Picture.- 9. Applications of Invariant Theory to the Rotation Group.- 10. Temperature Fields on the Euclidean Manifold E3.- 11. Velocity Vector Fields on the Euclidean Manifold E3.- 12. Covector Fields and Cartan's Exterior Differential - the Beauty of Differential Forms.- Part II. Ariadne's Thread in Gauge Theory: 13. The Commutative Weyl U(1)-Gauge Theory and the Electromagnetic Field.- 14. Symmetry Breaking.- 15. The Noncommutative Yang{Mills SU(N)-Gauge Theory.- 16. Cocycles and Observers.- 17. The Axiomatic Geometric Approach to Bundles.- Part III. Einstein's Theory of Special Relativity: 18. Inertial Systems and Einstein's Principle of Special Relativity.- 19. The Relativistic Invariance of the Maxwell Equations.- 20. The Relativistic Invariance of the Dirac Equation and the Electron Spin.- Part IV. Ariadne's Thread in Cohomology: 21. The Language of Exact Sequences.- 22. Electrical Circuits as a Paradigm in Homology and Cohomology.- 23. The Electromagnetic Field and the de Rham Cohomology.- Appendix.- Epilogue.- References.- List of Symbols.- Index

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Book SynopsisThis self-contained monograph presents matrix algorithms and their analysis. The new technique enables not only the solution of linear systems but also the approximation of matrix functions, e.g., the matrix exponential. Other applications include the solution of matrix equations, e.g., the Lyapunov or Riccati equation. The required mathematical background can be found in the appendix.The numerical treatment of fully populated large-scale matrices is usually rather costly. However, the technique of hierarchical matrices makes it possible to store matrices and to perform matrix operations approximately with almost linear cost and a controllable degree of approximation error. For important classes of matrices, the computational cost increases only logarithmically with the approximation error. The operations provided include the matrix inversion and LU decomposition.Since large-scale linear algebra problems are standard in scientific computing, the subject of hierarchical matrices is of interest to scientists in computational mathematics, physics, chemistry and engineering.Trade Review“Every line of the book reflects that the author is the leading expert for hierarchical matrices. … Hierarchical matrices: algorithms and analysis is without a doubt a beautiful, comprehensive introduction to hierarchical matrices that can serve as both a graduate level textbook and a valuable resource for future research.” (Thomas Mach, Mathematical Reviews, April, 2017)“The book ‘Hierarchical matrices: algorithms and analysis’ is a self-contained monograph which presents an efficient possibility to handle the numerical treatment of fully populated large scale matrices appearing in scientific computations, and therefore it is of interest to scientists in computational mathematics, physics, chemistry and engineering.” (Constantin Popa, zbMATH 1336.65041, 2016)Table of ContentsPreface.- Part I: Introductory and Preparatory Topics.- 1. Introduction.- 2. Rank-r Matrices.- 3. Introductory Example.- 4. Separable Expansions and Low-Rank Matrices.- 5. Matrix Partition.- Part II: H-Matrices and Their Arithmetic.- 6. Definition and Properties of Hierarchical Matrices.- 7. Formatted Matrix Operations for Hierarchical Matrices.- 8. H2-Matrices.- 9. Miscellaneous Supplements.- Part III: Applications.- 10. Applications to Discretised Integral Operators.- 11. Applications to Finite Element Matrices.- 12. Inversion with Partial Evaluation.- 13. Eigenvalue Problems.- 14. Matrix Functions.- 15. Matrix Equations.- 16. Tensor Spaces.- Part IV: Appendices.- A. Graphs and Trees.- B. Polynomials.- C. Linear Algebra and Functional Analysis.- D. Sinc Functions and Exponential Sums.- E. Asymptotically Smooth Functions.- References.- Index.

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Book Synopsis Kapitel 1. Grundbegriffe der Differentialgleichungen - Kapitel 2. Differentialgleichungen erster Ordnung - Kapitel 3. Differentialgleichungen zweiter Ordnung - Kapitel 4. Laplace-Transformationen.- Kapitel 5. System von linearen Differentialgleichungen.- Kapitel 6. Potenzreihenlösungen.- Kapitel 7. Numerische Methoden für Anfangswertprobleme.- Kapitel 8. Schießverfahren für lineare Randbedingungen.- Anhang A. Potenzreihen.- Anhang B. Einige elementare Integrationsformeln.- Anhang C. Tabelle der Laplace-Transformationen.          ?

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Book SynopsisThe Adomian decomposition method enables the accurate and efficient analytic solution of nonlinear ordinary or partial differential equations without the need to resort to linearization or perturbation approaches. It unifies the treatment of linear and nonlinear, ordinary or partial differential equations, or systems of such equations, into a single basic method, which is applicable to both initial and boundary-value problems. This volume deals with the application of this method to many problems of physics, including some frontier problems which have previously required much more computationally-intensive approaches. The opening chapters deal with various fundamental aspects of the decomposition method. Subsequent chapters deal with the application of the method to nonlinear oscillatory systems in physics, the Duffing equation, boundary-value problems with closed irregular contours or surfaces, and other frontier areas. The potential application of this method to a wide range of problems in diverse disciplines such as biology, hydrology, semiconductor physics, wave propagation, etc., is highlighted. For researchers and graduate students of physics, applied mathematics and engineering, whose work involves mathematical modelling and the quantitative solution of systems of equations. Trade Review`I recommend Adomian's new book to all researchers in the area of mathematical modeling and solving complex dynamical systems.' Foundations of Physics, 1994 Table of ContentsPreface. Foreword. 1. On Modelling Physical Phenomena. 2. The Decomposition Method for Ordinary Differential Equations. 3. The Decomposition Method in Several Dimensions. 4. Double Decomposition. 5. Modified Decomposition. 6. Applications of Modified Decomposition. 7. Decomposition Solutions for Neumann Boundary Conditions. 8. Integral Boundary Conditions. 9. Boundary Conditions at Infinity. 10. Integral Equations. 11. Nonlinear Oscillations in Physical Systems. 12. Solution of the Duffing Equation. 13. Boundary-Value Problems with Closed Irregular Contours or Surfaces. 14. Applications in Physics. Appendix I: Padé and Shanks Transform. Appendix II: On Staggered Summation of Double Decomposition Series. Appendix III: Cauchy Products of Infinite Series. Index.

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Book SynopsisChapter 1  Distributions.- Chapter 2  Fundamental Solutions of Linear Differential Operators.- Chapter 3  Boundary Value Problems of the Laplace Equations.- Chapter 4  Boundary Value Problem of Modified Helmholtz Equation.- Chapter 5  Boundary Value Problems of Helmholtz Equation.- Chapter 6  Boundary Value Problems of the Navier Equations.- Chapter 7  Boundary Value Problems of the Stokes Equations.- Chapter 8  Some Nonlinear Problems.- Chapter 9  Coercive and Symmetrical Coupling Methods of Finite Element Method and Boundary Element Method.

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Book SynopsisGlobally integrable quantum systems and their perturbations.- On two-dimensional Dirac operators with $delta$-shell interactions supported on unbounded curves with straight ends.- Attractor Subspace and Decoherence-Free Algebra of Quantum Dynamics.- Algebraic localization of generalized Wannier bases implies Roe triviality in any dimension.- Hearing the boundary conditions of the one-dimensional Dirac operator Bosonized Momentum Distribution of a Fermi Gas via Friedrichs Diagrams.- Self-adjointness and Domain of Generalized Spin–Boson Models with Mild Ultraviolet Divergences.- Random Linear Systems with Quadratic Constraints: from Random Matrix Theory to replicas and back.- New analytical and geometrical aspects on Trudinger-Moser type inequality in 2D.- Resolvent limits of exterior boundary value problems and singular perturbation of Laplace operator in 3D.- The Search for NLS Ground States on a hybrid domain: motivations, methods, and results.- From microscopic to macroscopic: the large number dynamics of agents and cells, possibly interacting with a chemical background.- Open problems and perspectives on solving Friedrichs systems by Krylov approximation.- Singularity: a Seventh Memo.

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Book SynopsisChapter 1 A comparison of the Coco-Russo scheme and -FEM for elliptic equations in arbitrary domains.- Chapter 2 A semi-implicit method for a degenerating convection-diffusion-reaction problem modeling secondary settling tanks.- Chapter 3 Multidimensional approximate Riemann solvers for hyperbolic nonconservative systems: a review.- Chapter 4 Challenges in Stochastic Galerkin Methods for Nonlinear Hyperbolic Systems with Uncertainty.- Chapter 5 On the role of momentum correction factor and general tube law in one-dimensional blood flow models for networks of vessels.- Chapter 6 Numerical modelling of the hemodynamic changes in the inferior vena cava in response to the Valsalva maneuver.

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Book SynopsisPreface.- The Autonomous Duffing Equation.- The Periodically Forced Duffing Equation.- Chaos in the Duffing Equation: With Some Simulations.- Topological Methods for the Detection of Chaos.- Applications to the Superlinear Duffing Equation.- Laser.- The Forced Pendulum.- Chaos in the Duffing-type Equation related to Tides.- Index.

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Book SynopsisTrade Review"Overall, the reviewer considers this text to offer a good and useful coverage of advanced mathematics for engineers. It gives useful and succinct coverage of the topics included." --IEEE PulseTable of Contents1. Prologue 2. Ordinary Differential Equations 3. Laplace and Fourier Transform Methods 4. Matrices and Linear Systems of Equations 5. Analytical Methods for Solving Partial Differential Equations 6.Difference Numerical Methods for Differential Equations 7. Finite Element Technique 8. Treatment of Experimental Results 9. Numerical Analysis 10. Introduction to Complex Analysis 11. Nondimensionalisation 12. Nonlinear Differential Equations 13. Integral Equations 14. Calculus of Variations

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Book SynopsisLinear non-autonomous equations arise as mathematical models in mechanics, chemistry, and biology. This book explores the preservation of invariant tori of dynamic systems under perturbation. It is a useful contribution to the literature on stability theory and provides a source of reference for postgraduates and researchers.Trade Review"This volume will be of great interest to researchers and students dealing with nonautonomous systems." - Zentralblatt fur Mathematik, Vol. 1026Table of ContentsExponentially Dichotomous Linear Systems of Differential Equations and Lyapunov Functions of Variable Sign. Exponential Dichotomy Criterion for Linear Systems in Terms of Quadratic Forms. Decomposition Over the Whole R Axis of Linear Systems of Differential Equations Exponentially Dichotomous on Semiaxes R+ and R_. Degeneracy of the Quadratic Form Possessing a Definite-Sign Derivative Along the Solutions of the System (1.1.1). Integral Representation of Weakly Regular Systems Bounded on the Whole R Axis. Complement to the Exponentially Dichotomous of Weakly Regular on R Linear Systems. Regularity of Linear Systems of the Block-Triangular Form. Perturbation of the Block-Triangular Form Linear Systems which are Regular and Weakly Regular on the Whole R Axis. Exponentially Dichotomous Linear Systems with Parameters. Comments and References. Linear Extension of Dynamical Systems on a Torus. Necessary Existence Conditions for Invariant Tori. The Green Function. Sufficient Existence Conditions for an Invariant Torus. Existence Conditions for an Exponentially Stable Invariant Torus. Uniqueness Conditions for the Green Function and its Properties. Sufficient Conditions for Exponential Dichotomy of the Invariant Torus. Necessary Conditions for Exponential Dichotomy of the Invariant Torus. Existence Criterion for the Green Function. The Non-Unique Green Function and the Properties of the System Implied by its Existence. Invariant Tori of Linear Extensions with Slowly Changing Phase. Preserving the Green Function Under Small Perturbations of Linear Expansions on a Torus. On the Smoothness of an Exponentially Stable Invariant Torus. On the Dependence of Green Functions on Parameters. Continuity and Differentiability of the Green Function. Invariant Tori of Linear Extensions with a Degenerate Matrix at the Derivatives. Bounded Invariant Manifolds of Dynamical Systems and their Smoothness. Comments and References. Splitability of Linear Extensions of Dynamical Systems on a Torus. Sufficient Conditions for Splitability of Linear Extensions of Dynamical Systems on a Torus. Reversibility of the Theorem on Splitability. On Triangulation and the Relationship of C'-Block Splitability of a Linear System with the Problem on r-frame Complementability up to the Periodic Basis in R^Tn. Reducing on Linearized Systems to a Diagonal Form. On the Relationship of Exponentially Dichotomous Linear Expansions with the Algebraic System Solvability. Three Block Divisibility of Linear Extensions and Lyapunov Functions of Variable Sign. Algebraic Problems of the K-Blocked Divisibility of Linear Extensions on a Torus. Comments and References. Problems of Perturbation Theory of Smooth Invariant Tori of Dynamical Systems. Solution Variations on the Manifold M. Exponential Stability and Dichotomy Conditions for Linear Extensions of Dynamical Systems on a Torus. Roughness Conditions for the Green Function of the Linear Extension of a Dynamical System on a Torus with the Index of Smoothness. A Theorem of Perturbation Theory of an Invariant Torus of a Dynamical System. Green Function for a Linear Matrix Equation. On the Problem of Structure of Some Regular Linear Extensions of Dynamical Systems on a Torus. Invariant Manifolds of Autonomous Differential Equations and Lyapunov Functions with Alternating Signs. Comments and References. Index.

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Book SynopsisNonlinear systems with random structures arise quite frequently as mathematical models in diverse disciplines. This monograph presents a systematic treatment of stability theory and the theory of stabilization of nonlinear systems with random structure in terms of new developments in the direct Lyapunov's method. The analysis focuses on dynamic systems with random Markov parameters. This high-level research text is recommended for all those researching or studying in the fields of applied mathematics, applied engineering, and physics-particularly in the areas of stochastic differential equations, dynamical systems, stability, and control theory.Trade Review"This volume will be of interest to researchers and students in stochastic stability theory." - Zentralblatt fur Mathematik, Vol. 1026Table of ContentsIntroductory Remarks. Random Variables and Probability Distributions. Probability Processes and their Mathematical Description. Random Differential Equations. System with Random Structure. Stability Analysis Using Scalar Lyapunov Functions. Stability Concepts for Stochastic Systems. Random Scalar Lyapunov Functions. Conditions of Stability in Probability. Converse Theorems. Stability in Mean Square. Stability in Mean Square of Linear Systems. Stability Analysis Using Multi-component Lyapunov Functions. Vector Lyapunov Functions. Stochastic Matrix-Valued Lyapunov Functions. Stability Analysis in General. Stability Analysis of Systems in Ito's Form. Stochastic Singularly Perturbed Systems. Large-Scale Singularly Perturbed Systems. Stability Analysis by the First-Order Approximation. Stability Criterion by the First-Order Approximation. Stability with Respect to the First-Order Approximation. Stability by First-Order Approximation of Systems with Random Delay. Convergence of Stochastic Approximation Procedure. Stabilization of Controlled Systems with Random Structure. Problems of Stabilization. Optimal Stabilization. Linear-Quadratic Optimal Stabilization. Sufficient Stabilization Conditions for Linear Systems. Optimal Solution Existence. The Small Parameter Method Algorithm. Applications. A Stochastic Version of the Lefschetz Problem. Stability in Probability of Oscillating Systems. Stability in Probability of Regulation Systems. Price Stability in a Stochastic Market Model. References. Index. Lyapunov Functions. Stability Analysis Using Multicomponent Lyapunov Functions. Stability Analysis by the First-order Approximation. Stabilization of Controlled Systems with Random Structure. Applications; References; Index.

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Book SynopsisEquations of Mathematical Diffraction Theory focuses on the comparative analysis and development of efficient analytical methods for solving equations of mathematical diffraction theory. Following an overview of some general properties of integral and differential operators in the context of the linear theory of diffraction processes, the authors provide estimates of the operator norms for various ranges of the wave number variation, and then examine the spectral properties of these operators. They also present a new analytical method for constructing asymptotic solutions of boundary integral equations in mathematical diffraction theory for the high-frequency case.Clearly demonstrating the close connection between heuristic and rigorous methods in mathematical diffraction theory, this valuable book provides you with the differential and integral equations that can easily be used in practical applications.Table of ContentsSome Preliminaries from Analysis and the Theory of Wave Processes. Integral Equations of Diffraction Theory for Obstacles in Unbounded Medium. Wave Fields in a Layer of Constant Thickness. Analytical Methods for Simply Connected Bounded Domains. Integral Equations in Diffraction by Linear Obstacles. Short-Wave Asymptotic Methods on the Basis of Multiple Integrals. Inverse Problems of the Short-Wave Diffraction. Ill-Posed Equations of Inverse Diffraction Problems for Arbitrary Boundary. Numerical Methods for Irregular Operator Equations.

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Book SynopsisThis second edition of the book, Nonlinear Random Vibration: Analytical Techniques and Applications, expands on the original edition with additional detailed steps in various places in the text. It is a first systematic presentation on the subject. Its features include:â a concise treatment of Markovian and non- Markovian solutions of nonlinear stochastic differential equations,â exact solutions of Fokker-Planck-Kolmogorov equations,â methods of statistical linearization,â statistical nonlinearization techniques,â methods of stochastic averaging,â truncated hierarchy techniques, andâ an appendix on probability theory.A special feature is its incorporation of detailed steps in many examples of engineering applications.Targeted audience: Graduates, research scientists and engineers in mechanical, aerospace, civil and environmental (earthquake, wind and transportation), automobile, naval, architectural, and mining engineering.Trade ReviewIn summary, the technical material in Prof. To’s 2012 second edition of Nonlinear Random Vibration: Analytical Techniques and Applications is well presented, of sufficient depth, detail, and quality, and supported by a good number of solved example problems.Robert M. KochNaval Undersea Warfare Center, Newport, RI, USAIn: Noise Control Engr. J. 61 (2), March-April 2013, pp 251-252Table of ContentsIntroduction. Markovian and Non-Markovian Solutions of Stochastic Nonlinear Differential Equations. Exact Solution of the Fokker-Planck-Kolmogorov Equation. Methods of Statistical Linearization. Statistical Nonlinearization Techniques. Methods of Stochastic Averaging. Truncated Hierarchy and other Techniques.

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Book SynopsisThis book contains expository papers and articles reporting on recent research by leading world experts in nonstandard mathematics, arising from the International Colloquium on Nonstandard Mathematics held at the University of Aveiro, Portugal in July 1994. Nonstandard mathematics originated with Abraham Robinson, and the body of ideas that have developed from this theory of nonstandard analysis now vastly extends Robinson''s work with infinitesimals. The range of applications includes measure and probability theory, stochastic analysis, differential equations, generalised functions, mathematical physics and differential geometry, moreover, the theory has implicaitons for the teaching of calculus and analysis. This volume contains papers touching on all of the abovbe topics, as well as a biographical note about Abraham Robinson based on the opening address given by W.A>J> Luxemburg - who knew Robinson - to the Aveiro conference which marked the 20th anniversary of Robinson'Table of ContentsThe infinitesimal rule of threeNonstandard methods in the precalculus curruculumDifference quotients and smoothnessContinuous maps with special propertiesSome nonstandard methods in geometric topologyDelayed bifurcations in perturbed systems analysis of slow passage of Suhl-thresholdFunctional analysis and NSANear-standard compact internal linear operatorsDiscrete Fredholm's equationsNonstandard theory of generalized functionsRepresenting distributions by nonstandard polynomialsContributions of nonstandard analysis to partial differential equationsLoeb measure theoryUnions of Loeb nullsets: the contextGredient lines and distributions of functionals in infinite dimensional Euclidean spacesNonstandard flat integral representation of the free Euclidean field and a large deviation bound for the exponential interactionNonstandard analysis in selective uniersesLattices and monadsA neometric surveyLong sequences and neocompact sets

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Book SynopsisThe First Pan-China Conference on Differential Equations was held in Kunming, China in June of 1997. Researchers from around the world attended-including representatives from the US, Canada, and the Netherlands-but the majority of the speakers hailed from China and Hong Kong. This volume contains the plenary lectures and invited talks presented at that conference, and provides an excellent view of the research on differential equations being carried out in China.Most of the subjects addressed arose from actual applications and cover ordinary and partial differential equations. Topics include:Table of ContentsPART I: ORDINARY DIFFERENTIAL EQUATIONSAdvances in the Asymptotic and Numerical Solution of Linear Ordinary Differential Equations, F.W.J. OlverSome Unsolved Problems in Asymptotics, R. WongPeriodic Solutions and Heteroclinic Cycles in the Convection Model of a Rotating Fluid Layer, J. Li and X.H. ZhaoThe Equivalence of Exponential Stability for Impulsive Time-Delay Differential Systems, Z.-H. Guan, Y.-C. Zhou, and X.-P. HeConditions for Identity of Bifurcations in Cubic Hamiltonian Systems with Symmetry or Nonsymmetry Perturbations, Z. Liu, H. Cao, and J. LiPART II: PARTIAL DIFFERENTIAL EQUATIONSLong Time Behavior for the Generalized Ginzburg-Landau Equations, B. GuoThe Inverse Scattering Transform for a Variable-Coefficient KdV Equations (with Applications to Shallow Water Waves), H.-H. DaiThe Semigroup Theory and Abstract Linear Equations, G. YangA Unified Approach Towards Nonlinear Parabolic Equations with Strong Reaction in Rn, Y.-W. QiGlobal Existence of Smooth Solution to Boltzmann-Poisson System in Semiconductor Physics, G. Cui and Y. WangAnalytical Methods for a Selection of Elliptic Singular Perturbation Problems, N.M. TemmeExponential Attractors of the Strongly Damped Nonlinear Wave Equations, Z. Dai and B. GuoGeneralized Isovorticity Principle for Ideal Magnetohydrodynamics, V.A. Vladimirov and K.I. IlinScroll Waves in Excitable Media and the Motion of Organization Center, Q. Lu and S. LiuTransport Equations for a General Class of Evolution Equations, M.Z. Guo and X.P. Wanga-Times Integrated Cosine Function, G. YangIdentifying Parameters in Elliptic Systems by Finite Element Methods with Multi-Level Initializing, Y.F. Seid and J. Zou TechniquesMonotone Difference Schemes for Two Dimensional Nonhomogeneous Conservation Laws, T. Tang and Z.-H. Teng

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Book SynopsisThe computational power currently available means that practitioners can find extremely accurate approximations to the solutions of more and more sophisticated mathematical models-providing they know the right analytical techniques. In relatively simple terms, this book describes a class of techniques that fulfill this need by providing closed-form solutions to many boundary value problems that arise in science and engineering. Boundary integral equation methods (BIEM''s) have certain advantages over other procedures for solving such problems: BIEM''s are powerful, applicable to a wide variety of situations, elegant, and ideal for numerical treatment. Certain fundamental constructs in BIEM''s are also essential ingredients in boundary element methods, often used by scientists and engineers.However, BIEM''s are also sometimes more difficult to use in plane cases than in their three-dimensional counterparts. Consequently, the full, detailed BIEM treatment of two-dimensional problTrade Review"The text is written clearly and the proofs are given in detail." M. Aron, Proceedings of the Edinburgh Mathematical Society, Vol. 44, 445-448, 2001 "…the book offers a comprehensive treatment of the subject matter and constitutes a very useful source of information for mathematicians and other scientists interested in boundary integral equation methods. M. Aron, Proceedings of the Edinburgh Mathematical Society, Vol. 44, 445-448, 2001Table of ContentsIntroduction. The Laplace Equation. Plane Strain. Bending of Elastic Plates. Which Method?NTI/Sales Copy

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Book SynopsisOne of the first things a student of partial differential equations learns is that it is impossible to solve elliptic equations by spatial marching. This new book describes how to do exactly that, providing a powerful tool for solving problems in fluid dynamics, heat transfer, electrostatics, and other fields characterized by discretized partial differential equations. Elliptic Marching Methods and Domain Decomposition demonstrates how to handle numerical instabilities (i.e., limitations on the size of the problem) that appear when one tries to solve these discretized equations with marching methods. The book also shows how marching methods can be superior to multigrid and pre-conditioned conjugate gradient (PCG) methods, particularly when used in the context of multiprocessor parallel computers. Techniques for using domain decomposition together with marching methods are detailed, clearly illustrating the benefits of these techniques for applications in engineering, applied mathemTrade Review"Together with an important historical perspective, this book uses the domain decomposition connection to develop and explore the nature of marching methods. Interesting analytical and anecdotal comparisons are made with direct methods and multigrid techniques, told by a scientist who has obviously has much experience with real practical problems."-Mathematical Reviews, 99aTable of ContentsBasic Marching Methods for 2D Elliptic ProblemsHigh-Order EquationsExtending the Mesh Size: Domain DecompositionBanded Approximations to Influence MatricesMarching Methods in 3DPerformance of the 2D GEM CodeVectorization and ParallelizationSemidirect Methods for Nonlinear Equations of Fluid DynamicsComparison to Multigrid MethodsAppendix A - Marching Schemes and Error Propagation for Various Discrete LaplaciansAppendix B - Tridiagonal Algorithm for Periodic Boundary ConditionsAppendix C - Gauss Elimination as a Direct SolverSubject IndexTOC for NTI/Flyer

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Book SynopsisIntended for researchers, numerical analysts, and graduate students in various fields of applied mathematics, physics, mechanics, and engineering sciences, Applications of Lie Groups to Difference Equations is the first book to provide a systematic construction of invariant difference schemes for nonlinear differential equations. A guide to methods and results in a new area of application of Lie groups to difference equations, difference meshes (lattices), and difference functionals, this book focuses on the preservation of complete symmetry of original differential equations in numerical schemes. This symmetry preservation results in symmetry reduction of the difference model along with that of the original partial differential equations and in order reduction for ordinary difference equations.A substantial part of the book is concerned with conservation laws and first integrals for difference models. The variational approach and Noether type theorems for diTrade ReviewThe book provides a systematic application of Lie groups to difference equations, difference meshes, and difference functionals. Besides the well-explained theoretical background and motivations, there is also a large number of concrete examples discussed in reasonable details. Due to the fairly broad introductory part, the book is indeed self-contained. The main ideas and concepts appear understandable not only to experts.—Vojtech Zadnik, Zentralblatt MATH 1236In recent years "difference geometry" and its applications to integrable systems and mathematical physics have attracted significant attention and this monograph will contribute to the ongoing developments in this general area. It is clearly written and largely self-contained … —Peter J. Vassiliou, Mathematical Reviews, 2012eTable of ContentsIntroduction. Finite differences and transformation groups in space of discrete variables. Invariance of finite difference equations and meshes. Invariant difference models of ordinary differential equations. Invariant difference models of partial differential equations. Combined models, admitting a transformation group. The discrete representation of a differential equation. Invariant variational problem and conservation laws for difference equations.

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Book SynopsisAnalyze Global Nonlinear Problems Using MetasolutionsMetasolutions of Parabolic Equations in Population Dynamics explores the dynamics of a generalized prototype of semilinear parabolic logistic problem. Highlighting the author's advanced work in the field, it covers the latest developments in the theory of nonlinear parabolic problems.The book reveals how to mathematically determine if a species maintains, dwindles, or increases under certain circumstances. It explains how to predict the time evolution of species inhabiting regions governed by either logistic growth or exponential growth. The book studies the possibility that the species grows according to the Malthus law while it simultaneously inherits a limited growth in other regions.The first part of the book introduces large solutions and metasolutions in the context of population dynamics. In a self-contained way, tTable of ContentsExistence of Large Solutions and Metasolutions. Dynamics. Uniqueness of the Large Solution. Metasolutions Do Arise Everywhere. Bibliography. Index.

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Book SynopsisUncover the Useful Interactions of Fixed Point Theory with Topological StructuresNonlinear Functional Analysis in Banach Spaces and Banach Algebras: Fixed Point Theory under Weak Topology for Nonlinear Operators and Block Operator Matrices with Applications is the first book to tackle the topological fixed point theory for block operator matrices with nonlinear entries in Banach spaces and Banach algebras. The book provides researchers and graduate students with a unified survey of the fundamental principles of fixed point theory in Banach spaces and algebras. The authors present several extensions of Schauder's and Krasnosel'skii's fixed point theorems to the class of weakly compact operators acting on Banach spaces and algebras, particularly on spaces satisfying the DunfordPettis property. They also address under which conditions a 2×2 block operator matrix with single- and multi-valued nonlinear entries will have a fixed point.Table of ContentsFixed Point Theory: Fundamentals. Fixed Point Theory under Weak Topology. Fixed Point Theory in Banach Algebras. Fixed Point Theory for BOM on Banach Spaces and Banach Algebras. Applications in Mathematical Physics and Biology: Existence of Solutions for Transport Equations. Exsistence of Solutions for Nonlinear Integral Equations. Two-Dimensional Boundary Value Problems.

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Book SynopsisFilling a gap in the literature, Delay Differential Evolutions Subjected to Nonlocal Initial Conditions reveals important results on ordinary differential equations (ODEs) and partial differential equations (PDEs). It presents very recent results relating to the existence, boundedness, regularity, and asymptotic behavior of global solutions for differential equations and inclusions, with or without delay, subjected to nonlocal implicit initial conditions.After preliminaries on nonlinear evolution equations governed by dissipative operators, the book gives a thorough study of the existence, uniqueness, and asymptotic behavior of global bounded solutions for differential equations with delay and local initial conditions. It then focuses on two important nonlocal cases: autonomous and quasi-autonomous. The authors next discuss sufficient conditions for the existence of almost periodic solutions, describe evolution systems with delay and nonlocal initial cTrade Review"This book will be useful to researchers and graduate students interested in delay evolution equations and inclusions subjected to nonlocal initial conditions." - Sotiris K. Ntouyas (Ioannina)Table of ContentsPreliminaries. Local Initial Conditions. Nonlocal Initial Conditions: The Autonomous Case. Nonlocal Initial Conditions: The Quasi-Autonomous Case. Almost Periodic Solutions. Evolution Systems with Nonlocal Initial Conditions. Delay Evolution Inclusions. Multivalued Reaction-Diffusion Systems. Viability for Nonlocal Evolution Inclusions. Bibliography. Index.

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Book SynopsisIterative processes are the tools used to generate sequences approximating solutions of equations describing real life problems. Intended for researchers in computational sciences and as a reference book for advanced computational method in nonlinear analysis, this book is a collection of the recent results on the convergence analysis of numerical algorithms in both finite-dimensional and infinite-dimensional spaces and presents several applications and connections with fixed point theory. It contains an abundant and updated bibliography and provides comparisons between various investigations made in recent years in the field of computational nonlinear analysis.The book also provides recent advancements in the study of iterative procedures and can be used as a source to obtain the proper method to use in order to solve a problem. The book assumes a basic background in Mathematical Statistics, Linear Algebra and Numerical Analysis and may be used as a self-study referTable of ContentsHalley’s method. Newton’s method for k-Fréchet differentiable operators. Nonlinear Ill-posed quations. Sixth-order iterative methods. Local convergence and basins of attraction of a two-step Newton like method for equations with solutions of multiplicity greater than one. Extending the Kantorovich theory for solving equations. Robust convergence for inexact Newton method. Inexact Gauss-Newton-like method for least square problems. Lavrentiev Regularization Methods for Ill-posed Equations. King-Werner-type methods of order 1+sqrt(2). Generalized equations and Newton’s method. Newton’s method for generalized equations using restricted domains. Secant-like methods. King-Werner-like methods free of derivatives. Müller’s method. Generalized Newton Method with applications. Newton-secant methods with values in a cone. Gauss-Newton method with applications to convex optimization. Directional Newton methods and restricted domains. Gauss-Newton method for convex optimization. Ball Convergence for eighth order method. Expanding Kantorovich’s theorem for solving generalized equations.

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Book SynopsisThis book is designed for a systematic understanding of nuclear diffusion theory along with fuzzy/interval/stochastic uncertainty. This will serve to be a benchmark book for graduate & postgraduate students, teachers, engineers and researchers throughout the globe. In view of the recent developments in nuclear engineering, it is important to study the basic concepts of this field along with the diffusion processes for nuclear reactor design. Also, it is known that uncertainty is a must in every field of engineering and science and, in particular, with regards to nuclear-related problems. As such, one may need to understand the nuclear diffusion principles/theories corresponding with reliable and efficient techniques for the solution of such uncertain problems. Accordingly this book aims to provide a new direction for readers with basic concepts of reactor physics as well as neutron diffusion theory. On the other hand, it also includes uncertainty (in terms ofTable of Contents Basic Reactor Principles Atomic Structure Binding energy Nuclear fusion Nuclear fission Radioactivity Principles, Production, and interaction of neutrons with matter Production of neutrons Neutron reactions and radiation Inelastic and elastic scattering of neutrons Maxwell-Boltzmann distribution Neutron diffusion theory Cross section of neutron reactions Rates of neutron reactions Fission neutrons Prompt neutrons Delayed neutrons Neutron transport and diffusion equation Fundamentals of Uncertainty Probabilistic uncertainty Non-probabilistic uncertainty Interval uncertainty Fuzzy uncertainty Uncertain Neutron diffusion Uncertain factors involved in neutron diffusion theory Modeling of uncertain neutron diffusion equations One group model Analytical methods Numerical methods Finite difference method Finite element method Conclusion Uncertain One Group Model Interval arithmetic and Fuzzy Finite Element Method (FFEM) Formulation of the uncertain stiffness matrices and force vectors Bare square homogeneous reactor Multi group model Uncertain factors involved in multi group neutron diffusion theory Formulation of uncertain multi group neutron diffusion equations Uncertain Multi Group Model Fuzzy finite element for coupled differential equations Fuzzy multi group neutron diffusion equation Case study Results and discussion Conclusion Point Kinetic Diffusion Theory of point kinetic neutron diffusion equation Case study Conclusion Stochastic Point Kinetic Diffusion Stochastic point kinetic model Euler-Maruyama method Example Hybridised uncertainty in point kinetic diffusion Development of stochastic point kinetic model with fuzzy parameters Fuzzy Euler-Maruyama method Case Study Conclusion Index

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Book SynopsisAlthough the analysis of scattering for closed bodies of simple geometric shape is well developed, structures with edges, cavities, or inclusions have seemed, until now, intractable to analytical methods. This two-volume set describes a breakthrough in analytical techniques for accurately determining diffraction from classes of canonical scatterers with comprising edges and other complex cavity features. It is an authoritative account of mathematical developments over the last two decades that provides benchmarks against which solutions obtained by numerical methods can be verified.The first volume, Canonical Structures in Potential Theory, develops the mathematics, solving mixed boundary potential problems for structures with cavities and edges. The second volume, Acoustic and Electromagnetic Diffraction by Canonical Structures, examines the diffraction of acoustic and electromagnetic waves from several classes of open structures with edges or cavities. Together these volumes present an authoritative and unified treatment of potential theory and diffraction-the first complete description quantifying the scattering mechanisms in complex structures.Table of ContentsMathematical Aspects of Potential Theory. Dual or Triple Series and Integral Equations. Electrostatic Potential Theory for Open Spherical Shells and Cavities. Open Spheroidal Conducting Shells and Cavities. Charged Toroidal Shells and Cavities. Potential Theory for Conical Structures with Edges. Two-Dimensional Potential Theory. Rigorous Solution Methods for more Complicated Structures.

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Book SynopsisInverse problems arise in many disciplines and hold great importance to practical applications. However, sound new methods are needed to solve these problems. Over the past few years, Japanese and Korean mathematicians have obtained a number of very interesting and unique results in inverse problems.Inverse Problems and Related Topics compiles papers authored by some of the top researchers in Korea and Japan. It presents a number of original and useful results and offers a unique opportunity to explore the current trends of research in inverse problems in these countries. Highlighting the existence and active work of several Japanese and Korean groups, it also serves as a guide to those seeking future scientific exchange with researchers in these countries.Trade Review"The aim of this book is to fill the gap between high-school mathematics and mathematics taught at university…the reader is shown what it means to prove something rigourously…This book is easy to read for anyone with a high-school mathematics background." - European Mathematical Society NewsletterTable of ContentsA Finite Difference Model for Calderón's Boundary Inverse Problem. Inverse Problems for Equations with Memory. Parameter Estimation of Elastic Media. The Probe Method and its Applications. Recent Progress in the Inverse Conductivity Problem with Single Measurement. A Moment Method on Inverse Problems for the Heat Equation. Some Remarks on Free Boundaries of Recirculation Euler Flows with Constant Vorticity. Algorithms for the Identification of Spatially Varying/Invariant Stiffness and Dampings in Flexible Beams. Numerical Solutions of the Cauchy Problem in Potential and Elastostatics. Inverse Source Problems in the Helmholtz Equations. A Numerical Method for a Magnetostatic Inverse Problem using the Edge Element. Exact Controllability Method and Multidimensional Linear Inverse Problems. Impedance Computed Tomo-Electrocardiography. An Inverse Problem for Free Channel Scattering. Surface Impedance Tensor and Boundary Value Problem. Aysmptotics for the Spectral and Weyl Functions of the Operator-Value Sturm-Liouville Problem. Exact Controllability Method and Multidimensional Linear Inverse Problems

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Book SynopsisThe theory of holomorphic functions of several complex variables emerged from the attempt to generalize the theory in one variable to the multidimensional situation. Research in this area has led to the discovery of many sophisticated facts, structures, ideas, relations, and applications. This deepening of knowledge, however, has also revealed more and more paradoxical differences between the structures of the two theories. The authors of this Research Note were driven by the quest to construct a theory in several complex variables that has the same structure as the one-variable theory. That is, they sought a reproducing kernel for the whole class that is universal and from same class. Integral Theorems for Functions and Differential Forms in Cm documents their success. Their highly original approach allowed them to obtain new results and refine some well-known results from the classical theory of several complex variables. The 'hyperholomorphic" theory they developed proved to be a kind of direct sum of function theories for two Dirac-type operators of Clifford analysis considered in the same domain.In addition to new results and methods, this work presents a first-look at a brand new setting, based upon the natural language of differential forms, for complex analysis. Integral Theorems for Functions and Differential Forms in Cm reveals a deep link between the fields of several complex variables theory and Clifford analysis. It will have a strong influence on researchers in both areas, and undoubtedly will change the general viewpoint on the methods and ideas of several complex variables theory.Trade Review"…the book will be interesting to specialists in complex analysis and its applications".- Mathematical Reviews, 2003a"This well-written book is a valuable contribution to the broad field of interactions between complex analysis and partial differential equations...Moreover, the book can be used for individual studies, because fundamental concepts and important theorems are explained in detail."-Mathematical Reviews, Issue 94aTable of ContentsIntroduction. Differential Forms. Differential Forms with Co-Efficients in 2x2 Matrices. Hyperholomorphic Functions and Differential Forms in Cm. Cauchy's Theorem. Morera's Theorem. Cauchy's Integral Representation. Hyperholomorphic D-problem. Complex Hodge-Dolbeault System. Relations with Clifford Analysis.

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Book SynopsisWith contributions from some of the leading authorities in the field, the work in Differential Equations: Inverse and Direct Problems stimulates the preparation of new research results and offers exciting possibilities not only in the future of mathematics but also in physics, engineering, superconductivity in special materials, and other scientific fields.Exploring the hypotheses and numerical approaches that relate to pure and applied mathematics, this collection of research papers and surveys extends the theories and methods of differential equations. The book begins with discussions on Banach spaces, linear and nonlinear theory of semigroups, integrodifferential equations, the physical interpretation of general Wentzell boundary conditions, and unconditional martingale difference (UMD) spaces. It then proceeds to deal with models in superconductivity, hyperbolic partial differential equations (PDEs), blowup of solutions, reaction-diffusion equation with memory, and Navier-Stokes equations. The volume concludes with analyses on Fourier-Laplace multipliers, gradient estimates for Dirichlet parabolic problems, a nonlinear system of PDEs, and the complex Ginzburg-Landau equation.By combining direct and inverse problems into one book, this compilation is a useful reference for those working in the world of pure or applied mathematics.Trade Review"…Almost all of the fourteen contributions contain original results; they do not just survey or explain results already published elsewhere. They cover a wide scope of up-to-date topics from the field of differential equations. … The book will be an interesting and stimulating read for research workers in the field."-EMS Newsletter, June 2007Table of ContentsDegenerate first order identification problems in Banach spaces. A non-isothermal dynamical Ginzburg-Landau model of superconductivity. Some global in time results for integrodifferential parabolic inverse problems. Fourth order ordinary differential operators with general Wentzell boundary conditions. Study of elliptic differential equations in UMD spaces. Degenerate integrodifferential equations of parabolic type. Exponential attractors for semiconductor equations. Convergence to stationary states of solutions to the semilinear equa-tion of viscoelasticity. Asymptotic behavior of a phase field system with dynamic boundary conditions. The power potential and nonexistence of positive solutions. The Model-Problem associated to the Stefan Problem with Surface Tension: an Approach via Fourier-Laplace Multipliers. Identification problems for nonautonomous degenerate integrodifferential equations of parabolic type with Dirichlet boundary conditions. Existence results for a phase transition model based on microscopic movements. Strong L2-wellposedness in the complex Ginzburg-Landau equation.

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Book SynopsisPart II of the Selected Works of Ivan Georgievich Petrowsky, contains his major papers on second order Partial differential equations, systems of ordinary. Differential equations, the theory, of Probability, the theory of functions, and the calculus of variations. Many of the articles contained in this book have Profoundly, influenced the development of modern mathematics. Of exceptional value is the article on the equation of diffusion with growing quantity of the substance. This work has found extensive application in biology, genetics, economics and other branches of natural science. Also of great importance is Petrowsky's work on a Problem which still remains unsolved - that of the number of limit cycles for ordinary differential equations with rational right-hand sides.Table of ContentsForeword, Editor’s Preface, Ivan Georgievich Petrowsky, PETROWSKY’S ARTICLES ON PARTIAL DIFFERENTIAL EQUATIONS, PETROWSKY’S ARTICLES ON ORDINARY DIFFERENTIAL EQUATIONS, PETROWSKY’S ARTICLES ON THE THEORY OF PROBABILITY AND OTHER PROBLEMS OF ANALYSIS, APPENDIX: COMMENTARIES, Index

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Book SynopsisThe monograph is devoted to one of the most important trends in contemporary mathematical physics, the investigation of evolution equations of many-particle systems of statistical mechanics. The book systematizes rigorous results obtained in this field in recent years, and it presents contemporary methods for the investigation of evolution equations of infinite-particle systems. The book is intended for experts in statistical physics, mathematical physics, and probability theory and for students of universities specialized in mathematics and physics.Trade Review"This book may be useful for advanced graduate students and for scientists who are interested in mathematical problems of the statistical mechanics and rare ed gasesow.?"Oleg A. Sinkevich in: Zentralblatt Math 1/2010<

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Book SynopsisThis textbook is for the standard, one-semester, junior-senior course that often goes by the title "Elementary Partial Differential Equations" or "Boundary Value Problems". The audience consists of students in mathematics, engineering, and the sciences. The topics include derivations of some of the standard models of mathematical physics and methods for solving those equations on unbounded and bounded domains, and applications of PDE's to biology. The text differs from other texts in its brevity; yet it provides coverage of the main topics usually studied in the standard course, as well as an introduction to using computer algebra packages to solve and understand partial differential equations.For the 3rd edition the section on numerical methods has been considerably expanded to reflect their central role in PDE's. A treatment of the finite element method has been included and the code for numerical calculations is now written for MATLAB. Nonetheless the brevity of the text has been maintained. To further aid the reader in mastering the material and using the book, the clarity of the exercises has been improved, more routine exercises have been included, and the entire text has been visually reformatted to improve readability.Trade Review“The aim of this book is to provide the reader with basic ideas encountered in partial differential equations. … The mathematical content is highly motivated by physical problems and the emphasis is on motivation, methods, concepts and interpretation rather than formal theory. The textbook is a valuable material for undergraduate science and engineering students.” (Marius Ghergu, zbMATH 1310.35001, 2015)Table of ContentsPreface to the Third Edition.- To the Students.- 1: The Physical Origins of Partial Differential Equations.- 1.1 PDE Models.- 1.2 Conservation Laws.- 1.3 Diffusion.- 1.4 Diffusion and Randomness.- 1.5 Vibrations and Acoustics.- 1.6 Quantum Mechanics*.- 1.7 Heat Conduction in Higher Dimensions.- 1.8 Laplace’s Equation.- 1.9 Classification of PDEs.- 2. Partial Differential Equations on Unbounded Domains.- 2.1 Cauchy Problem for the Heat Equation.- 2.2 Cauchy Problem for the Wave Equation.- 2.3 Well-Posed Problems.- 2.4 Semi-Infinite Domains.- 2.5 Sources and Duhamel’s Principle.- 2.6 Laplace Transforms.- 2.7 Fourier Transforms.- 3. Orthogonal Expansions.- 3.1 The Fourier Method.- 3.2 Orthogonal Expansions.- 3.3 Classical Fourier Series.-4. Partial Differential Equations on Bounded Domains.- 4.1 Overview of Separation of Variables.- 4.2 Sturm–Liouville Problems - 4.3 Generalization and Singular Problems.- 4.4 Laplace's Equation.- 4.5 Cooling of a Sphere.- 4.6 Diffusion inb a Disk.- 4.7 Sources on Bounded Domains.- 4.8 Poisson's Equation*.-5. Applications in the Life Sciences.-5.1 Age-Structured Models.- 5.2 Traveling Waves Fronts.- 5.3 Equilibria and Stability.- References.- Appendix A. Ordinary Differential Equations.- Index.

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Book SynopsisMethod of Variation of Parameters for Dynamic Systems presents a systematic and unified theory of the development of the theory of the method of variation of parameters, its unification with Lyapunov's method and typical applications of these methods. No other attempt has been made to bring all the available literature into one volume. This book is a clear exposition of this important topic in control theory, which is not covered by any other text. Such an exposition finally enables the comparison and contrast of the theory and the applications, thus facilitating further development in this fascinating field.Table of Contents1. Ordinary Differential Equations 2. Integrodifferential Equations 3. Differential Equations with Delay 4. Difference Equations 5. Stochastic Differential Equations 6. Abstract Differential Equations 7. Impulsive Differential Equations Equations 3. Differential Equations with Delay 4. Difference Equations 5. Abstract Differential Equations 6. Impulsive Differential Equations 7. Stochastic Differential Equations

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Book SynopsisThis volume comprises selected papers presented at the Volterra Centennial Symposium and is dedicated to Volterra and the contribution of his work to the study of systems - an important concept in modern engineering. Vito Volterra began his study of integral equations at the end of the nineteenth century and this was a significant development in the theory of integral equations and nonlinear functional analysis. Volterra series are of interest and use in pure and applied mathematics and engineering.Table of Contents1. Retrospective of Vito Volterra and His Influence on Nonlinear Systems Theory 2. Volterra Integral Equations at Wisconsin 3. Stability and Asymptotic Behaviour of Solutions of Equations with Aftereffect 4. Generalized Halay Inequalities for Volterra Functional Differential Equations and Discretized Versions 5. Stochastic Convolutions with Kernels Arising in Volterra Equations 6. An Example of Lp-Regularity for Hyperbolic Integrodifferential Equations 7. The Present Status of UAS for Volterra and Delay Equations 8. Myopic Maps and Volterra Series Approximations 9. State Space Theory for Abstract Volterra Operators

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