Hyperbolic Partial Differential Equations Springer, Berlin

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zukunftsorientiert - praxisnah - anwendungsbezogen

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Polynomial Chaos Methods for Hyperbolic Partial Differential Equations: Numerical Techniques for Fluid Dynamics Problems in the Presence of Uncertaint

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Numerical Methods For Elliptic And Parabolic Partial Differential Equations

Książki Obcojęzyczne>Angielskie>Mathematics & science>Mathematics>Calculus & mathematical analysis>Differential calculus & equationsKsi...

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Non - Elliptic Partial Differential Equations

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Remarks On Strongly Elliptic Partial Differential Equations (Classic Reprint)

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Some Domain Decomposition Algorithms For Nonselfadjoint Elliptic And Parabolic Partial Differential Equations

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Functional Spaces For The Theory Of Elliptic Partial Differential Equations

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Computational Partial Differential Equations Using MATLAB (R) Taylor & Francis Ltd

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In this popular text for an Numerical Analysis course, the authors introduce several major methods of solving various partial differential equations (PDEs) including elliptic, parabolic, and hyperbolic equations. It covers traditional techniques includi

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Essential Partial Differential Equations Springer International Publishing AG

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This volume provides an introduction to the analytical and numerical aspects of partial differential equations (PDEs). It unifies an analytical and computational approach for these; the qualitative behaviour of solutions being established using classical concepts: maximum principles and energy methods.§§Notable inclusions are the treatment of irregularly shaped boundaries, polar coordinates and the use of flux-limiters when approximating hyperbolic conservation laws. The numerical analysis of difference schemes is rigorously developed using discrete maximum principles and discrete Fourier analysis. A novel feature is the inclusion of a chapter containing projects, intended for either individual or group study, that cover a range of topics such as parabolic smoothing, travelling waves, isospectral matrices, and the approximation of multidimensional advection-diffusion problems.§§The underlying theory is illustrated by numerous examples and there are around 300 exercises, designed to promote and test understanding. They are starred according to level of difficulty. Solutions to odd-numbered exercises are available to all readers while even-numbered solutions are available to authorised instructors.§§Written in an informal yet rigorous style, Essential Partial Differential Equations is designed for mathematics undergraduates in their final or penultimate year of university study, but will be equally useful for students following other scientific and engineering disciplines in which PDEs are of practical importance. The only prerequisite is a familiarity with the basic concepts of calculus and linear algebra.§

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Finite Volume Methods for Hyperbolic Problems Cambridge University Press

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This book contains an introduction to hyperbolic partial differential equations and a powerful class of numerical methods for approximating their solution, including both linear problems and nonlinear conservation laws. These equations describe a wide range of wave propagation and transport phenomena arising in nearly every scientific and engineering discipline. Several applications are described in a self-contained manner, along with much of the mathematical theory of hyperbolic problems. High-resolution versions of Godunov's method are developed, in which Riemann problems are solved to determine the local wave structure and limiters are then applied to eliminate numerical oscillations. These methods were originally designed to capture shock waves accurately, but are also useful tools for studying linear wave-propagation problems, particularly in heterogenous material. The methods studied are implemented in the CLAWPACK software package and source code for all the examples presented can be found on the web, along with animations of many of the simulations. This provides an excellent learning environment for understanding wave propagation phenomena and finite volume methods.

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Optimal Shape Design for Elliptic Systems Springer-Verlag Berlin and Heidelberg GmbH & Co....

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The study of optimal shape design can be arrived at by asking the following question: "What is the best shape for a physical system?" This book is an applications-oriented study of such physical systems; in particular, those which can be described by an elliptic partial differential equation and where the shape is found by the minimum of a single criterion function. There are many problems of this type in high-technology industries. In fact, most numerical simulations of physical systems are solved not to gain better understanding of the phenomena but to obtain better control and design. Problems of this type are described in Chapter 2. Traditionally, optimal shape design has been treated as a branch of the calculus of variations and more specifically of optimal control. This subject interfaces with no less than four fields: optimization, optimal control, partial differential equations (PDEs), and their numerical solutions-this is the most difficult aspect of the subject. Each of these fields is reviewed briefly: PDEs (Chapter 1), optimization (Chapter 4), optimal control (Chapter 5), and numerical methods (Chapters 1 and 4).

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Numerical Methods for Engineering Applications 2e John Wiley & Sons Inc

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State-of-the-art numerical methods for solving complex engineering problems Great strides in computer technology have been made in the years since the popular first edition of this book was published. Several excellent software packages now help engineers solve complex problems. Making the most of these programs requires a working knowledge of the numerical methods on which the programs are based. Numerical Methods for Engineering Application provides that knowledge. While it avoids intense mathematical detail, Numerical Methods for Engineering Application supplies more in-depth explanations of methods than found in the typical engineer's numerical "cookbook." It offers complete coverage of most commonly encountered algebraic, interpolation, and integration problems. Ordinary differential equations are examined in great detail, as are three common types of partial differential equations--parabolic, elliptic, and hyperbolic. The author also explores a wide range of methods for solving initial and boundary value problems.This complete guide to numerical methods for solving engineering problems on computers provides: Practical advice on how to select the best method for a given problem Valuable insights into how each method works and why it is the best choice Complete algorithms and source code for all programs covered Code from the book and problem-solving programs designed by the author available from the author's website Numerical Methods for Engineering Application is a valuable working resource for engineers and applied physicists. It also serves as an excellent upper-level text for physics and engineering students in courses on modern numerical methods.

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Catastrophe Theory and Its Applications Dover Publications Inc.

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Preface1 Smooth and sudden changes 1. Catastrophes 2. The Zeeman catastrophe machine 3. Gravitational catastrophe machines 4. Catastrophe theory2 Multidimensional geometry 1. Set-theoretic notation 2. Euclidean space 3. Linear transformations 4. Matrices 5. Quadratic forms 6. Two-variable cubic forms 7. Polynomial geometry3 Multidimensional calculus 1. Distance in Euclidean space 2. The derivative as tangent 3. Contours 4. Partial derivatives 5. Higher derivatives 6. Taylor series 7. Truncated algebra 8. The Inverse Function Theorem 9. The Implicit Function Theorem4 Critical points and transversality 1. Critical points 2. The Morse Lemma 3. Functions of a single variable 4. Functions of several variables 5. The Splitting Lemma 6. Structural stability 7. Manifolds 8. Transversality 9. Transversality and stability 10. Transversality for mappings 11. Codimension5 Machines revisited 1. The Zeeman machine 2. The canonical cusp catastrophe 3. Dynamics of the Zeeman machine 4. The gravitational machines 5. Formulation of a general problem6 Structural stability 1. Equivalence of families 2. Structural stabillty of families 3. Physical interpretations of structural stability 4. The Morse and Splitting Lemmas for families 5. Catastrophe geometry7 Thom's classification theorem 1. Functions and families of functions 2. One-parameter families 3. Non-transversaliity and symmetry 4. Two-parameter families 5. "Three-, four- and five-parameter families" 6. Higher catastrophes 7. Thom's theorem8 Determinacy and unfoldings 1. Determine and strong determinacy 2. One-variable jet spaces 3. Infinitesimal changes of variable 4. Weaker determinacy conditions 5. Transformations that move the origin 6. Tangency and transversality 7. Codimension and unfoldings 8. Transversality and universality 9. Strong equivalence of unfoldings 10. Numbers associated with singularities 11. Inequalities 12. Summary of results and calculation methods 13. Examples and calculations 14. Compulsory remarks on terminology9 The first seven catastrophe geometries 1. The objects of study 2. The fold catastrophe 3. The cusp catastrophe 4. The swallowtail catastrophe 5. The butterfly catastrophe 6. The elliptic umbilic 7. The hyperbolic umbilic 8. The parabolic umbilic 9. Ruled surfaces10 Stability of ships Static equilibrium 1. Buoyancy 2. Equilibrium 3. Stability 4. The vertical-sided ship 5. Geometry of the buoyancy locus 6. Metacentres Ship shapes 7. The elliptical ship 8. The rectangular ship 9. Three dimensions 10. Oil-rigs 11. Comparison with current methods11. The geometry of fluids Background on fluid mechanics 1. What we are describing 2. Stream functions 3. Examples of flows 4. Rotation 5. Complex variable methods Stability and experiment 6. Changes of variable 7. Heuristic programme 8. Experimental realization Combining polymer molecules 9. Non-Newtonian behaviour 10. Extensional flows Degenerate flows 11. The six-roll mill 12. The non-local bifurcation set of the elliptic umbilic 13. The six-roll mill with polymer solution 14. The 2n-roll mill12 Optics and scattering theory Ray optics 1. Caustics 2. The rainbow 3. Variational principles 4. Scattering Wave optics 5. Asymptotic solutions of wave equations 6. Oscillatory integrals 7. Universal unfoldings 8. Orders of caustics Applications 9. Scattering from a crystal lattice 10. Other caustics 11. Mirages 12. Sonic booms 13. Giant ocean waves13 Elastic structures General theory 1. Objects under stress 2. Elastic equilibria 3. Infinite-dimensional peculiarities Euler struts 4. Finite element vision 5. Classical (1744) variational version 6. Perturbation analysis 7. Modern functional analysis 8. The buckling of a spring 9. The pinned strut The geometry of collapse 10. Imperfection sensitivity 11. "(r, s)-Stability" 12. Optimization 13. Symmetry: rods and shells Buckling plates 14. The von Kármán equations 15. Unfolding a double eigenvalue Dynamics 16. Soft modes 17. Stiffness14 Thermodynamics and phase transitions Equations of state 1. van der Waals' equation 2. Ferromagnetism Thermodynamic potentials 3. Entropy 4. Transforming the maximum entropy principle 5. Legendre transformations 6. Explicit potentials 7. The Landau theory Fluctuations and critical exponents 8. Classical exponents 9. Topological tinkering 10. The rôle of fluctuations 11. Spatial variation 12. Partition functions   13. Renormalization group 14. Structural stability of renormalization The rôle of symmetry 15. Even functions 16. The shapes of rotating stars 17. Symmetry breaking 18. Tricritical points 19. Crystal symmetries 20. Spectrum singularities15 Laser physics Preliminaries 1. Atoms 2. Field 3. Interaction 4. Measurement The laser catastrophe 5. Unfolded Hamiltonian 6. Equations of motion 7. Mean field approximation 8. Boundary conditions 9. Non-equilibrium stationary manifold Experiments 10. Laser transition 11. Optical bistability 12. Photocount distributions Analytic correspondence 13. Equilibrium boundary conditions 14. Equilibrium manifold 15. Thermodynamic phase transition 16. Critical behaviour 17. Analytic correspondence of experiments &

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Numerical Methods in Computational Finance John Wiley & Sons Inc

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This book is a detailed and step-by-step introduction to the mathematical foundations of ordinary and partial differential equations, their approximation by the finite difference method and applications to computational finance. The book is structured so that it can be read by beginners, novices and expert users.Part A Mathematical Foundation for One-Factor ProblemsChapters 1 to 7 introduce the mathematical and numerical analysis concepts that are needed to understand the finite difference method and its application to computational finance.Part B Mathematical Foundation for Two-Factor ProblemsChapters 8 to 13 discuss a number of rigorous mathematical techniques relating to elliptic and parabolic partial differential equations in two space variables. In particular, we develop strategies to preprocess and modify a PDE before we approximate it by the finite difference method, thus avoiding ad-hoc and heuristic tricks.Part C The Foundations of the Finite Difference Method (FDM)Chapters 14 to 17 introduce the mathematical background to the finite difference method for initial boundary value problems for parabolic PDEs. It encapsulates all the background information to construct stable and accurate finite difference schemes.Part D Advanced Finite Difference Schemes for Two-Factor ProblemsChapters 18 to 22 introduce a number of modern finite difference methods to approximate the solution of two factor partial differential equations. This is the only book we know of that discusses these methods in any detail.Part E Test Cases in Computational FinanceChapters 23 to 26 are concerned with applications based on previous chapters. We discuss finite difference schemes for a wide range of one-factor and two-factor problems.This book is suitable as an entry-level introduction as well as a detailed treatment of modern methods as used by industry quants and MSc/MFE students in finance. The topics have applications to numerical analysis, science and engineering.More on computational finance and the author's online courses, see www.datasim.nl.

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Semiclassical Soliton Ensembles for the Focusing Nonlinear Princeton University Press

Inne 1

This book represents the first asymptotic analysis, via completely integrable techniques, of the initial value problem for the focusing nonlinear Schrodinger equation in the semiclassical asymptotic regime. This problem is a key model in nonlinear optical physics and has increasingly important applications in the telecommunications industry. The authors exploit complete integrability to establish pointwise asymptotics for this problem's solution in the semiclassical regime and explicit integration for the underlying nonlinear, elliptic, partial differential equations suspected of governing the semiclassical behavior. In doing so they also aim to explain the observed gradient catastrophe for the underlying nonlinear elliptic partial differential equations, and to set forth a detailed, pointwise asymptotic description of the violent oscillations that emerge following the gradient catastrophe. To achieve this, the authors have extended the reach of two powerful analytical techniques that have arisen through the asymptotic analysis of integrable systems: the Lax-Levermore-Venakides variational approach to singular limits in integrable systems, and Deift and Zhou's nonlinear Steepest-Descent/Stationary Phase method for the analysis of Riemann-Hilbert problems. In particular, they introduce a systematic procedure for handling certain Riemann-Hilbert problems with poles accumulating on curves in the plane. This book, which includes an appendix on the use of the Fredholm theory for Riemann-Hilbert problems in the Holder class, is intended for researchers and graduate students of applied mathematics and analysis, especially those with an interest in integrable systems, nonlinear waves, or complex analysis.

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