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Author: Peter D. Lax
Publisher: American Mathematical Soc.
ISBN: 0821835769
Category : Mathematics
Languages : en
Pages : 234
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Book Description
The theory of hyperbolic equations is a large subject, and its applications are many: fluid dynamics and aerodynamics, the theory of elasticity, optics, electromagnetic waves, direct and inverse scattering, and the general theory of relativity. This book is an introduction to most facets of the theory and is an ideal text for a second-year graduate course on the subject. The first part deals with the basic theory: the relation of hyperbolicity to the finite propagation of signals, the concept and role of characteristic surfaces and rays, energy, and energy inequalities. The structure of solutions of equations with constant coefficients is explored with the help of the Fourier and Radon transforms. The existence of solutions of equations with variable coefficients with prescribed initial values is proved using energy inequalities. The propagation of singularities is studied with the help of progressing waves. The second part describes finite difference approximations of hyperbolic equations, presents a streamlined version of the Lax-Phillips scattering theory, and covers basic concepts and results for hyperbolic systems of conservation laws, an active research area today. Four brief appendices sketch topics that are important or amusing, such as Huygens' principle and a theory of mixed initial and boundary value problems. A fifth appendix by Cathleen Morawetz describes a nonstandard energy identity and its uses. -- Back cover.
Author: Peter D. Lax
Publisher: American Mathematical Soc.
ISBN: 0821835769
Category : Mathematics
Languages : en
Pages : 234
Get Book Here
Book Description
The theory of hyperbolic equations is a large subject, and its applications are many: fluid dynamics and aerodynamics, the theory of elasticity, optics, electromagnetic waves, direct and inverse scattering, and the general theory of relativity. This book is an introduction to most facets of the theory and is an ideal text for a second-year graduate course on the subject. The first part deals with the basic theory: the relation of hyperbolicity to the finite propagation of signals, the concept and role of characteristic surfaces and rays, energy, and energy inequalities. The structure of solutions of equations with constant coefficients is explored with the help of the Fourier and Radon transforms. The existence of solutions of equations with variable coefficients with prescribed initial values is proved using energy inequalities. The propagation of singularities is studied with the help of progressing waves. The second part describes finite difference approximations of hyperbolic equations, presents a streamlined version of the Lax-Phillips scattering theory, and covers basic concepts and results for hyperbolic systems of conservation laws, an active research area today. Four brief appendices sketch topics that are important or amusing, such as Huygens' principle and a theory of mixed initial and boundary value problems. A fifth appendix by Cathleen Morawetz describes a nonstandard energy identity and its uses. -- Back cover.
Author: Serge Alinhac
Publisher: Springer Science & Business Media
ISBN: 0387878238
Category : Mathematics
Languages : en
Pages : 159
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Book Description
This excellent introduction to hyperbolic differential equations is devoted to linear equations and symmetric systems, as well as conservation laws. The book is divided into two parts. The first, which is intuitive and easy to visualize, includes all aspects of the theory involving vector fields and integral curves; the second describes the wave equation and its perturbations for two- or three-space dimensions. Over 100 exercises are included, as well as "do it yourself" instructions for the proofs of many theorems. Only an understanding of differential calculus is required. Notes at the end of the self-contained chapters, as well as references at the end of the book, enable ease-of-use for both the student and the independent researcher.
Author: Sylvie Benzoni-Gavage
Publisher: Oxford University Press, USA
ISBN: 019921123X
Category : Mathematics
Languages : en
Pages : 535
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Book Description
Authored by leading scholars, this comprehensive text presents a view of the multi-dimensional hyperbolic partial differential equations, with a particular emphasis on problems in which modern tools of analysis have proved useful. It is useful to graduates and researchers in both hyperbolic PDEs and compressible fluid dynamics.
Author: Mitsuru Ikawa
Publisher: American Mathematical Soc.
ISBN: 9780821810217
Category : Mathematics
Languages : en
Pages : 218
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Book Description
The familiar wave equation is the most fundamental hyperbolic partial differential equation. Other hyperbolic equations, both linear and nonlinear, exhibit many wave-like phenomena. The primary theme of this book is the mathematical investigation of such wave phenomena. The exposition begins with derivations of some wave equations, including waves in an elastic body, such as those observed in connection with earthquakes. Certain existence results are proved early on, allowing the later analysis to concentrate on properties of solutions. The existence of solutions is established using methods from functional analysis. Many of the properties are developed using methods of asymptotic solutions. The last chapter contains an analysis of the decay of the local energy of solutions. This analysis shows, in particular, that in a connected exterior domain, disturbances gradually drift into the distance and the effect of a disturbance in a bounded domain becomes small after sufficient time passes. The book is geared toward a wide audience interested in PDEs. Prerequisite to the text are some real analysis and elementary functional analysis. It would be suitable for use as a text in PDEs or mathematical physics at the advanced undergraduate and graduate level.
Author: Jeffrey Rauch
Publisher: American Mathematical Soc.
ISBN: 0821872915
Category : Mathematics
Languages : en
Pages : 386
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Book Description
This book introduces graduate students and researchers in mathematics and the sciences to the multifaceted subject of the equations of hyperbolic type, which are used, in particular, to describe propagation of waves at finite speed. Among the topics carefully presented in the book are nonlinear geometric optics, the asymptotic analysis of short wavelength solutions, and nonlinear interaction of such waves. Studied in detail are the damping of waves, resonance, dispersive decay, and solutions to the compressible Euler equations with dense oscillations created by resonant interactions. Many fundamental results are presented for the first time in a textbook format. In addition to dense oscillations, these include the treatment of precise speed of propagation and the existence and stability questions for the three wave interaction equations. One of the strengths of this book is its careful motivation of ideas and proofs, showing how they evolve from related, simpler cases. This makes the book quite useful to both researchers and graduate students interested in hyperbolic partial differential equations. Numerous exercises encourage active participation of the reader. The author is a professor of mathematics at the University of Michigan. A recognized expert in partial differential equations, he has made important contributions to the transformation of three areas of hyperbolic partial differential equations: nonlinear microlocal analysis, the control of waves, and nonlinear geometric optics.
Author: Horst Reinhard Beyer
Publisher: Springer
ISBN: 3540711295
Category : Mathematics
Languages : en
Pages : 291
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Book Description
This book introduces the treatment of linear and nonlinear (quasi-linear) abstract evolution equations by methods from the theory of strongly continuous semigroups. The theoretical part is accessible to graduate students with basic knowledge in functional analysis, with only some examples requiring more specialized knowledge from the spectral theory of linear, self-adjoint operators in Hilbert spaces. Emphasis is placed on equations of the hyperbolic type which are less often treated in the literature.
Author: Andreas Meister
Publisher: Vieweg+Teubner Verlag
ISBN: 9783322802293
Category : Mathematics
Languages : en
Pages : 0
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Book Description
The book gives an introduction to the fundamental properties of hyperbolic partial differential equations und their appearance in the mathematical modelling of various problems from practice. It shows in an unique manner concepts for the numerical treatment of such equations starting from basic algorithms up actual research topics in this area. The numerical methods discussed are central and upwind schemes for structured and unstructured grids based on ENO and WENO reconstructions, pressure correction schemes like SIMPLE and PISO as well as asymptotic-induced algorithms for low-Mach number flows.
Author: John A. Trangenstein
Publisher: Cambridge University Press
ISBN: 052187727X
Category : Mathematics
Languages : en
Pages : 0
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Book Description
Numerical Solution of Hyperbolic Partial Differential Equations is a new type of graduate textbook, with both print and interactive electronic components (on CD). It is a comprehensive presentation of modern shock-capturing methods, including both finite volume and finite element methods, covering the theory of hyperbolic conservation laws and the theory of the numerical methods. The range of applications is broad enough to engage most engineering disciplines and many areas of applied mathematics. Classical techniques for judging the qualitative performance of the schemes are used to motivate the development of classical higher-order methods. The interactive CD gives access to the computer code used to create all of the text's figures, and lets readers run simulations, choosing their own input parameters; the CD displays the results of the experiments as movies. Consequently, students can gain an appreciation for both the dynamics of the problem application, and the growth of numerical errors.
Author: J.W. Thomas
Publisher: Springer Science & Business Media
ISBN: 1489972781
Category : Mathematics
Languages : en
Pages : 451
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Book Description
What makes this book stand out from the competition is that it is more computational. Once done with both volumes, readers will have the tools to attack a wider variety of problems than those worked out in the competitors' books. The author stresses the use of technology throughout the text, allowing students to utilize it as much as possible.
Author: Lars Hörmander
Publisher: Springer Science & Business Media
ISBN: 9783540629214
Category : Mathematics
Languages : en
Pages : 308
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Book Description
In this introductory textbook, a revised and extended version of well-known lectures by L. Hörmander from 1986, four chapters are devoted to weak solutions of systems of conservation laws. Apart from that the book only studies classical solutions. Two chapters concern the existence of global solutions or estimates of the lifespan for solutions of nonlinear perturbations of the wave or Klein-Gordon equation with small initial data. Four chapters are devoted to microanalysis of the singularities of the solutions. This part assumes some familiarity with pseudodifferential operators which are standard in the theory of linear differential operators, but the extension to the more exotic classes of opertors needed in the nonlinear theory is presented in complete detail.
