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Product Code:  CLN/14 
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Electronic ISBN:  9781470431143 
Product Code:  CLN/14.E 
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Book DetailsCourant Lecture NotesVolume: 14; 2006; 217 ppMSC: Primary 35;Peter D. Lax is the winner of the 2005 Abel Prize
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 secondyear 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 LaxPhillips 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.
Peter D. Lax is the winner of the 2005 Abel Prize. Read more here.ReadershipGraduate students and research mathematicians interested in hyperbolic equations.

Table of Contents

Chapters

Chapter 1. Basic notions

Chapter 2. Finite speed of propagation of signals

Chapter 3. Hyperbolic equations with constant coefficients

Chapter 4. Hyperbolic equations with variable coefficients

Chapter 5. Pseudodifferential operators and energy inequalities

Chapter 6. Existence of solutions

Chapter 7. Waves and rays

Chapter 8. Finite difference approximation to hyperbolic equations

Chapter 9. Scattering theory

Chapter 10. Hyperbolic systems of conservation laws

Appendix A. Huygens’ principle for the wave equation on odddimensional spheres

Appendix B. Hyperbolic polynomials

Appendix C. The multiplicity of eigenvalues

Appendix D. Mixed initial and boundary value problems

Appendix E. Energy decay for starshaped obstacles


Additional Material

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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 secondyear 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 LaxPhillips 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.
Peter D. Lax is the winner of the 2005 Abel Prize. Read more here.
Graduate students and research mathematicians interested in hyperbolic equations.

Chapters

Chapter 1. Basic notions

Chapter 2. Finite speed of propagation of signals

Chapter 3. Hyperbolic equations with constant coefficients

Chapter 4. Hyperbolic equations with variable coefficients

Chapter 5. Pseudodifferential operators and energy inequalities

Chapter 6. Existence of solutions

Chapter 7. Waves and rays

Chapter 8. Finite difference approximation to hyperbolic equations

Chapter 9. Scattering theory

Chapter 10. Hyperbolic systems of conservation laws

Appendix A. Huygens’ principle for the wave equation on odddimensional spheres

Appendix B. Hyperbolic polynomials

Appendix C. The multiplicity of eigenvalues

Appendix D. Mixed initial and boundary value problems

Appendix E. Energy decay for starshaped obstacles