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Differential equations, Hyperbolic --- Differential equations, Hyperbolic. --- Hyperbolic differential equations --- Differential equations, Partial --- Equacions diferencials hiperbòliques --- Equacions en derivades parcials --- Lleis de conservació (Matemàtica)

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Differential equations --- Differential equations, Hyperbolic --- Differential equations, Nonlinear --- Cauchy problem --- Differential equations, Partial --- Nonlinear differential equations --- Nonlinear theories --- Hyperbolic differential equations --- Differential equations, Hyperbolic. --- Équations différentielles hyperboliques. --- Differential equations, Nonlinear. --- Équations différentielles non linéaires. --- Cauchy problem. --- Cauchy, Problème de.

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This masterly exposition of the mathematical theory of hyperbolic system for conservation laws brings out the intimate connection with continuum thermodynamics, by emphasising issues in which the analysis may reveal something about the physics and, in return, the underlying physical structure may direct and drive the analysis. The reader is expected to have a certain mathematical sophistication and to be familiar with (at least) the rudiments of the qualitative theory of partial differential equations, whereas the required notions from continuum physics are introduced from scratch. The target group of readers would consist of (a) experts in the mathematical theory of hyperbolic systems of conservation laws who wish to learn about the connection with classical physics; (b) specialists in continuum mechanics who may need analytical tools; (c) experts in numerical analysis who wish to learn the underlying mathematical theory; and (d) analysts and graduate students who seek introduction to the theory of hyperbolic systems of conservation laws. The 2nd edition contains a new chapter recounting the exciting recent developments on the vanishing viscosity method. Numerous new sections have been incorporated in preexisting chapters, to introduce newly derived results or present older material, omitted in the first edition, whose relevance and importance has been underscored by current trends in research. In addition, a substantal portion of the original text has been revamped so as to streamline the exposition, enrich the collection of examples and improve the notation. The bibliography has been updated and expanded as well, now comprising over one thousand titles. .

Conservation laws (Physics) --- Differential equations, Hyperbolic. --- Field theory (Physics) --- Classical field theory --- Continuum physics --- Physics --- Continuum mechanics --- Hyperbolic differential equations --- Differential equations, Partial --- Physical laws --- Differential equations, partial. --- Thermodynamics. --- Mechanics. --- Partial Differential Equations. --- Classical Mechanics. --- Classical mechanics --- Newtonian mechanics --- Dynamics --- Quantum theory --- Chemistry, Physical and theoretical --- Mechanics --- Heat --- Heat-engines --- Partial differential equations --- Partial differential equations.

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This text is a concise introduction to the partial differential equations which change from elliptic to hyperbolic type across a smooth hypersurface of their domain. These are becoming increasingly important in diverse sub-fields of both applied mathematics and engineering, for example:   • The heating of fusion plasmas by electromagnetic waves • The behaviour of light near a caustic • Extremal surfaces in the space of special relativity • The formation of rapids; transonic and multiphase fluid flow • The dynamics of certain models for elastic structures • The shape of industrial surfaces such as windshields and airfoils • Pathologies of traffic flow • Harmonic fields in extended projective space   They also arise in models for the early universe, for cosmic acceleration, and for possible violation of causality in the interiors of certain compact stars. Within the past 25 years, they have become central to the isometric embedding of Riemannian manifolds and the prescription of Gauss curvature for surfaces: topics in pure mathematics which themselves have important applications.   Elliptic−Hyperbolic Partial Differential Equations is derived from a mini-course given at the ICMS Workshop on Differential Geometry and Continuum Mechanics held in Edinburgh, Scotland in June 2013. The focus on geometry in that meeting is reflected in these notes, along with the focus on quasilinear equations. In the spirit of the ICMS workshop, this course is addressed both to applied mathematicians and to mathematically-oriented engineers. The emphasis is on very recent applications and methods, the majority of which have not previously appeared in book form.

Mathematics. --- Partial Differential Equations. --- Mathematical Applications in the Physical Sciences. --- Differential equations, partial. --- Mathématiques --- Differential equations, Elliptic. --- Differential equations, Hyperbolic. --- Mathematics --- Physical Sciences & Mathematics --- Calculus --- Hyperbolic differential equations --- Elliptic differential equations --- Elliptic partial differential equations --- Linear elliptic differential equations --- Partial differential equations. --- Mathematical physics. --- Differential equations, Partial --- Differential equations, Linear --- Partial differential equations --- Physical mathematics --- Physics --- Differential equations, Partial.

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Differential equations, Hyperbolic --- Nonlinear waves --- Singularities (Mathematics) --- Wave equation --- Ondes non linéaires --- Singularités (Mathématiques) --- Equations d'onde --- Numerical solutions --- Solutions numériques --- 517.95 --- -Nonlinear waves --- -#KVIV:BB --- Differential equations, Partial --- Wave-motion, Theory of --- Geometry, Algebraic --- Nonlinear theories --- Waves --- Hyperbolic differential equations --- Partial differential equations --- 517.95 Partial differential equations --- #KVIV:BB --- Numerical analysis --- Analyse microlocale --- Equations aux derivees partielles non lineaires