Quasilinear Hyperbolic Systems, Compressible Flows, and Waves: 1st Edition (Hardback) book cover

Quasilinear Hyperbolic Systems, Compressible Flows, and Waves

1st Edition

By Vishnu D. Sharma

CRC Press

282 pages | 49 B/W Illus.

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Hardback: 9781439836903
pub: 2010-04-29
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Filled with practical examples, Quasilinear Hyperbolic Systems, Compressible Flows, and Waves presents a self-contained discussion of quasilinear hyperbolic equations and systems with applications. It emphasizes nonlinear theory and introduces some of the most active research in the field.

After linking continuum mechanics and quasilinear partial differential equations, the book discusses the scalar conservation laws and hyperbolic systems in two independent variables. Using the method of characteristics and singular surface theory, the author then presents the evolutionary behavior of weak and mild discontinuities in a quasilinear hyperbolic system. He also explains how to apply weakly nonlinear geometrical optics in nonequilibrium and stratified gas flows and demonstrates the power, generality, and elegance of group theoretic methods for solving Euler equations of gasdynamics involving shocks. The final chapter deals with the kinematics of a shock of arbitrary strength in three dimensions.

With a focus on physical applications, this text takes readers on a journey through this fascinating area of applied mathematics. It provides the essential mathematical concepts and techniques to understand the phenomena from a theoretical standpoint and to solve a variety of physical problems.

Table of Contents

Hyperbolic Systems of Conservation Laws



Scalar Hyperbolic Equations in One Dimension

Breakdown of Smooth Solutions

Entropy Conditions Revisited

Riemann Problem for Nonconvex Flux Function


Asymptotic Behavior

Hyperbolic Systems in One Space Dimension

Genuine Nonlinearity

Weak Solutions and Jump Condition

Entropy Conditions

Riemann Problem

Shallow Water Equations

Evolution of Weak Waves in Hyperbolic Systems

Waves and Compatibility Conditions

Evolutionary Behavior of Acceleration Waves

Interaction of Shock Waves with Weak Discontinuities

Weak Discontinuities in Radiative Gasdynamics

One-Dimensional Weak Discontinuity Waves

Weak Nonlinear Waves in an Ideal Plasma

Relatively Undistorted Waves

Asymptotic Waves for Quasilinear Systems

Weakly Nonlinear Geometrical Optics

Far Field Behavior

Energy Dissipated across Shocks

Evolution Equation Describing Mixed Nonlinearity

Singular Ray Expansions

Resonantly Interacting Waves

Self-Similar Solutions Involving Discontinuities and Their Interaction

Waves in Self-Similar Flows

Imploding Shocks in a Relaxing Gas

Exact Solutions of Euler Equations via Lie Group Analysis

Kinematics of a Shock of Arbitrary Strength

Shock Wave through an Ideal Gas in 3-Space Dimensions

An Alternative Approach Using the Theory of Distributions

Kinematics of a Bore over a Sloping Beach



About the Author

Vishnu D. Sharma is chair professor in the Department of Mathematics at the Indian Institute of Technology, Bombay (IITB). Dr. Sharma is also president of the Indian Society of Theoretical and Applied Mechanics.

About the Series

Monographs and Surveys in Pure and Applied Mathematics

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Subject Categories

BISAC Subject Codes/Headings:
SCIENCE / Physics