Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems
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Summary
Nonlinearity plays a major role in the understanding of most physical, chemical, biological, and engineering sciences.
Nonlinear problems fascinate scientists and engineers, but often elude exact treatment. However elusive they may be, the solutions do exist-if only one perseveres in seeking them out.
Self-Similarity and Beyond presents a myriad of approaches to finding exact solutions for a diversity of nonlinear problems. These include group-theoretic methods, the direct method of Clarkson and Kruskal, traveling waves, hodograph methods, balancing arguments, embedding special solutions into a more general class, and the infinite series approach.
The author's approach is entirely constructive. Numerical solutions either motivate the analysis or confirm it, therefore they are treated alongside the analysis whenever possible. Many examples drawn from real physical situations-primarily fluid mechanics and nonlinear diffusion-illustrate and emphasize the central points presented.
Accessible to a broad base of readers, Self-Similarity and Beyond illuminates a variety of productive methods for meeting the challenges of nonlinearity. Researchers and graduate students in nonlinearity, partial differential equations, and fluid mechanics, along with mathematical physicists and numerical analysts, will re-discover the importance of exact solutions and find valuable additions to their mathematical toolkits.
Table of Contents
INTRODUCTION
FIRST ORDER PARTIAL DIFFERENTIAL EQUATIONS
Linear Partial Differential Equations of First Order
Quasilinear Partial Differential Equation of First Order
Reduction of ut = unux+H(x,t,u)=0 to form Ut + UnnUx=0
Initial Value problem for ut+ g(u)Ux+lh(u)=0
Initial Value problem for ut+ ua+ux+ lub=0
EXACT SIMILARITY SOLUTIONS OF NONLINEAR PDES
Reduction of PDEs by Infinitesimal Transformation
System of Partial Differential Equations
Self-Similar Solutions of the Second Kind-Viscous Gravity Currents
A Nonlinear Heat Equation in Three Dimensions
Similarity Solution of Burgers Equation by Direct Method
Exact Free Surface Flows for Shallow Water Equations via Direct Similarity Approach
Multi-Pronged Approach to Exact Solutions of Nonlinear PDEs-An Example from Gas Dynamics
EXACT TRAVELLING WAVE SOLUTIONS
Travelling Waves Solutions
Simple Waves in 1-D Gas Dynamics
Elementary Nonlinear Diffusive Travelling Waves
Travelling Waves for Higher Order Diffusive Systems
Simple Wave Flows in Multi-Dimensional Systems of Homogeneous Partial Differential Equations
Travelling Waves for Nonhomogeneous Hyperbolic or Dispersive Systems
Exact Hydromagnetic Travellng Waves
Exact Simple Waves on shear Flows in a Copressible Barotropic Medium
EXACT LINEARIZATION OF NONLINEAR PDES
Introduction
Comments on the Solution of Linear PDEs
Burgers Equation in One and Higher Dimensions
Nonlinear Degenerate Diffusion Equation ut=[f(u)ux-1
One-Dimensional Motion of an Ideal Compressible Isentropic Gas in the Hodograph Plane
Born-Infeld Equation
Water Waves up a Uniformly Sloping Beach
Simple Waves on Shear Flows
C-Integrable Nonlinear PDEs
NONLINEARIZATION AND EMBEDDING OF SPECIAL SOLUTIONS
Introduction
Exact Nonlinearization of N Wave Solutions for Generalised Burgers Equations
Burgers Equation in Cylindrical Coordinates with Axisymetry
Nonplanar Burgers Equation-A Composite Solution
Modified Burgers Equation
Embedding of Similarity Solution in a Larger Class
ASYMPTOTIC SOLUTION BY BALANCING ARGUMENTS
Asymptotic Solution by Balancing Arguments-Examples from ODEs
Asymptotic Solution of Nonplanar Burgers Equation with N Wave Initial Conditions
Asymptotic Profiles with Finite Mass in 1-D Contaminant Transport through Porous Media
SERIES SOLUTIONS OF NONLINEAR PDES
Introduction
Analysis of Expansion of a Gas Sphere (Cylinder) into Vacuum
Collapse of a Spherical or cylindrical Cavity
Converging shock Wave from a Spherical or cylindrical Piston
Asymptotic Solutions by Balancing Arguments
Editorial Reviews
"The main theme of this book is exact solutions to nonlinear partial differential equations and systematic methods for finding them. All techniques are demonstrated with plenty of worked-out examples, which are predominantly drawn from fluid mechanics, reaction-diffusion systems, and nonlinear diffusion…Throughout the book, notation is kept elementary, and the sections are largely independent of each other. This work will serve as a valuable reference for anybody working in applied mathematics and respective areas of application, as well as a source of nontrivial, yet elementary, examples of solutions to nonlinear partial differential equations for teaching purposes."
- Mathematical Reviews, Issue 2002
"The methods and their limitations are clearly explained, and are complemented by a number of solved examples focusing on equations in fluid mechanics and nonlinear diffusion."
-European Mathematical Society Newsletter, No. 42, December 2001
