Self-Similarity and beyond
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Self-Similarity and beyond : Exact Solutions of Nonlinear Problems

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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.show more

Product details

  • Hardback | 336 pages
  • 163.3 x 242.1 x 23.1mm | 696.71g
  • Taylor & Francis Inc
  • Chapman & Hall/CRC
  • Boca Raton, FL, United States
  • English
  • 10 black & white illustrations, 4 black & white tables
  • 1584882115
  • 9781584882114

Review quote

"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 diffusionThroughout 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 2001show more

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 Argumentsshow more