Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems

by Sachdev; P.L.

• ISBN13: 9781584882114
• ISBN10: 1584882115
• Edition: 1st
• Format: Hardcover
• Copyright: 2000-08-24
• Publisher: Chapman & Hall/

Summary

Nonlinearity plays a major role in the understanding of most physical, chemical, biological, and engineering sciences. This volume presents a myriad of approaches to finding exact solutions of a variety of nonlinear problems. The author's approach is entirely constructive. The presentation includes numerical solutions that motivate or confirm the methods developed and contains no abstract analysis, leading to a work that is readily accessible to a broad base of readers.

Table of Contents

Introduction	1	(4)
First-Order Partial Differential Equations	5	(46)
Linear Partial Differential Equations of First Order	5	(4)
Quasilinear Partial Differential Equations of First Order	9	(18)
Reduction of ut + unux + H(x,t,u) = 0	27	(5)
Initial Vaulue Problem for ut + g(u)ux + λh(u) = 0	32	(9)
Initial Value Problem for ut + uαux + λu&betal = 0	41	(10)
Exact Similarity Solutions of Nonlinear PDEs	51	(1)
PART A	51	(34)
Reduction of PDEs by Infinitesimal Transformations	51	(12)
Systems of Partial Differential Equations	63	(10)
Self-Similar Solutions of the Second Kind - Viscous Gravity Currents	73	(12)
PART B	85	(220)
Introduction	85	(1)
A Nonlinear Heat Equation in Three Dimensions	85	(3)
Similarity Solution of Burgers Equation by the Direct Method	88	(9)
Exact Free Surface Flows for Shallow-Water Equations via the Direct Similarity Approach	97	(9)
Multipronged Approach to Exact Solutions of Nonlinear PDEs and Example from Gasdynamics	106	(21)
Exact Travelling Wave Solutions	127	(48)
Travelling Wave Solutions	127	(1)
Simple Waves in 1-D Gasdynamics	128	(3)
Elementary Nonlinear Diffusive Travelling Waves	131	(11)
Travelling Waves for Higher-Order Diffusive Systems	142	(6)
Simple Wave Flows in Multidimensional Systems of Homogeneous Partial Differential Equations	148	(6)
Travelling Waves for Nonhomogeneous Hyperbolic or Dispersive Systems	154	(7)
Exact Hydromagnetic Travelling Waves	161	(6)
Exact Simple Waves on Shear Flows in a Compressible Barotropic Medium	167	(8)
Exact Linearisation of Nonlinear PDEs	175	(38)
Introduction	175	(1)
Comments on the Solution of Linear PDEs	176	(2)
Burgers Equation in One and Higher Dimensions	178	(4)
Nonlinear Degenerate Diffusion Equation ut = [f(u)ux-1]x	182	(3)
One-Dimensional Motion of an Ideal Compressible Isentropic Gas in the Hodograph Plane	185	(5)
The Born-Infeld Equation	190	(3)
Water Waves up a Uniformly Sloping Beach	193	(8)
Simple Waves on Shear Flows	201	(5)
C-Integrable Nonlinear PDEs	206	(7)
Nonlinearisation and Embedding of Special Solutions	213	(44)
Introduction	213	(3)
Exact Nonlinearisation of N Wave Solutions for Generalised Burgers Equations	216	(9)
Burgers Equation in Cylindrical Coordinates with Axisymmetry	225	(3)
Nonplanar Burgers Equation - A Composite Solution	228	(11)
Modified Burgers Equation	239	(6)
Embedding of Similarity Solution in a Larger Class	245	(12)
Asymptotic Solutions by Balancing Arguments	257	(28)
Asymptotic Solution by Balancing Arguments - Examples from ODEs	257	(6)
Asymptotic Solution of Nonplanar Burgers Equation with N Wave Initial Conditions	263	(5)
Asymptotic Profiles with Finite Mass in 1-D Contaminant Transport through Porous Media	268	(17)
Series Solutions of Nonlinear PDEs	285	(20)
Introduction	285	(1)
Analysis of Expansion of a Gas Sphere (Cylinder) into Vacuum	286	(6)
Collapse of a Spherical or Cylindrical Cavity	292	(5)
Converging Shock Wave from a Spherical or Cylindrical Piston	297	(8)
References	305	(12)
Index	317