Nonlinear Partial Differential Equations for Scientists and Engineers Second Edition by Lokenath Debnath ISBN 978-0817643232 0817643230
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Product details:
ISBN 10: 0817643230
ISBN 13: 978-0817643232
Author: Lokenath Debnath
This expanded, revised edition is a thorough and systematic treatment of linear and nonlinear partial differential equations and their varied applications. It contains updated modern examples and applications from diverse fields. Methods and properties of solutions, along with their physical significance, make the book useful for a diverse readership including graduates, researchers, and professionals in mathematics, physics and engineering.
Table of contents:
1 Linear Partial Differential Equations.
1.1 Introduction
1.2 Basic Concepts and Definitions
1.3 The Linear Superposition Principle.
1.4 Some Important Classical Linear Model Equations.
1.5 Second-Order Linear Equations and Method of Characteristics
1.6 The Method of Separation of Variables.
1.7 Fourier Transforms and Initial Boundary-Value Problems
1.8 Multiple Fourier Transforms and Partial Differential Equations
1.9 Laplace Transforms and Initial Boundary-Value Problems.
1.10 Hankel Transforms and Initial Boundary-Value Problems
1.11 Green’s Functions and Boundary-Value Problems.
1.12 Sturm-Liouville Systems and Some General Results
1.13 Energy Integrals and Higher Dimensional Equations
1.14 Fractional Partial Differential Equations…..
1.15 Exercises
2 Nonlinear Model Equations and Variational Principles
2.1 Introduction
2.2 Basic Concepts and Definitions
2.3 Some Nonlinear Model Equations
2.4 Variational Principles and the Euler-Lagrange Equations
2.5 The Variational Principle for Nonlinear Klein-Gordon Equations.
2.6 The Variational Principle for Nonlinear Water Waves.
2.7 The Euler Equation of Motion and Water Wave Problems
2.8 Exercises
3 First-Order, Quasi-Linear Equations and Method of Characteristics
3.1 Introduction
3.2 The Classification of First-Order Equations
3.3 The Construction of a First-Order Equation
3.4 The Geometrical Interpretation of a First-Order Equation
3.5 The Method of Characteristics and General Solutions
3.6 Exercises
4 First-Order Nonlinear Equations and Their Applications.
4.1 Introduction
4.2 The Generalized Method of Characteristics
4.3 Complete Integrals of Certain Special Nonlinear Equations.
4.4 The Hamilton-Jacobi Equation and Its Applications.
4.5 Applications to Nonlinear Optics.
4.6 Exercises
5 Conservation Laws and Shock Waves.
5.1 Introduction
5.2 Conservation Laws
5.3 Discontinuous Solutions and Shock Waves
5.4 Weak or Generalized Solutions
5.5 Exercises
6 Kinematic Waves and Real-World Nonlinear Problems
6.1 Introduction
6.2 Kinematic Waves.
6.3 Traffic Flow Problems.
6.4 Flood Waves in Long Rivers
6.5 Chromatographic Models and Sediment Transport in Rivers
6.6 Glacier Flow
6.7 Roll Waves and Their Stability Analysis.
6.8 Simple Waves and Riemann’s Invariants
6.9 The Nonlinear Hyperbolic System and Riemann’s Invariants
6.10 Generalized Simple Waves and Generalized Riemann’s Invariants
6.11 Exercises
7 Nonlinear Dispersive Waves and Whitham’s Equations.
7.1 Introduction
7.2 Linear Dispersive Waves
7.3 Initial-Value Problems and Asymptotic Solutions
7.4 Nonlinear Dispersive Waves and Whitham’s Equations
7.5 Whitham’s Theory of Nonlinear Dispersive Waves
7.6 Whitham’s Averaged Variational Principle
7.7 Whitham’s Instability Analysis of Water Waves.
7.8 Whitham’s Equation: Peaking and Breaking of Waves
8 Nonlinear Diffusion-Reaction Phenomena.
8.1 Introduction
8.2 Burgers’ Equation and the Plane Wave Solution
8.3 Traveling Wave Solutions and Shock-Wave Structure
8.4 The Exact Solution of the Burgers Equation
8.5 The Asymptotic Behavior of the Burgers Solution
8.6 The N-Wave Solution
8.7 Burgers’ Initial- and Boundary-Value Problem
8.8 Fisher’s Equation and Diffusion-Reaction Process
8.9 Traveling Wave Solutions and Stability Analysis.
8.10 Perturbation Solutions of the Fisher Equation
8.11 Method of Similarity Solutions of Diffusion Equations
8.12 Nonlinear Reaction-Diffusion Equations
8.13 Brief Summary of Recent Work..
8.14 Exercises
9 Solitons and the Inverse Scattering Transform
9.1 Introduction
9.2 The History of the Solitons and Soliton Interactions
9.3 The Boussinesq and Korteweg-de Vries Equations.
9.4 Solutions of the KdV Equation: Solitons and Cnoidal Waves
9.5 The Lie Group Method and Similarity Analysis of the KdV Equation
9.6 Conservation Laws and Nonlinear Transformations
9.7 The Inverse Scattering Transform (IST) Method
9.8 Bäcklund Transformations and the Nonlinear Superposition Principle.
9.9 The Lax Formulation and the Zakharov and Shabat Scheme.
9.10 The AKNS Method
9.11 Asymptotic Behavior of the Solution of the KdV-Burgers Equation
9.12 Strongly Dispersive Nonlinear Equations and Compactons
9.13 Exercises
10 The Nonlinear Schrödinger Equation and Solitary Waves.
10.1 Introduction
10.2 The One-Dimensional Linear Schrödinger Equation..
10.3 The Nonlinear Schrödinger Equation and Solitary Waves
10.4 Properties of the Solutions of the Nonlinear Schrödinger Equation
10.5 Conservation Laws for the NLS Equation
10.6 The Inverse Scattering Method for the Nonlinear Schrödinger Equation
10.7 Examples of Physical Applications in Fluid Dynamics and Plasma Physics
10.8 Applications to Nonlinear Optics.
10.9 Exercises
11 Nonlinear Klein-Gordon and Sine-Gordon Equations.
11.1 Introduction
11.2 The One-Dimensional Linear Klein-Gordon Equation.
11.3 The Two-Dimensional Linear Klein-Gordon Equation.
11.4 The Three-Dimensional Linear Klein-Gordon Equation
11.5 The Nonlinear Klein-Gordon Equation and Averaging Techniques
11.6 The Klein-Gordon Equation and the Whitham Averaged Variational Principle
11.7 The Sine-Gordon Equation: Soliton and Antisoliton Solutions
11.8 The Solution of the Sine-Gordon Equation by Separation of Variables
11.9 Bäcklund Transformations for the Sine-Gordon Equation
11.10The Solution of the Sine-Gordon Equation by the Inverse Scattering Method
11.11The Similarity Method for the Sine-Gordon Equation
11.12Nonlinear Optics and the Sine-Gordon Equation..
11.13Exercises
12 Asymptotic Methods and Nonlinear Evolution Equations
12.1 Introduction
12.2 The Reductive Perturbation Method and Quasi-Linear Hyperbolic Systems
12.3 Quasi-Linear Dissipative Systems
12.4 Weakly Nonlinear Dispersive Systems and the Korteweg-de Vries Equation.
12.5 Strongly Nonlinear Dispersive Systems and the NLS Equation
12.6 The Perturbation Method of Ostrovsky and Pelinovsky
12.7 The Method of Multiple Scales
12.8 Asymptotic Expansions and Method of Multiple Scales.
12.9 Derivation of the NLS Equation and Davey-Stewartson Evolution Equations
13 Tables of Integral Transforms
13.1 Fourier Transforms
13.2 Fourier Sine Transforms.
13.3 Fourier Cosine Transforms
13.4 Laplace Transforms..
13.5 Hankel Transforms
13.6 Finite Hankel Transforms
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Tags: Lokenath Debnath, Nonlinear Partial, Differential Equations, Scientists and Engineers