NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

This book charts a clear and systematic roadmap for nonlinear partial differential equations (NLPDES). Beginning from the definition of a partial differential equation to the recent developments of nonlinear partial differential equations, this book will be a valuable resource for advanced postgraduate students and researchers in applied mathematics, physics, nonlinear optics, and other engineering disciplines where knowledge of nonlinear differential equations is a must.

The book begins with an introductory chapter that briefly describes the developments of linear as well as nonlinear partial differential equations. Several nonlinear partial differential equations that have emerged in various fields have also been discussed. Chapter 2 introduces several analytical techniques, including the traveling wave solutions and the similarity solutions of the nonlinear partial differential equations. In Chapter 3, approximate analytical solutions and semi-analytic solutions are presented, in which solutions of non-integrable or non-autonomous nonlinear partial differential equations are investigated after suitable approximation. Some recent breakthroughs in semi-analytical approaches such as the Variational iteration method (VIM), Adomian decomposition method (ADM), Homotopy Analysis method (HAM), and Homotopy Perturbation method (HPM) are also explained with examples. Chapter 4 deals with modern advancements in NLPDE, Painlevé tests, the Inverse Scattering Method, the Lax Pair Method, Darboux Transformation, Bäcklund Transformation, and the Hirota Direct Method. The focus of this comprehensive monograph is to check the integrability and find analytical solutions for important NLPDEs according to recent developments.

Contents:

• Introduction:
  • Introduction
  • Differential Equation
  • Classification of First-Order PDE
  • Types of Solutions
  • Initial Conditions and Boundary Conditions
  • Formation of PDE
  • Solution of Linear PDEs of Order One
  • Integral Surfaces Passing Through a Given Curve
  • Cauchy Problem and Characteristics
  • Nonlinear PDE: Charpit's Method
  • Second-Order PDE
  • Method of Separation of Variables
  • d'Alembert's Solution of Wave Equation
  • Higher-Order Linear PDE
  • Non-Homogeneous PDE
  • Solving Non-homogeneous PDEs by Separation of Variable
  • Equations in Physics
  • Some Nonlinear Model Equations
  • Dispersive Waves
• Traveling Wave Solution Method:
  • Introduction
  • Solutions of NLPDEs
  • Traveling Wave Solution
  • Solution of KdV Equation and KdV-like Equations
  • Hyperbolic Tangent (Tanh) Method
  • Tanh–Coth Method
  • (G'/G) Expansion Method
  • Modified (G'/G) Expansion Method
  • Cosine Expansion Method
  • Sine Expansion Method
  • Sine–Cosine Expansion Method
  • Sech Expansion Method
  • Sec–Tanh Expansion Method
  • Jacobian Elliptic Function Expansion Method
  • Modified Expansion Function Method
  • Generalized Riccati Equation Expansion Method
  • Generalized Kudryashov Method
  • Similarity Solution
• Approximate Analytical Solutions:
  • Introduction
  • Conservation Laws and Integrals of the Motions
  • Conservation Law of the Gardner Equation
  • Analytical Solution of KdV-Type Equations
  • Adomian Decomposition Method (ADM)
  • Homotopy Analysis Method (HAM)
  • Homotopy Perturbation Method (HPM)
  • Variational Iteration Method (VIM)
• Integrability and Some Modern Methods:
  • Introduction
  • Integrability
  • Painlevé Analysis
  • Series Solutions
  • Lax Pair
  • Lax Compatibility Equation
  • Lax Pair in Operator Form
  • How to Construct [L, M]
  • Lax Pair of Some Nonlinear Evolution Equations
  • Lax Pairs in Matrix Form
  • Inverse Scattering Method
  • Solution of KdV Equation
  • Hirota Direct Method
  • Some Important Information
  • Hirota's Condition for the Existence of Soliton Solutions
  • Lump Soliton
  • Breather Soliton
  • Bell Polynomial
  • Wronskian Technique
  • Definition
  • Plücker relation
  • Solution of the KdV Equation
  • Solutions of the Non-Planar KdV Equation
  • Solution of KP Equation
  • Double–Wronskian Method
  • Double–Wronskian Solutions of the Boussinesq Systems
  • Soliton Solution in Double–Wronskian Form
  • Darboux Transformation (DT)
  • Darboux Theorem
  • Darboux Covariant Theorem
  • DT for ODE
  • DT for Generalized Lax Equation
  • MKdV Equation Using Matrix Form
  • Solution of mKdV Equation
  • Bäcklund Transformation (BT)
  • Soliton Solution of the Sine-Gordon Equation
  • Liouville Equation

Readership: Researchers and postgraduate students in applied mathematics, physics, nonlinear optics, and engineering.