Applied Partial Differential Equations 1st Edition by John Ockendon, Sam Howison, Andrew Lacey, Alexander Movchan ISBN 0198527713 9780198527718

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Product details:
ISBN 10: 0198527713
ISBN 13: 9780198527718
Author: John Ockendon, Sam Howison, Andrew Lacey, Alexander Movchan

Applied Partial Differential Equations 1st Edition Table of contents:

Chapter 1: Introduction to Partial Differential Equations

• Definition and Classification of Partial Differential Equations
• The Importance of PDEs in Science and Engineering
• Basic Solutions to PDEs: General and Particular Solutions
• The Role of Boundary and Initial Conditions
• Overview of Common PDEs in Applications
• Methods of Solving PDEs: Analytical vs. Numerical

Chapter 2: First-Order Partial Differential Equations

• The General Form of First-Order PDEs
• Linear and Nonlinear First-Order PDEs
• The Method of Characteristics
• Applications: Traffic Flow, Fluid Dynamics, and Conservation Laws
• Solution Techniques for First-Order PDEs
• Example Problems and Their Solutions

Chapter 3: Second-Order Partial Differential Equations

• The General Form of Second-Order PDEs
• Classification: Elliptic, Parabolic, and Hyperbolic Equations
• Solution Techniques for Second-Order Equations
• Separation of Variables Method
• The Heat Equation, Wave Equation, and Laplace Equation
• Applications to Diffusion, Vibrations, and Electrostatics
• Example Problems and Their Solutions

Chapter 4: Boundary and Initial Value Problems

• The Importance of Boundary and Initial Conditions in Applications
• Solving Boundary Value Problems (BVPs)
• Types of Boundary Conditions: Dirichlet, Neumann, and Robin
• The Role of Eigenvalue Problems in BVPs
• Applications to Heat Flow, Fluid Mechanics, and Electromagnetics
• Time-Dependent Problems and Initial Conditions
• Examples and Techniques for Solving BVPs

Chapter 5: The Method of Separation of Variables

• Introduction to the Separation of Variables Method
• Solving Linear PDEs Using Separation of Variables
• Applications to the Heat Equation, Wave Equation, and Laplace’s Equation
• Fourier Series and Fourier Transforms in Solutions
• Vibrating Membranes and Quantum Mechanics
• Example Problems and Their Solutions

Chapter 6: Fourier Series and Transforms

• Fourier Series: Definition and Applications
• Fourier Transforms for Solving PDEs
• The Heat Equation and Fourier Transforms
• The Role of Spectral Methods in Solving PDEs
• Applications in Signal Processing and Acoustics
• Solving Non-Constant Coefficient Problems Using Fourier Techniques
• Example Problems and Their Solutions

Chapter 7: Green’s Functions and Integral Equations

• Introduction to Green’s Functions in Solving PDEs
• Green’s Functions for Elliptic, Parabolic, and Hyperbolic Equations
• The Method of Integral Equations for Solving PDEs
• Applications in Electromagnetics, Heat Transfer, and Structural Mechanics
• Numerical Methods for Calculating Green’s Functions
• Example Problems and Their Solutions

Chapter 8: Numerical Methods for Solving PDEs

• Introduction to Numerical PDE Solving Techniques
• Finite Difference Method (FDM) for Time and Space Discretization
• The Finite Element Method (FEM) for Complex Geometries
• The Boundary Element Method (BEM) and Meshless Methods
• Stability, Convergence, and Accuracy of Numerical Solutions
• Applications in Computational Fluid Dynamics and Structural Analysis
• Example Problems and Their Solutions

Chapter 9: Applications of Elliptic Equations

• The Poisson Equation and Electrostatics
• Laplace’s Equation and Potential Theory
• Solutions in Bounded and Unbounded Domains
• Applications to Fluid Flow, Heat Conduction, and Gravity
• Numerical Techniques for Solving Elliptic Equations
• Example Problems and Their Solutions

Chapter 10: Applications of Parabolic Equations

• The Heat Equation and Heat Transfer Problems
• Diffusion in Chemical and Biological Systems
• The Black-Scholes Equation in Finance
• Time-Dependent Boundary Conditions and Their Effects
• Numerical Methods for Parabolic Equations
• Example Problems and Their Solutions

Chapter 11: Applications of Hyperbolic Equations

• The Wave Equation and Vibrational Analysis
• Acoustics and Electromagnetic Wave Propagation
• Shock Waves and Nonlinear Hyperbolic PDEs
• Numerical Methods for Hyperbolic Equations
• The Method of Characteristics for Solving Hyperbolic PDEs
• Example Problems and Their Solutions

Chapter 12: Nonlinear Partial Differential Equations

• The Nature of Nonlinear PDEs and Their Applications
• Characteristics of Nonlinear Waves and Solitons
• Shock Waves and Discontinuities in Nonlinear PDEs
• Methods for Solving Nonlinear PDEs: Perturbation and Numerical Techniques
• Applications in Fluid Mechanics, Optics, and Traffic Flow
• Example Problems and Their Solutions

Chapter 13: Advanced Topics in PDEs

• Multidimensional PDEs and Their Applications
• Nonlinear Dynamics and Chaos Theory in PDEs
• Stability Analysis and Bifurcation Theory
• PDEs in Material Science and Biophysics
• Control Theory and Optimal Design Problems
• Emerging Areas: PDEs in Climate Modeling and Astrophysics

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John Ockendon,Sam Howison,Andrew Lacey,Alexander Movchan,Partial Differential Equations