Partial Differential Equations: Analytical Methods and Applications

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Preface
1. Introduction
1.1 Basic Definitions
1.2 Examples
2. First-Order Equations
2.1 Linear First-Order Equations
2.1.1 General Solution
2.1.2 Initial Condition
2.2 Quasilinear First-Order Equations
2.2.1 Characteristic Curves
2.2.2 Examples
3. Second-Order Equations
3.1 Classification of Second-Order Equations
3.2 Canonical Forms
3.2.1 Hyperbolic Equations
3.2.2 Elliptic Equations
3.2.3 Parabolic Equations
4. The Sturm-Liouville Problem
4.1 General Consideration
4.2 Examples of Sturm-Liouville Problems
5. One-Dimensional Hyperbolic Equations
5.1 Wave Equation
5.2 Boundary and Initial Conditions
5.3 Longitudinal Vibrations of a Rod and Electrical Oscillations
5.3.1 Rod Oscillations: Equations and Boundary Conditions
5.3.2 Electrical Oscillations in a Circuit
5.4 Traveling Waves: D'Alembert Method
5.5 Cauchy Problem for Nonhomogeneous Wave Equation
5.5.1 D'Alembert's Formula
5.5.2 Green's Function
5.5.3 Well-Posedness of the Cauchy Problem
5.6 Finite Intervals: The Fourier Method for Homogeneous Equations
5.7 The Fourier Method for Nonhomogeneous Equations
5.8 The Laplace Transform Method: Simple Cases
5.9 Equations with Nonhomogeneous Boundary Conditions
5.10 The Consistency Conditions and Generalized Solutions
5.11 Energy in the Harmonics
5.12 Dispersion of Waves
5.12.1 Cauchy Problem in an Infinite Region
5.12.2 Propagation of a Wave Train
5.13 Wave Propagation on an Inclined Bottom: Tsunami Effect
6. One-Dimensional Parabolic Equations
6.1 Heat Conduction and Diffusion: Boundary Value Problems
6.1.1 Heat Conduction
6.1.2 Diffusion Equation
6.1.3 One-dimensional Parabolic Equations and Initial and Boundary Conditions
6.2 The Fourier Method for Homogeneous Equations
6.3 Nonhomogeneous Equations
6.4 Green's Function and Duhamel's Principle
6.5 The Fourier Method for Nonhomogeneous Equations with Nonhomogeneous Boundary Conditions
6.6 Large Time Behavior of Solutions
6.7 Maximum Principle
6.8 The Heat Equation in an Infinite Region
7. Elliptic Equations
7.1 Elliptic Differential Equations and Related Physical Problems
7.2 Harmonic Functions
7.3 Boundary Conditions
7.3.1 Example of an Ill-posed Problem
7.3.2 Well-posed Boundary Value Problems
7.3.3 Maximum Principle and its Consequences
7.4 Laplace Equation in Polar Coordinates
7.5 Laplace Equation and Interior BVP for Circular Domain
7.6 Laplace Equation and Exterior BVP for Circular Domain
7.7 Poisson Equation: General Notes and a Simple Case
7.8 Poisson Integral
7.9 Application of Bessel Functions for the Solution of Poisson Equations in a Circle
7.10 Three-dimensional Laplace Equation for a Cylinder
7.11 Three-dimensional Laplace Equation for a Ball
7.11.1 Axisymmetric Case
7.11.2 Non-axisymmetric Case
7.12 BVP for Laplace Equation in a Rectangular Domain
7.13 The Poisson Equation with Homogeneous Boundary Conditions
7.14 Green's Function for Poisson Equations
7.14.1 Homogeneous Boundary Conditions
7.14.2 Nonhomogeneous Boundary Conditions
7.15 Some Other Important Equations
7.15.1 Helmholtz Equation
7.15.2 Schrödinger Equation
8. Two-Dimensional Hyperbolic Equations
8.1 Derivation of the Equations of Motion
8.1.1 Boundary and Initial Conditions
8.2 Oscillations of a Rectangular Membrane
8.2.1 The Fourier Method for Homogeneous Equations with Homogeneous Boundary Conditions
8.2.2 The Fourier Method for Nonhomogeneous Equations with Homogeneous Boundary Conditions
8.2.3 The Fourier Method for Nonhomogeneous Equations with Nonhomogeneous Boundary Conditions
8.3 Small Transverse Oscillations of a Circular Membrane
8.3.1 The Fourier Method for Homogeneous Equations with Homogeneous Boundary Conditions
8.3.2 Axisymmetric Oscillations of a Membrane
8.3.3 The Fourier Method for Nonhomogeneous Equations with Homogeneous Boundary Conditions
8.3.4 Forced Axisymmetric Oscillations
8.3.5 The Fourier Method for Equations with Nonhomogeneous Boundary Conditions
9. Two-Dimensional Parabolic Equations
9.1 Heat Conduction within a Finite Rectangular Domain
9.1.1 The Fourier Method for the Homogeneous Heat Equation (Free Heat Exchange)
9.1.2 The Fourier Method for Nonhomogeneous Heat Equation with Homogeneous Boundary Conditions
9.2 Heat Conduction within a Circular Domain
9.2.1 The Fourier Method for the Homogeneous Heat Equation
9.2.2 The Fourier Method for the Nonhomogeneous Heat Equation
9.2.3 The Fourier Method for the Nonhomogeneous Heat Equation with Nonhomogeneous Boundary Conditions
9.3 Heat Conduction in an Infinite Medium
9.4 Heat Conduction in a Semi-Infinite Medium
10. Nonlinear Equations
10.1 Burgers Equation
10.1.1 Kink Solution
10.1.2 Symmetries of the Burger's Equation
10.2 General Solution of the Cauchy Problem
10.2.1 Interaction of Kinks
10.3 Korteweg-de Vries Equation
10.3.1 Symmetry Properties of the KdV Equation
10.3.2 Cnoidal Waves
10.3.3 Solitons
10.3.4 Bilinear Formulation of the KdV Equation
10.3.5 Hirota's Method
10.3.6 Multisoliton Solutions
10.4 Nonlinear Schrödinger Equation
10.4.1 Symmetry Properties of NSE
10.4.2 Solitary Waves
Appendix A: Fourier Series, Fourier and Laplace Transforms
A.1 Periodic Processes and Periodic Functions
A.2 Fourier Formulas
A.3 Convergence of Fourier Series
A.4 Fourier Series for Non-periodic Functions
A.5 Fourier Expansions on Intervals of Arbitrary Length
A.6 Fourier Series in Cosine or in Sine Functions
A.7 Examples
A.8 The Complex Form of the Trigonometric Series
A.9 Fourier Series for Functions of Several Variables
A.10 Generalized Fourier Series
A.11 The Gibbs Phenomenon
A.12 Fourier Transforms
A.13 Laplace Transforms
A.14 Applications of Laplace Transform for ODE
Appendix B: Bessel and Legendre Functions
B.1 Bessel Equation
B.2 Properties of Bessel Functions
B.3 Boundary Value Problems and Fourier-Bessel Series
B.4 Spherical Bessel Functions
B.5 The Gamma Function
B.6 Legendre Equation and Legendre Polynomials
B.7 Fourier-Legendre Series in Legendre Polynomials
B.8 Associated Legendre Functions
B.9 Fourier-Legendre Series in Associated Legendre Functions
B.10 Airy Functions
Appendix C: Sturm-Liouville Problem and Auxiliary Functions for One and Two Dimensions
C.1 Eigenvalues and Eigenfunctions of 1D Sturm-Liouville Problem for Different Types of Boundary Conditions
C.2 Auxiliary Functions
Appendix D: The Sturm-Liouville Problem for Circular and Rectangular Domains
D.1 The Sturm-Liouville Problem for a Circle
D.2 The Sturm-Liouville Problem for the Rectangle
Appendix E: The Heat Conduction and Poisson Equations for Rectangular Domains – Examples
E.1 The Laplace and Poisson Equations for a Rectangular Domain with Nonhomogeneous Boundary Conditions – Examples
E.2 The Heat Conduction Equations with Nonhomogeneous Boundary Conditions – Examples
Bibliography
Index