Partial Differential Equations Analytical Methods and Applicatns 1st edition by Victor, Tatyana, Alexander 1138339830 9781138339835

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