Partial Differential Equations: Analytical Solution Techniques

Preface

Chapter 1. The Diffusion Equation

1.2.1 Similarity (Invariance)
1.2.2 Qualitative Behavior; Diffusion
1.2.3 Laplace Transforms
1.2.4 Fourier Transforms

1.4.1 Green's Function of the First Kind
1.4.2 Homogeneous Boundary-Value Problems
1.4.3 Inhomogeneous Boundary Condition $u=g(t)$
1.4.4 Green's Function of the Second Kind
1.4.5 Homogeneous Boundary-Value Problems
1.4.6 Inhomogeneous Boundary Condition
1.4.7 The General Boundary-Value Problem

1.5.1 Green's Function of the First Kind
1.5.2 Connection with Separation of Variables
1.5.3 Connection with Laplace Transform Solution
1.5.4 Uniqueness of Solutions
1.5.5 Inhomogeneous Boundary Conditions
1.5.6 Higher-Dimensional Problems

1.6.1 The Cole-Hopf Transformation
1.6.2 Initial-Value Problem on
1.6.3 Boundary-Value Problem on

Chapter 2. Laplace's Equation

2.1.1 Incompressible Irrotational Flow
2.1.2 Two-Dimensional Incompressible Flow

2.2.1 Mapping of Harmonic Functions
2.2.2 Transformation of Boundary Conditions
2.2.3 Example, Solution in a ``Simpler'' Transformed Domain

2.3.1 Point Source in Three Dimensions
2.3.2 Fundamental Solution in Two-Dimensions; Descent
2.3.3 Effect of Lower Derivative Terms
2.3.4 Potential Due to a Dipole

2.4.1 Volume Distribution of Sources
2.4.2 Surface and Line Distribution of Sources or Dipoles
2.4.3 An Example: Flow Over a Nonlifting Body of Revolution
2.4.4 Limiting Surface Values for Source and Dipole Distributions

2.5.1 Gauss' Integral Theorem
2.5.2 Energy Theorem and Corollaries
2.5.3 Uniqueness Theorems
2.5.4 Mean-Value Theorem
2.5.5 Surface Distribution of Sources and Dipoles
2.5.6 Potential Due to Dipole Distribution of Unit Strength

2.6.1 Green's Function
2.6.2 Neumann's Function

2.8.1 Upper Half-Plane, (Two Dimensions)
2.8.1 Upper Half-Space, (Three Dimensions)
2.8.3 Interior (Exterior) of Unit Sphere or Circle

Dipole-Green's Functions
2.11. Series Representations; Connection with Separation of Variables
2.12. Solutions in Terms of Integral Equations

2.12.1 Dirichlet's Problem
2.12.2 Neumann's Problem

Review Problems
Problems
References

Chapter 3. The Wave Equation

3.2.1 Assumptions
3.2.2 Hydrostatic Balance
3.2.3 Conservation of Mass
3.2.4 Conservation of Momentum in the X direction
3.2.5 Smooth Solutions
3.2.6 Energy Conservation
3.2.7 Initial-Value Problem
3.2.8 Signaling Problem
3.2.9 Small-Amplitude Theory

3.3.1 Conservation Laws
3.3.2 One-Dimensional Ideal Gas
3.3.3 Signaling Problem for One-Dimensional Flow
3.3.4 Inviscid, Non-Heat-Conducting Gas; Analogy with Shallow-Water Waves
3.3.5 Small-Disturbance Theory in One-Dimensional Flow (Signaling Problem)
3.3.6 Small Disturbance Theory in Three Dimensional, Inviscid Non-Heat-Conducting Flow

3.4.1 Fundamental Solution
3.4.2 General Initial-Value Problem on
3.4.3 An Example

3.5.1 Green's Function of the First Kind
3.5.2 Homogeneous Boundary Condition, Nonzero Initial Conditions
3.5.3 Inhomogeneous Boundary Condition $u (0, t) =g (t)$
3.5.4 An Example
3.5.5 A Second Example: Solutions with a Fixed Interface; Reflected and Transmitted Waves
3.5.6 Green's Function of the Second Kind

3.6.1 Green's Function of the First Kind on
3.6.2 The Inhomogeneous Problem, Nonzero Initial Conditions
3.6.3 Inhomogeneous Boundary Conditions
3.6.4 Uniqueness of the General Initial- and Boundary-Value Problem of the First Kind

3.7.1 Transformation to D'Alembert Form: Removal of Lower-Derivative Terms
3.7.2 Fundamental Solution; Stability
3.7.3 Green's Functions; Initial- and Boundary-Value Problems

3.8.1 Uniform Waves
3.8.2 General Initial-Value Problem
3.8.3 Group Velocity
3.8.4 Dispersion

3.9.1 Fundamental Solution
3.9.2 Arbitrary Source Distribution
3.9.3 Initial-Value Problems for the Homogeneous Equation

3.10.1 The Bursting Balloon
3.10.2 Source Distribution over the Plane
3.10.3 Perturbation of a Uniform Flow

Chapter 4. Linear Second-Order Equations with Two Independent Variables

4.2.1 The Hyperbolic Problem,
4.2.2 Hyperbolic Examples
4.2.3 The Parabolic Problem,
4.2.4 The Elliptic Problem,

4.3.1 Cauchy's Problem
4.3.2 Characteristics as Carriers of Discontinuities in the Second Derivative

4.4. Solution of Hyperbolic Equations in Terms of Characteristics

4.4.1 Cauchy Data on a Spacelike Arc
4.4.2 Cauchy Problem; the Numerical Method of Characteristics
4.4.3 Goursat's Problem; Boundary Conditions on a Timelike Arc
4.4.4 Characteristic Boundary-Value Problem
4.4.5 Well-Posedness
4.4.6 The General Solution of Cauchy's Problem; the Riemann Function
4.4.7 Weak Solutions; Propagation of Discontinuities in P and Q; Stability

4.5. Hyperbolic Systems of Two First-Order Equations

4.5.1 The Perturbation of a Quasilinear System near a Known Solution
4.5.2 Characteristics
4.5.3 Transformation on Characteristic Variables
4.5.4 Numerical Solutions; Propagation of Discontinuities
4.5.5 Connection with the Second-Order Equation
4.5.6 Perturbation of the Dam-Breaking Problem

Problems
References

Chapter 5. Quasilinear First-Order Equations

5.1.1 Flow of Water in a Conduit with Friction
5.1.2 Traffic Flow

5.2.1 Geometrical Aspects of Solutions
5.2.2 Characteristic Curves; the Solution Surface

5.3.1 Shock Speed for a System of Integral Conservation Laws
5.3.2 Formal Definition of a Weak Solution
5.3.3 The Correct Shock and Interface Conditions
5.3.4 Constant Speed Shocks; Nonuniqueness of Weak Solutions
5.3.5 An Example of Shock Fitting for the Scalar Problem
5.3.6 Exact Solution of Burgers' Equation: Shock Layer, Corner Layer

5.4.1 The Initial-Value Problem
5.4.2 The Characteristic Manifold; Existence and Uniqueness of Solutions
5.4.3 A Linear Example
5.4.4 A Quasilinear Example

Chapter 6. Nonlinear First-Order Equations

6.1.1 Huyghens' Construction; the Eikonal Equation
6.1.2 The Equation for Light Rays
6.1.3 Fermat's Principle

6.2.1 The Variation of a Functional
6.2.2 A Variational Principle; The Euler-Lagrange Equations
6.2.3 Hamiltonian Form of the Variational Problem
6.2.4 Field of Extremals from a Point; The Hamilton-Jacobi Equation
6.2.5 Extremals from a Manifold; Transversality
6.2.6 Canonical Transformations

6.3.1 The Geometry of Solutions
6.3.2 Focal Strips and Characteristic Strips
6.3.3 The Initial-Value Problem
6.3.4 Example Problems for the Eikonal Equation

6.4.1 Envelope Surfaces Associated with the Complete Integral
6.4.2 Relationsip between Characteristic Strips and the Complete Integral
6.4.3 The Complete Integral of the Hamilton-Jacobi Equation

Chapter 7. Quasilinear Hyperbolic Systems

7.1.1 Transformation to Characteristic Variables
7.1.2 The Cauchy Problem; the Numerical Method of Characteristics

7.2.1 Characteristic Curves and the Normal Form
7.2.2 Unsteady Nonisentropic Flow
7.2.3 A Semilinear Example

7.3.1 Characteristic Coordinates
7.3.2 The Hodograph Transformation
7.3.3 The Riemann Invariants
7.3.4 Applications of the Riemann Invariants

7.4.1 Characteristic Coordinates; Riemann Invariants
7.4.2 Simple Wave Solutions
7.4.3 Solutions with Bores

7.5.1 One-Dimensional Unsteady Flow
7.5.2 Steady Irrotational Two-Dimensional Flow

Chapter 8. Perturbation Solutions

8.1.1 Order Symbols
8.1.2 Definition of an Asymptotic Expansion
8.1.3 Asymptotic Expansion of a Given Function
8.1.4 Asymptotic Expansion of the Root of an Algebraic Equation
8.1.5 Asymptotic Expansion of a Definite Integral

8.2.1 Green's Function for an Ordinary Differential Equation
8.2.2 Eigenvalues and Eigenfunctions of a Perturbed Self-Adjoint Operator
8.2.3 A Boundary Perturbation Problem

8.3.1 An Ordinary Differential Equation
8.3.2 A Second Example
8.3.3 Interior Dirichlet Problems for Elliptic Equations
8.3.4 Slender Body Theory; a Problem with a Boundary Singularity
8.3.5 Burgers' Equation for

8.4.1 The Oscillator with a Weak Nonlinear Damping; Regular Expansion
8.4.2 The Multiple Scale Expansion
8.4.3 Near-Identity Averaging Transformations
8.4.4 Evolution Equations for a Weakly Nonlinear Problem

Index

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Table of Contents

Partial Differential Equations: Analytical Solution Techniques