Chapter
1.4.3 The Residue Theorem
1.5 Application to Real Integrals
1.5.1 Principal Values; Jordan's Lemma
2.1.1 Batchelor's Trailing Vortex
2.2.1 Different Forms for gamma(z)
2.3 Functions Defined by Differential Equations
2.3.3 Hypergeometric Functions
2.3.4 Confluent Hypergeometric Functions
2.3.5 Chebyshev Functions; Worked Example
2.4 Integral Representations
2.4.1 Physical Problem: Heat Conduction
2.4.2 Bessel Function Integrals
2.4.3 Hypergeometric Integrals
2.4.4 Airy Function Integrals
3 Eigenvalue Problems and Eigenfunction Expansions
3.2 Synge's Setup for Rayleigh's Criterion
3.3 Sturm–Liouville Problems
3.3.2 Orthogonality of Eigenfunctions
3.3.3 Synge's Proof of Rayleigh's Criterion
3.4 Expansions in Eigenfunctions
3.4.1 The Nonhomogeneous Problem; Solvability Condition
3.5.1 Heat Conduction in a Nonuniform Rod
3.5.2 Waves on Shallow Water
3.6 Nonstandard Eigenvalue Problems
3.6.2 Sturm–Liouville Problems with Weight Changing Sign (Counterrotating Cylinders)
3.6.3 Singular Sturm–Liouville Problems
3.7 Fourier–Bessel Series
3.7.1 Worked Bessel Function Examples
3.8 Continuous versus Discrete Spectra
4 Green's Functions for Boundary-Value Problems
4.1.1 Sources and Fundamental Solutions
4.1.2 Conduction of Heat in a Spherical Shell with Sources
4.2.1 Second-Order Problems
4.2.2 Higher-Order Problems
4.2.3 Adjoint and Self-Adjoint Problems
Separated Boundary Conditions
4.3 Connections with Distributions
4.4 First-Order System: Green's Matrices
4.5 Generalized Green's Functions
4.6 Expansions in Eigenfunctions
4.6.1 Delta Function Representation
4.7 Appendix: Linear Ordinary Differential Equations
4.7.1 Fundamental Solutions and the Wronskian Matrix
The Initial-Value Problem
4.7.2 Variation of Parameters
4.7.3 The Adjoint Equation; Lagrange's Identity
Separated Boundary Conditions
5 Laplace Transform Methods
5.2 The Laplace Transform and Its Inverse
Limiting Behavior: Initial-Value and Final-Value Theorems
5.2.2 Completion of the gamma Contour for Laplace Inversion
5.3.1 An Ordinary Differential Equation
5.3.2 Translating Plate in a Fluid
5.3.3 Heat Conduction in a Strip
5.3.5 A Scattering Problem
5.3.6 Conduction of Heat in a Spherical Shell
5.3.7 Boundary Layer Evolution for MHD Flow: Hartmann Layer
5.4 Bilateral Laplace Transform
5.4.1 Inverse Bilateral Laplace Transform
A Modified Bessel Function
Time-Dependent Boundary Layer with Suction
6 Fourier Transform Methods
6.2 The Fourier Transform and Its Inverse
6.2.3 Special Properties of Fourier Transforms
6.2.4 Cosine and Sine Transforms
6.3.1 Example 1: The Ekman Layer
6.3.2 Example 2: Heat Conduction in a Strip
6.3.3 Example 3: Heat Conduction in a Half-Plane
6.3.4 Example 4: Sound Waves
6.3.5 Example 5: Diffusion in a Force Field
6.3.6 Example 6: An Integro-Differential Equation
6.3.7 Example 7: Thermal Wake in a Small-Prandtl-Number Fluid
6.3.8 Example 8: Fundamental Solution for Stokes Flow
6.5.1 Stability of Flow Near a Stagnation Point
6.5.2 Stability of Jeffery–Hamel Flows
7 Particular Physical Problems
7.3 The Far Momentum Wake
7.4 Kelvin–Helmholtz Instability
Case 1 – Plane-Wave Modes
Case 2 – Localized Initial Disturbance
7.5 The Boundary Layer Signal Problem
7.6 Stability of Plane Couette Flow
7.7 Generalized Transform Techniques
7.7.2 A Boundary-Layer Example
8 Asymptotic Expansions of Integrals
8.2 Asymptotic Expansions
8.4 Laplace-Type Integrals; Watson's Lemma
8.4.2 Application: Early-Time Heat Transfer
8.5 Generalized Laplace Integrals: Laplace's Method
8.6 Method of Steepest Descent
8.6.1 Application: A Special Function
8.6.2 Application: The Oscillating Plate
8.6.3 Application: Lee Waves
8.6.4 Application: Sound Waves
8.6.5 Application: Two-dimensional Laminar Wake
8.7 Method of Stationary Phase; Kelvin's Results
Partial Differential Equations in Fluid Dynamics
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