Introduction / 1.: |
Parameter Perturbations / 1.1.: |
An Algebraic Equation / 1.1.1.: |
The van der Pol Oscillator / 1.1.2.: |
Coordinate Perturbations / 1.2.: |
The Bessel Equation of Zeroth Order / 1.2.1.: |
A Simple Example / 1.2.2.: |
Order Symbols and Gauge Functions / 1.3.: |
Asymptotic Expansions and Sequences / 1.4.: |
Asymptotic Series / 1.4.1.: |
Asymptotic Expansions / 1.4.2.: |
Uniqueness of Asymptotic Expansions / 1.4.3.: |
Convergent versus Asymptotic Series / 1.5.: |
Nonuniform Expansions / 1.6.: |
Elementary Operations on Asymptotic Expansions / 1.7.: |
Exercises |
Straightforward Expansions and Sources of Nonuniformity / 2.: |
Infinite Domains / 2.1.: |
The Duffing Equation / 2.1.1.: |
A Model for Weak Nonlinear Instability / 2.1.2.: |
Supersonic Flow Past a Thin Airfoil / 2.1.3.: |
Small Reynolds Number Flow Past a Sphere / 2.1.4.: |
A Small Parameter Multiplying the Highest Derivative / 2.2.: |
A Second-Order Example / 2.2.1.: |
High Reynolds Number Flow Past a Body / 2.2.2.: |
Relaxation Oscillations / 2.2.3.: |
Unsymmetrical Bending of Prestressed Annular Plates / 2.2.4.: |
Type Change of a Partial Differential Equation / 2.3.: |
Long Waves on Liquids Flowing down Incline Planes / 2.3.1.: |
The Presence of Singularities / 2.4.: |
Shift in Singularity / 2.4.1.: |
The Earth-Moon-Spaceship Problem / 2.4.2.: |
Thermoelastic Surface Waves / 2.4.3.: |
Turning Point Problems / 2.4.4.: |
The Role of Coordinate Systems / 2.5.: |
The Method of Strained Coordinates / 3.: |
The Method of Strained Parameters / 3.1.: |
The Lindstedt-Poincare Method / 3.1.1.: |
Transition Curves for the Mathieu Equation / 3.1.2.: |
Characteristic Exponents for the Mathieu Equation (Whittaker's Method) / 3.1.3.: |
The Stability of the Triangular Points in the Elliptic Restricted Problem of Three Bodies / 3.1.4.: |
Characteristic Exponents for the Triangular Points in the Elliptic Restricted Problem of Three Bodies / 3.1.5.: |
A Simple Linear Eigenvalue Problem / 3.1.6.: |
A Quasi-Linear Eigenvalue Problem / 3.1.7.: |
The Quasi-Linear Klein-Gordon Equation / 3.1.8.: |
Lighthill's Technique / 3.2.: |
A First-Order Differential Equation / 3.2.1.: |
The One-Dimensional Earth-Moon-Spaceship Problem / 3.2.2.: |
A Solid Cylinder Expanding Uniformly in Still Air / 3.2.3.: |
Expansions by Using Exact Characteristics--Nonlinear Elastic Waves / 3.2.4.: |
Temple's Technique / 3.3.: |
Renormalization Technique / 3.4.: |
Limitations of the Method of Strained Coordinates / 3.4.1.: |
The Methods of Matched and Composite Asymptotic Expansions / 3.5.1.: |
The Method of Matched Asymptotic Expansions / 4.1.: |
Introduction--Prandtl's Technique / 4.1.1.: |
Higher Approximations and Refined Matching Procedures / 4.1.2.: |
A Second-Order Equation with Variable Coefficients / 4.1.3.: |
Reynolds' Equation for a Slider Bearing / 4.1.4.: |
The Method of Composite Expansions / 4.1.5.: |
A Second-Order Equation with Constant Coefficients / 4.2.1.: |
An Initial Value Problem for the Heat Equation / 4.2.2.: |
Limitations of the Method of Composite Expansions / 4.2.4.: |
Variation of Parameters and Methods of Averaging / 5.: |
Variation of Parameters / 5.1.: |
Time-Dependent Solutions of the Schrodinger Equation / 5.1.1.: |
A Nonlinear Stability Example / 5.1.2.: |
The Method of Averaging / 5.2.: |
Van der Pol's Technique / 5.2.1.: |
The Krylov-Bogoliubov Technique / 5.2.2.: |
The Generalized Method of Averaging / 5.2.3.: |
Struble's Technique / 5.3.: |
The Krylov-Bogoliubov-Mitropolski Technique / 5.4.: |
The Duffiing Equation / 5.4.1.: |
The Klein-Gordon Equation / 5.4.2.: |
The Method of Averaging by Using Canonical Variables / 5.5.: |
The Mathieu Equation / 5.5.1.: |
A Swinging Spring / 5.5.3.: |
Von Zeipel's Procedure / 5.6.: |
Averaging by Using the Lie Series and Transforms / 5.6.1.: |
The Lie Series and Transforms / 5.7.1.: |
Generalized Algorithms / 5.7.2.: |
Simplified General Algorithms / 5.7.3.: |
A Procedure Outline / 5.7.4.: |
Algorithms for Canonical Systems / 5.7.5.: |
Averaging by Using Lagrangians / 5.8.: |
A Model for Dispersive Waves / 5.8.1.: |
A Model for Wave-Wave Interaction / 5.8.2.: |
The Nonlinear Klein-Gordon Equation / 5.8.3.: |
The Method of Multiple Scales / 6.: |
Description of the Method / 6.1.: |
Many-Variable Version (The Derivative-Expansion Procedure) / 6.1.1.: |
The Two-Variable Expansion Procedure / 6.1.2.: |
Generalized Method--Nonlinear Scales / 6.1.3.: |
Applications of the Derivative-Expansion Method / 6.2.: |
Forced Oscillations of the van der Pol Equation / 6.2.1.: |
Parametric Resonances--The Mathieu Equation / 6.2.4.: |
The van der Pol Oscillator with Delayed Amplitude Limiting / 6.2.5.: |
Limitations of the Derivative-Expansion Method / 6.2.6.: |
Limitations of This Technique / 6.3.: |
Generalized Method / 6.4.: |
A General Second-Order Equation with Variable Coefficients / 6.4.1.: |
A Linear Oscillator with a Slowly Varying Restoring Force / 6.4.3.: |
An Example with a Turning Point / 6.4.4.: |
The Duffing Equation with Slowly Varying Coefficients / 6.4.5.: |
Reentry Dynamics / 6.4.6.: |
Advantages and Limitations of the Generalized Method / 6.4.7.: |
Asymptotic Solutions of Linear Equations / 7.: |
Second-Order Differential Equations / 7.1.: |
Expansions Near an Irregular Singularity / 7.1.1.: |
An Expansion of the Zeroth-Order Bessel Function for Large Argument / 7.1.2.: |
Liouville's Problem / 7.1.3.: |
Higher Approximations for Equations Containing a Large Parameter / 7.1.4.: |
Homogeneous Problems with Slowly Varying Coefficients / 7.1.5.: |
Reentry Missile Dynamics / 7.1.7.: |
Inhomogeneous Problems with Slowly Varying Coefficients / 7.1.8.: |
Successive Liouville-Green (WKB) Approximations / 7.1.9.: |
Systems of First-Order Ordinary Equations / 7.2.: |
Expansions Near an Irregular Singular Point / 7.2.1.: |
Asymptotic Partitioning of Systems of Equations / 7.2.2.: |
Subnormal Solutions / 7.2.3.: |
Systems Containing a Parameter / 7.2.4.: |
Homogeneous Systems with Slowly Varying Coefficients / 7.2.5.: |
The Langer Transformation / 7.3.: |
Problems with Two Turning Points / 7.3.3.: |
Higher-Order Turning Point Problems / 7.3.4.: |
Higher Approximations / 7.3.5.: |
An Inhomogeneous Problem with a Simple Turning Point--First Approximation / 7.3.6.: |
An Inhomogeneous Problem with a Simple Turning Point--Higher Approximations / 7.3.7.: |
An Inhomogeneous Problem with a Second-Order Turning Point / 7.3.8.: |
Turning Point Problems about Singularities / 7.3.9.: |
Turning Point Problems of Higher Order / 7.3.10.: |
Wave Equations / 7.4.: |
The Born or Neumann Expansion and The Feynman Diagrams / 7.4.1.: |
Renormalization Techniques / 7.4.2.: |
Rytov's Method / 7.4.3.: |
A Geometrical Optics Approximation / 7.4.4.: |
A Uniform Expansion at a Caustic / 7.4.5.: |
The Method of Smoothing / 7.4.6.: |
References and Author Index |
Subject Index |
Introduction / 1.: |
Parameter Perturbations / 1.1.: |
An Algebraic Equation / 1.1.1.: |