Partial Differential Equations

Transcription

1 Partial Differential Equations Analytical Solution Techniques J. Kevorkian University of Washington Wadsworth & Brooks/Cole Advanced Books & Software Pacific Grove, California

2 C H A P T E R 1 The Diffusion Equation Heat Conduction Fundamental Solution Similarity (Invariance) Qualitative Behavior; Diffusion Laplace Transforms Fourier Transforms Initial-Value Problem (Cauchy Problem) on the Infinite Domain; Superposition Initial- and Boundary-Value Problems on the Semi-Infinite Domain; Green's Functions Green's Function of the First Kind Homogeneous Boundary-Value Problems Inhomogeneous Boundary Condition u = g{t) Green's Function ofthe Second Kind Homogeneous Boundary-Value Problems Inhomogeneous Boundary Condition The General Boundary-Value Problem Initial- and Boundary-Value Problems on the Finite Domain; Green's Functions Green's Function of the First Kind Connection with Separation of Variables Connection with Laplace Transform Solution Uniqueness of Solutions Inhomogeneous Boundary Conditions Higher-Dimensional Problems 31 vii

3 1.6 Burgers' Equation The Cole-Hopf Transformation Initial- Value Problem on oo < x < oo Boundary- Value Problem on 0 < x < oo 33 Review Problems 35 Problems 39 References 47 C H A P T E R 2 Laplace's Equation Applications Incompressible Irrotational Flow Two-Dimensional Incompressible Flow The Two-Dimensional Problem; Conformal Mapping Mapping of Harmonie Functions Transformation of Boundary Conditions Example, Solution in a "Simpler" TransformedDomain Fundamental Solution; Dipole Potential Point Source in Three Dimensions Fundamental Solution in Two-Dimensions;Descent Effect of Lower Derivative Terms Potential Due to a Dipole Potential Due to Volume, Surface, and Line Distribution of Sources and Dipoles Volume Distribution of Sources Surface and Line Distribution of Sources or Dipoles An Example: Flow Over a Nonlifting Body of Revolution Limiting Surface Valuesfor Source and Dipole Distributions Green's Formula and Applications Gauss'Integral Theorem Energy Theorem and Corollaries Uniqueness Theorems Mean- Value Theorem Surface Distribution of Sources and Dipoles Potential Due to Dipole Distribution of Unit Strength Green's and Neumann's Functions Green's Function Neumann 's Function 82

4 t; ix 2.7 Dirichlet's and Neumann's Problems Examples of Green's and Neumann's Functions Upper Half-Plane, y >0 (Two Dimensions) Upper Half-Space, z >0 (Three Dimensions) Inferior (Exterior) of Unit Sphere or Circle Estimates; Harnack's Inequality Connection between Green's Function and Conformal Mapping (Two Dimensions); Dipole-Green's Functions Series Representations; Connection with Separation of Variables Solutions in Terms of Integral Equations Dirichlet's Problem Neumann's Problem 98 Review Problems 99 Problems 106 References 116 C H A P T E R 3 The Wave Equation The Vibrating String Shallow-Water Waves Assumptions Hydrostatic Balance Conservation ofmass Conservation of Momentum in the X direction Smooth Solutions Energy Conservation Initial-Value Problem Signaling Problem Small-Amplitude Theory Compressible Flow Conservation Laws One-Dimensional Ideal Gas Signaling Problem for One-Dimensional Flow Inviscid, Non-Heat-Conducting Gas; Analogy with Shallow-Water Waves Small-Disturbance Theory in One-Dimensional Flow (Signaling Problem) 134

5 x Small Disturbance Theory in Three Dimensional, Inviscid Non-Heat-Conducting Flow The One-Dimensional Problem in the Infinite Domain Fundamental Solution General Initial- Value Problem on oo < x < oo An Example Initial- and Boundary-Value Problems on the Semi-Infinite Interval; Green's Functions Green's Function of the First Kind Homogeneous Boundary Condition, Nonzero Initial Conditions Inhomogeneous Boundary Condition u(0,f) = g(t) An Example A Second Example: Solutions with a Fixed Interface; Reflected and Transmitted Waves Green's Function of the Second Kind Initial- and Boundary-Value Problems on the Finite Interval; Green's Functions Green 's Function of the First Kind on0< x<\ The Inhomogeneous Problem, Nonzero Initial Conditions Inhomogeneous Boundary Conditions Uniqueness of the General Initial- and Boundary- Value Problem of the First Kind Effect of Lower-Derivative Terms Transformation to D'Alembert Form: Removal of Lower- Derivative Terms Fundamental Solution; Stability Green 's Functions; Initial- and Boundary- Value Problems Dispersive Waves on the Infinite Interval Uniform Waves General Initial-Value Problem Group Velocity Dispersion The Three-Dimensional Wave Equation; Acoustics Fundamental Solution Arbitrary Source Distribution Initial- Value Problems for the Homogeneous Equation Examples in Acoustics and Aerodynamics The Bursting Balloon Source Distribution over the Plane Perturbation of a Uniform Flow 189 Problems 192 References 205

6 v xi C H A P T E R 4 Linear Second-Order Equations with Two Independent Variables A General Transformation of Variables Classification The Hyperbolic Problem, A > 0; A = C = Hyperbolic Examples The Parabolic Problem, Ä = 0; C = The Elliptic Problem, A<0;B = 0,A = C The Role of Characteristics in Hyperbolic Equations Cauchy's Problem Characteristics as Carriers of Discontinuities in the Second Derivative 216 AA Solution of Hyperbolic Equations in Terms of Characteristics Cauchy Data on a Spacelike Are Cauchy Problem; the Numerical Method of Characteristics Goursat's Problem; Boundary Conditions on a Timelike Are Characteristic Boundary- Value Problem Well-Posedness The General Solution ofcauchy's Problem; the Riemann Function Weak Solutions; Propagation of Discontinuities in P and Q; Stability Hyperbolic Systems of Two First-Order Equations The Perturbation ofa Quasilinear System near a Known Solution Characteristics Transformation to Characteristic Variables Numerical Solutions; Propagation of Discontinuities Connection with the Second-Order Equation Perturbation of the Dam-Breaking Problem 245 Problems 250 References 260 C H A P T E R 5 Quasilinear First-Order Equations The Scalar Conservation Law; Quasilinear Equations Flow of Water in a Conduit with Friction TrafßcFlow Continuously Differentiable Solution of the Quasilinear Equation in Two Independent Variables Geometrical Aspects of Solutions Characteristic Curves; the Solution Surface 269

7 xii 5.3 Weak Solutions: Shocks, Fans, and Interfaces Shock Speedfor a System ofintegral Conservation Laws Formal Definition of a Weak Solution The Correct Shock and Interface Conditions Constant Speed Shocks; Nonuniqueness of Weak Solutions An Example of Shock Fittingfor the Scalar Problem Exact Solution of Burgers' Equation: Shock Layer, Corner Layer The Quasilinear Equation in n Independent Variables The Initial-Value Problem The Characteristic Manifold; Existence and Uniqueness of Solutions A Linear Example A Quasilinear Example 316 Problems 317 References 321 C H A P T E R 6 Nonlinear First-Order Equations Geometrical Optics: A Nonlinear Equation Huyghens' Construction; the Eikonal Equation The Equation for Light Rays Fermat's Principle Applications Leading to the Hamilton-Jacobi Equation The Variation of a Functional A Variational Principle; The Euler-Lagrange Equations Hamiltonian Form of the Variational Problem Field of Extremais from a Point; The Hamilton-Jacobi Equation Extremais from a Manifold; Transversality Canonical Transformations The Nonlinear Equation The Geometry of Solutions Focal Strips and Characteristic Strips The Inital-Value Problem Example Problems for the Eikonal Equation The Complete Integral; Solutions by Envelope Formation Envelope Surfaces Associated with the Complete Integral Relationship between Characteristic Strips and the Complete Integral The Complete Integral of the Hamilton-Jacobi Equation 375 Problems 379 References 385

8 xiii C H A P T E R 7 Quasilinear Hyperbolic Systems The Quasilinear Second-Order Hyperbolic Equation Transformation to Characteristic Variables The Cauchy Problem; the Numerical Method ofcharacteristics Systems of n First-Order Equations Characteristic Curves and the Normal Form Unsteady Nonisentropic Flow A Semilinear Example Systems of Two First-Order Equations Characteristic Coordinates The Hodograph Transformation The Riemann Invariants Applications ofthe Riemann Invariants Shallow-Water Waves Characteristic Coordinates; Riemann Invariants Simple Wave Solutions Solutions with Bores Compressible Flow Problems One-Dimensional Unsteady Flow Steady Irrotational Two-Dimensional Flow 449 Problems 453 References 457 C H A P T E R 8 Perturbatio«! Solutions Asymptotic Expansions Order Symbols Definition of an Asymptotic Expansion Asymptotic Expansion of a Given Function Asymptotic Expansion of the Root of an Algebraic Equation Asymptotic Expansion of a Definite Integral Regulär Perturbations Green s Function for an Ordinary Differential Equation Eigenvalues and Eigenfunctions of a Perturbed Self-Adjoint Operator A Boundary Perturbation Problem 477

9 xiv 8.3 Matched Asymptotic Expansions An Ordinary Differential Equation A Second Example Inferior Dirichlet Problemsfor Elliptic Equations Slender Body Theory; a Problem with a Boundary Singularity Burgers' Equation for e « Cumulative Perturbations; Solution Valid in the Far Field The Oscillator with a Weak Nonlinear Damping; Regulär Expansion The Multiple Scale Expansion Near-Identity Averaging Transformations Evolution Equationsfor a Weakly Nonlinear Problem 513 Problems 526 References 536 Index 538