Partial Differential Equations - GBV

Partial Differential Equations Analytical Solution Techniques

J. Kevorkian University of Washington

Wadsworth & Brooks/Cole Advanced Books & Software Pacific Grove, California

Contents

C H A P T E R 1

The Diffusion Equation 1

1.1 Heat Conduction 1 1.2 Fundamental Solution 5

1.2.1 Similarity (Invariance) 5

1.2.2 Qualitative Behavior; Diffusion 9 1.2.3 Laplace Transforms 10 1.2.4 Fourier Transforms 11

1.3 Initial-Value Problem (Cauchy Problem) on the Infinite Domain; Superposition 12

1.4 Initial- and Boundary-Value Problems on the Semi-Infinite Domain; Green's Functions 14

1.4.1 Green's Function of the First Kind 14 1.4.2 Homogeneous Boundary-Value Problems 15 1.4.3 Inhomogeneous Boundary Condition u = g{t) 18 1.4.4 Green's Function ofthe Second Kind 20 1.4.5 Homogeneous Boundary-Value Problems 21 1.4.6 Inhomogeneous Boundary Condition 21 1.4.7 The General Boundary-Value Problem 22

1.5 Initial- and Boundary-Value Problems on the Finite Domain; Green's Functions 22

1.5.1 Green's Function of the First Kind 22 1.5.2 Connection with Separation of Variables 24 1.5.3 Connection with Laplace Transform Solution 27 1.5.4 Uniqueness of Solutions 29 1.5.5 Inhomogeneous Boundary Conditions 30 1.5.6 Higher-Dimensional Problems 31

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1.6 Burgers' Equation 31 1.6.1 The Cole-Hopf Transformation 32 1.6.2 Initial- Value Problem on — oo < x < oo 32 1.6.3 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 48

2.1 Applications 48 2.1.1 Incompressible Irrotational Flow 49

2.1.2 Two-Dimensional Incompressible Flow 49

2.2 The Two-Dimensional Problem; Conformal Mapping 51 2.2.1 Mapping of Harmonie Functions 51 2.2.2 Transformation of Boundary Conditions 52

2.2.3 Example, Solution in a "Simpler" TransformedDomain 53

2.3 Fundamental Solution; Dipole Potential 57 2.3.1 Point Source in Three Dimensions 58

2.3.2 Fundamental Solution in Two-Dimensions;Descent 59 2.3.3 Effect of Lower Derivative Terms 61 2.3.4 Potential Due to a Dipole 62

2.4 Potential Due to Volume, Surface, and Line Distribution of Sources and Dipoles 64

2.4.1 Volume Distribution of Sources 65 2.4.2 Surface and Line Distribution of Sources or Dipoles 66 2.4.3 An Example: Flow Over a Nonlifting Body of Revolution 67 2.4.4 Limiting Surface Valuesfor Source and Dipole Distributions 68

2.5 Green's Formula and Applications 75 2.5.1 Gauss'Integral Theorem 76 2.5.2 Energy Theorem and Corollaries 76 2.5.3 Uniqueness Theorems 77 2.5.4 Mean- Value Theorem 77 2.5.5 Surface Distribution of Sources and Dipoles 78 2.5.6 Potential Due to Dipole Distribution of Unit Strength 79

2.6 Green's and Neumann's Functions 81 2.6.1 Green's Function 81 2.6.2 Neumann 's Function 82

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2.7 Dirichlet's and Neumann's Problems 83 2.8 Examples of Green's and Neumann's Functions 84

2.8.1 Upper Half-Plane, y >0 (Two Dimensions) 84 2.8.2 Upper Half-Space, z > 0 (Three Dimensions) 85

2.8.3 Inferior (Exterior) of Unit Sphere or Circle 87

2.9 Estimates; Harnack's Inequality 89 2.10 Connection between Green's Function and Conformal Mapping

(Two Dimensions); Dipole-Green's Functions 90 2.11 Series Representations; Connection with Separation

of Variables 92 2.12 Solutions in Terms of Integral Equations 94

2.12.1 Dirichlet's Problem 94 2.12.2 Neumann's Problem 98

Review Problems 99 Problems 106 References 116

C H A P T E R 3

The Wave Equation 117

3.1 The Vibrating String 117 3.2 Shallow-Water Waves 120

3.2.1 Assumptions 120 3.2.2 Hydrostatic Balance 121 3.2.3 Conservation ofMass 122 3.2.4 Conservation of Momentum in the X direction 123 3.2.5 Smooth Solutions 123 3.2.6 Energy Conservation 124 3.2.7 Initial-Value Problem 125 3.2.8 Signaling Problem 125 3.2.9 Small-Amplitude Theory 126

3.3 Compressible Flow 130 3.3.1 Conservation Laws 130 3.3.2 One-Dimensional Ideal Gas 130 3.3.3 Signaling Problem for One-Dimensional Flow 132 3.3.4 Inviscid, Non-Heat-Conducting Gas; Analogy with

Shallow-Water Waves 133 3.3.5 Small-Disturbance Theory in One-Dimensional Flow

(Signaling Problem) 134

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3.3.6 Small Disturbance Theory in Three Dimensional, Inviscid Non-Heat-Conducting Flow 136

3.4 The One-Dimensional Problem in the Infinite Domain 138 3.4.1 Fundamental Solution 138 3.4.2 General Initial- Value Problem on — oo < x < oo 140 3.4.3 An Example 143

3.5 Initial- and Boundary-Value Problems on the Semi-Infinite Interval; Green's Functions 148

3.5.1 Green's Function of the First Kind 148 3.5.2 Homogeneous Boundary Condition, Nonzero Initial Conditions 149 3.5.3 Inhomogeneous Boundary Condition u(0,f) = g(t) 153 3.5.4 An Example 154 3.5.5 A Second Example: Solutions with a Fixed Interface; Reflected and

Transmitted Waves 160 3.5.6 Green's Function of the Second Kind 162

3.6 Initial- and Boundary-Value Problems on the Finite Interval; Green's Functions 163

3.6.1 Green 's Function of the First Kind on0< x<\ 163 3.6.2 The Inhomogeneous Problem, Nonzero Initial Conditions 164 3.6.3 Inhomogeneous Boundary Conditions 166 3.6.4 Uniqueness of the General Initial- and Boundary- Value Problem of the

First Kind 166

3.7 Effect of Lower-Derivative Terms 168 3.7.1 Transformation to D'Alembert Form: Removal of Lower-

Derivative Terms 168 3.7.2 Fundamental Solution; Stability 169 3.7.3 Green 's Functions; Initial- and Boundary- Value Problems 170

3.8 Dispersive Waves on the Infinite Interval 170 3.8.1 Uniform Waves 171 3.8.2 General Initial-Value Problem 173 3.8.3 Group Velocity 174 3.8.4 Dispersion 176

3.9 The Three-Dimensional Wave Equation; Acoustics 178 3.9.1 Fundamental Solution 178 3.9.2 Arbitrary Source Distribution 181 3.9.3 Initial- Value Problems for the Homogeneous Equation 183

3.10 Examples in Acoustics and Aerodynamics 184 3.10.1 The Bursting Balloon 184 3.10.2 Source Distribution over the Plane 186 3.10.3 Perturbation of a Uniform Flow 189

Problems 192 References 205

C H A P T E R 4

Contents xi

Linear Second-Order Equations with Two Independent Variables 206

4.1 A General Transformation of Variables 206 4.2 Classification 208

4.2.1 The Hyperbolic Problem, A > 0; A = C = 0 208 4.2.2 Hyperbolic Examples 210 4.2.3 The Parabolic Problem, Ä = 0; C = 0 211 4.2.4 The Elliptic Problem, A<0;B = 0,A = C 212

4.3 The Role of Characteristics in Hyperbolic Equations 214 4.3.1 Cauchy's Problem 214 4.3.2 Characteristics as Carriers of Discontinuities in the

Second Derivative 216

AA Solution of Hyperbolic Equations in Terms of Characteristics 218 4.4.1 Cauchy Data on a Spacelike Are 219 4.4.2 Cauchy Problem; the Numerical Method of Characteristics 221 4.4.3 Goursat's Problem; Boundary Conditions on a Timelike Are 223 4.4.4 Characteristic Boundary- Value Problem 225 4.4.5 Well-Posedness 225 4.4.6 The General Solution ofCauchy's Problem; the Riemann Function 227 4.4.7 Weak Solutions; Propagation of Discontinuities in P and Q;

Stability 231

4.5 Hyperbolic Systems of Two First-Order Equations 233 4.5.1 The Perturbation ofa Quasilinear System near a Known Solution 234

4.5.2 Characteristics 236 4.5.3 Transformation to Characteristic Variables 238 4.5.4 Numerical Solutions; Propagation of Discontinuities 241 4.5.5 Connection with the Second-Order Equation 243 4.5.6 Perturbation of the Dam-Breaking Problem 245


C H A P T E R 5

Quasilinear First-Order Equations 261

5.1 The Scalar Conservation Law; Quasilinear Equations 261 5.1.1 Flow of Water in a Conduit with Friction 262 5.1.2 TrafßcFlow 266

5.2 Continuously Differentiable Solution of the Quasilinear Equation in Two Independent Variables 267

5.2.1 Geometrical Aspects of Solutions 267 5.2.2 Characteristic Curves; the Solution Surface 269

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5.3 Weak Solutions: Shocks, Fans, and Interfaces 277 5.3.1 Shock Speedfor a System ofIntegral Conservation Laws 278 5.3.2 Formal Definition of a Weak Solution 280 5.3.3 The Correct Shock and Interface Conditions 282 5.3.4 Constant Speed Shocks; Nonuniqueness of Weak Solutions 287 5.3.5 An Example of Shock Fittingfor the Scalar Problem 300 5.3.6 Exact Solution of Burgers' Equation: Shock Layer, Corner Layer 303

5.4 The Quasilinear Equation in n Independent Variables 310 5.4.1 The Initial-Value Problem 310 5.4.2 The Characteristic Manifold; Existence and Uniqueness of Solutions 312 5.4.3 A Linear Example 313

5.4.4 A Quasilinear Example 316


C H A P T E R 6

Nonlinear First-Order Equations 322

6.1 Geometrical Optics: A Nonlinear Equation 322 6.1.1 Huyghens' Construction; the Eikonal Equation 322 6.1.2 The Equation for Light Rays 324

6.1.3 Fermat's Principle 327

6.2 Applications Leading to the Hamilton-Jacobi Equation 328 6.2.1 The Variation of a Functional 328 6.2.2 A Variational Principle; The Euler-Lagrange Equations 331 6.2.3 Hamiltonian Form of the Variational Problem 332 6.2.4 Field of Extremais from a Point; The Hamilton-Jacobi Equation 341 6.2.5 Extremais from a Manifold; Transversality 345 6.2.6 Canonical Transformations 347

6.3 The Nonlinear Equation 358 6.3.1 The Geometry of Solutions 358 6.3.2 Focal Strips and Characteristic Strips 360 6.3.3 The Inital-Value Problem 364 6.3.4 Example Problems for the Eikonal Equation 366

6.4 The Complete Integral; Solutions by Envelope Formation 369 6.4.1 Envelope Surfaces Associated with the Complete Integral 369 6.4.2 Relationship between Characteristic Strips and the

Complete Integral 371 6.4.3 The Complete Integral of the Hamilton-Jacobi Equation 375


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C H A P T E R 7

Quasilinear Hyperbolic Systems 386

7.1 The Quasilinear Second-Order Hyperbolic Equation 386 7.1.1 Transformation to Characteristic Variables 387 7.1.2 The Cauchy Problem; the Numerical Method

ofCharacteristics 389

7.2 Systems of n First-Order Equations 392 7.2.1 Characteristic Curves and the Normal Form 393 7.2.2 Unsteady Nonisentropic Flow 397

7.2.3 A Semilinear Example 402

7.3 Systems of Two First-Order Equations 404 7.3.1 Characteristic Coordinates 404

7.3.2 The Hodograph Transformation 406 7.3.3 The Riemann Invariants 409 7.3.4 Applications ofthe Riemann Invariants 413

7.4 Shallow-Water Waves 419 7.4.1 Characteristic Coordinates; Riemann Invariants 420 7.4.2 Simple Wave Solutions 421 7.4.3 Solutions with Bores 431

7.5 Compressible Flow Problems 440 7.5.1 One-Dimensional Unsteady Flow 440 7.5.2 Steady Irrotational Two-Dimensional Flow 449


C H A P T E R 8

Perturbatio«! Solutions 458

8.1 Asymptotic Expansions 458 8.1.1 Order Symbols 459 8.1.2 Definition of an Asymptotic Expansion 460 8.1.3 Asymptotic Expansion of a Given Function 461 8.1.4 Asymptotic Expansion of the Root of an Algebraic Equation 462 8.1.5 Asymptotic Expansion of a Definite Integral 466

8.2 Regulär Perturbations 468 8.2.1 Green s Function for an Ordinary Differential Equation 468 8.2.2 Eigenvalues and Eigenfunctions of a Perturbed

Self-Adjoint Operator 472 8.2.3 A Boundary Perturbation Problem 477

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8.3 Matched Asymptotic Expansions 478 8.3.1 An Ordinary Differential Equation 478 8.3.2 A Second Example 485 8.3.3 Inferior Dirichlet Problemsfor Elliptic Equations 489 8.3.4 Slender Body Theory; a Problem with a Boundary Singularity 494 8.3.5 Burgers' Equation for e « 1 499

8.4 Cumulative Perturbations; Solution Valid in the Far Field 504 8.4.1 The Oscillator with a Weak Nonlinear Damping;

Regulär Expansion 504 8.4.2 The Multiple Scale Expansion 506 8.4.3 Near-Identity Averaging Transformations 508 8.4.4 Evolution Equationsfor a Weakly Nonlinear Problem 513


Index 538

Partial Differential Equations - GBV

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