Partial Differential Equations: Graduate Level Problems and ...

Partial Di?erential Equations: Graduate Level Problems

and Solutions

Igor Yanovsky

1

Partial Di?erential Equations

Igor Yanovsky, 2005

2

Disclaimer: This handbook is intended to assist graduate students with qualifying

examination preparation. Please be aware, however, that the handbook might contain,

and almost certainly contains, typos as well as incorrect or inaccurate solutions. I can

not be made responsible for any inaccuracies contained in this handbook.

Partial Di?erential Equations

Igor Yanovsky, 2005

3

Contents

1 Trigonometric Identities

6

2 Simple Eigenvalue Problem

8

3 Separation of Variables:

Quick Guide

9

4 Eigenvalues of the Laplacian:

Quick Guide

9

5 First-Order Equations

5.1 Quasilinear Equations . . . . . . . . . . . . . . .

5.2 Weak Solutions for Quasilinear Equations . . . .

5.2.1 Conservation Laws and Jump Conditions

5.2.2 Fans and Rarefaction Waves . . . . . . . .

5.3 General Nonlinear Equations . . . . . . . . . . .

5.3.1 Two Spatial Dimensions . . . . . . . . . .

5.3.2 Three Spatial Dimensions . . . . . . . . .

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10

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12

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6 Second-Order Equations

14

6.1 Classi?cation by Characteristics . . . . . . . . . . . . . . . . . . . . . . . 14

6.2 Canonical Forms and General Solutions . . . . . . . . . . . . . . . . . . 14

6.3 Well-Posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

7 Wave Equation

7.1 The Initial Value Problem . . . . . . . . . .

7.2 Weak Solutions . . . . . . . . . . . . . . . .

7.3 Initial/Boundary Value Problem . . . . . .

7.4 Duhamel¡¯s Principle . . . . . . . . . . . . .

7.5 The Nonhomogeneous Equation . . . . . . .

7.6 Higher Dimensions . . . . . . . . . . . . . .

7.6.1 Spherical Means . . . . . . . . . . .

7.6.2 Application to the Cauchy Problem

7.6.3 Three-Dimensional Wave Equation .

7.6.4 Two-Dimensional Wave Equation . .

7.6.5 Huygen¡¯s Principle . . . . . . . . . .

7.7 Energy Methods . . . . . . . . . . . . . . .

7.8 Contraction Mapping Principle . . . . . . .

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8 Laplace Equation

8.1 Green¡¯s Formulas . . . . . . . . . . . . . . . . . . . . .

8.2 Polar Coordinates . . . . . . . . . . . . . . . . . . . .

8.3 Polar Laplacian in R2 for Radial Functions . . . . . .

8.4 Spherical Laplacian in R3 and Rn for Radial Functions

8.5 Cylindrical Laplacian in R3 for Radial Functions . . .

8.6 Mean Value Theorem . . . . . . . . . . . . . . . . . . .

8.7 Maximum Principle . . . . . . . . . . . . . . . . . . .

8.8 The Fundamental Solution . . . . . . . . . . . . . . . .

8.9 Representation Theorem . . . . . . . . . . . . . . . . .

8.10 Green¡¯s Function and the Poisson Kernel . . . . . . . .

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23

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30

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31

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42

Partial Di?erential Equations

Igor Yanovsky, 2005

4

8.11 Properties of Harmonic Functions . . . . . . . . . . . . . . . . . . . . . .

8.12 Eigenvalues of the Laplacian . . . . . . . . . . . . . . . . . . . . . . . . .

44

44

9 Heat Equation

45

9.1 The Pure Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . . 45

9.1.1 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . 45

9.1.2 Multi-Index Notation . . . . . . . . . . . . . . . . . . . . . . . . 45

9.1.3 Solution of the Pure Initial Value Problem . . . . . . . . . . . . . 49

9.1.4 Nonhomogeneous Equation . . . . . . . . . . . . . . . . . . . . . 50

9.1.5 Nonhomogeneous Equation with Nonhomogeneous Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

9.1.6 The Fundamental Solution . . . . . . . . . . . . . . . . . . . . . 50

10 Schro?dinger Equation

52

11 Problems: Quasilinear Equations

54

12 Problems: Shocks

75

13 Problems: General Nonlinear Equations

86

13.1 Two Spatial Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

13.2 Three Spatial Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 93

14 Problems: First-Order Systems

102

15 Problems: Gas Dynamics Systems

15.1 Perturbation . . . . . . . . . . . .

15.2 Stationary Solutions . . . . . . . .

15.3 Periodic Solutions . . . . . . . . .

15.4 Energy Estimates . . . . . . . . . .

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127

127

128

130

136

16 Problems: Wave Equation

16.1 The Initial Value Problem . . . .

16.2 Initial/Boundary Value Problem

16.3 Similarity Solutions . . . . . . . .

16.4 Traveling Wave Solutions . . . .

16.5 Dispersion . . . . . . . . . . . . .

16.6 Energy Methods . . . . . . . . .

16.7 Wave Equation in 2D and 3D . .

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139

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187

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196

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216

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232

242

249

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17 Problems: Laplace Equation

17.1 Green¡¯s Function and the Poisson Kernel . . .

17.2 The Fundamental Solution . . . . . . . . . . .

17.3 Radial Variables . . . . . . . . . . . . . . . .

17.4 Weak Solutions . . . . . . . . . . . . . . . . .

17.5 Uniqueness . . . . . . . . . . . . . . . . . . .

17.6 Self-Adjoint Operators . . . . . . . . . . . . .

17.7 Spherical Means . . . . . . . . . . . . . . . .

17.8 Harmonic Extensions, Subharmonic Functions

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Partial Di?erential Equations

Igor Yanovsky, 2005

5

18 Problems: Heat Equation

255

18.1 Heat Equation with Lower Order Terms . . . . . . . . . . . . . . . . . . 263

18.1.1 Heat Equation Energy Estimates . . . . . . . . . . . . . . . . . . 264

19 Contraction Mapping and Uniqueness - Wave

271

20 Contraction Mapping and Uniqueness - Heat

273

21 Problems: Maximum Principle - Laplace and Heat

279

21.1 Heat Equation - Maximum Principle and Uniqueness . . . . . . . . . . . 279

21.2 Laplace Equation - Maximum Principle . . . . . . . . . . . . . . . . . . 281

22 Problems: Separation of Variables - Laplace Equation

282

23 Problems: Separation of Variables - Poisson Equation

302

24 Problems: Separation of Variables - Wave Equation

305

25 Problems: Separation of Variables - Heat Equation

309

26 Problems: Eigenvalues of the Laplacian - Laplace

323

27 Problems: Eigenvalues of the Laplacian - Poisson

333

28 Problems: Eigenvalues of the Laplacian - Wave

338

29 Problems: Eigenvalues of the Laplacian - Heat

346

29.1 Heat Equation with Periodic Boundary Conditions in 2D

(with extra terms) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360

30 Problems: Fourier Transform

365

31 Laplace Transform

385

32 Linear Functional Analysis

32.1 Norms . . . . . . . . . . . .

32.2 Banach and Hilbert Spaces

32.3 Cauchy-Schwarz Inequality

32.4 Ho?lder Inequality . . . . . .

32.5 Minkowski Inequality . . . .

32.6 Sobolev Spaces . . . . . . .

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