Partial Differential Equations: Graduate Level Problems and ...

Partial Differential Equations: Graduate Level Problems and Solutions

Igor Yanovsky

1

Partial Differential 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 Differential 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

10

5.1 Quasilinear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

5.2 Weak Solutions for Quasilinear Equations . . . . . . . . . . . . . . . . . 12

5.2.1 Conservation Laws and Jump Conditions . . . . . . . . . . . . . 12

5.2.2 Fans and Rarefaction Waves . . . . . . . . . . . . . . . . . . . . . 12

5.3 General Nonlinear Equations . . . . . . . . . . . . . . . . . . . . . . . . 13

5.3.1 Two Spatial Dimensions . . . . . . . . . . . . . . . . . . . . . . . 13

5.3.2 Three Spatial Dimensions . . . . . . . . . . . . . . . . . . . . . . 13

6 Second-Order Equations

14

6.1 Classification by Characteristics . . . . . . . . . . . . . . . . . . . . . . . 14

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

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

7 Wave Equation

23

7.1 The Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 23

7.2 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

7.3 Initial/Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . 24

7.4 Duhamel's Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

7.5 The Nonhomogeneous Equation . . . . . . . . . . . . . . . . . . . . . . . 24

7.6 Higher Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7.6.1 Spherical Means . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7.6.2 Application to the Cauchy Problem . . . . . . . . . . . . . . . . 26

7.6.3 Three-Dimensional Wave Equation . . . . . . . . . . . . . . . . . 27

7.6.4 Two-Dimensional Wave Equation . . . . . . . . . . . . . . . . . . 28

7.6.5 Huygen's Principle . . . . . . . . . . . . . . . . . . . . . . . . . . 28

7.7 Energy Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

7.8 Contraction Mapping Principle . . . . . . . . . . . . . . . . . . . . . . . 30

8 Laplace Equation

31

8.1 Green's Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

8.2 Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

8.3 Polar Laplacian in R2 for Radial Functions . . . . . . . . . . . . . . . . 32 8.4 Spherical Laplacian in R3 and Rn for Radial Functions . . . . . . . . . . 32 8.5 Cylindrical Laplacian in R3 for Radial Functions . . . . . . . . . . . . . 33

8.6 Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

8.7 Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

8.8 The Fundamental Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 34

8.9 Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

8.10 Green's Function and the Poisson Kernel . . . . . . . . . . . . . . . . . . 42

Partial Differential Equations

Igor Yanovsky, 2005

4

8.11 Properties of Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . 44 8.12 Eigenvalues of the Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . 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 Condi-

tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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

127

15.1 Perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

15.2 Stationary Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

15.3 Periodic Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

15.4 Energy Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

16 Problems: Wave Equation

139

16.1 The Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 139

16.2 Initial/Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . 141

16.3 Similarity Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

16.4 Traveling Wave Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 156

16.5 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

16.6 Energy Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

16.7 Wave Equation in 2D and 3D . . . . . . . . . . . . . . . . . . . . . . . . 187

17 Problems: Laplace Equation

196

17.1 Green's Function and the Poisson Kernel . . . . . . . . . . . . . . . . . . 196

17.2 The Fundamental Solution . . . . . . . . . . . . . . . . . . . . . . . . . . 205

17.3 Radial Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

17.4 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

17.5 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223

17.6 Self-Adjoint Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

17.7 Spherical Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

17.8 Harmonic Extensions, Subharmonic Functions . . . . . . . . . . . . . . . 249

Partial Differential 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

393

32.1 Norms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

32.2 Banach and Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . 393

32.3 Cauchy-Schwarz Inequality . . . . . . . . . . . . . . . . . . . . . . . . . 393

32.4 H?older Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

32.5 Minkowski Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

32.6 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download