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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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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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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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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