TOC
Table of Contents
0. Preface
0.1. What one needs to know?
Chapter 1. Introduction
1.1. PDE Motivations and Context
1.2. Initial and Boundary Value Problems
1.3. Classification of equations
1.4. Origin of some equations
1.P. Problems to Chapter 1
Chapter 2. $1$-dimensional waves
2.1. First order PDEs
2.2. First order PDEs (continued)
2.3. Homogeneous $1$D Wave equation
2.4. 1D Wave equation reloaded: characteristic coordinates
2.5. Wave equation reloaded (continued)
2.6. 1D Wave equation: IBVP
2.7. Energy integral
2.8. Hyperbolic first order systems with one spatial variable
Chapter 3. Heat equation in 1D
3.1. 1D Heat equation
3.2. Heat equation (Miscellaneous)
Appendix 3.A. Intro into project: Random Walks
Chapter 4. Separation of variables and Fourier Series
4.1. Separation of variables (the first blood)
4.2. Eigenvalue problem
4.3. Orthogonal systems
4.4. Ortogonal systems and Fourier series
4.5. Other Fourier series
Appendix 4.A. Calculation of negative eigenvalues in Robin problem
Appendix 4.B. Multidimensional Fourier series
Appendix 4.C. Harmonic Oscillator
Chapter 5. Fourier transform
5.1. Fourier transform, Fourier integral
Appendix 5.1.A. Justification
Appendix 5.1.B. Discussion: pointwise convergence of Fourier integrals and series
5.2. Properties of Fourier transform
5.3. Applications of Fourier transform to PDEs
Chapter 6. Separation of variables
6.1. Separation of variables for heat equation
6.2. Separation of variables: Misc equations
6.3. Laplace operator in different coordinates
6.4. Laplace operator in the disk: separation of variables
6.5. Laplace operator in the disk. II
Appendix 6.A. Linear second order ODEs
Chapter 7. Laplace equation
7.1. General properties of Laplace equation
7.2. Potential theory and around
7.3. Green function
Chapter 8. Separation of variables
8.1. Separation of variable in spherical coordinates
8.2. Separation of variable in polar and cylindrical coordinates
8.3. Helmholtz equation in the cylinder
8.4. Laplace equation in the cylinder
Appendix 8.A. Separation of variable in elliptic and parabolic coordinates
Chapter 9. Wave equation
9.1. Wave equation: energy method
9.2. Wave equation: energy method
Chapter 10. Variational methods
10.1. Functionals, extremums and variations
10. Functionals, extremums and variations. More
10.3. Variational methods in physics
10.P. Problems
Chapter 11. Distributions and weak solutions
11.1. Distributions
11.2. Distributions: more
11.3. Applications of distributions
11.4. Weak solutions
Chapter 12. Non-linear equations
12.1 Burgers equation
Chapter 13. Eigenvalues and eigenfunctions
13.1. Variational theory
13.2. Asymptotic distribution of eigenvalues
13.3. Properties of eigenfunctions
13.4. Definitions and classification
13.5. Continuous spectrum and scattering
Chapter 14. Miscellaneous
14.1. Conservation laws
14.2. Maxwell equations
14.3. Some quantum mechanical operators
General appendices
A.1. Field theory
A.2. Landau's notations ($O$, $o$, $\asymp$, $\sim$)