##[Table of Contents](#toc) * [0. Preface](./Chapter0/S0.html) * [0.1. What one needs to know?](./Chapter0/S0.1.html) ###[Chapter 1. Introduction](#chapter-1) * [1.1. PDE Motivations and Context](./Chapter1/S1.1.html) * [1.2. Initial and Boundary Value Problems](./Chapter1/S1.2.html) * [1.3. Classification of equations](./Chapter1/S1.3.html) * [1.4. Origin of some equations](./Chapter1/S1.4.html) * [1.P. Problems to Chapter 1](./Chapter1/S1.P.html) ###[Chapter 2. $1$-dimensional waves](#chapter-2) * [2.1. First order PDEs](./Chapter2/S2.1.html) * [2.2. First order PDEs (continued)](./Chapter2/S2.2.html) * [2.3. Homogeneous $1$D Wave equation](./Chapter2/S2.3.html) * [2.4. 1D Wave equation reloaded: characteristic coordinates](./Chapter2/S2.4.html) * [2.5. Wave equation reloaded (continued)](./Chapter2/S2.5.html) * [2.6. 1D Wave equation: IBVP](#section-2.6.html) * [2.7. Energy integral](./Chapter2/S2.7.html) * [2.8. Hyperbolic first order systems with one spatial variable](./Chapter2/S2.8.html) ###[Chapter 3. Heat equation in 1D](#chapter-3) * [3.1. 1D Heat equation](./Chapter3/S3.1.html) * [3.2. Heat equation (Miscellaneous)](./Chapter3/S3.2.html) * [Appendix 3.A. Intro into project: Random Walks](./Chapter3/S3.A.html) ###[Chapter 4. Separation of variables and Fourier Series](#chapter-4) * [4.1. Separation of variables (the first blood)](./Chapter4/S4.1.html) * [4.2. Eigenvalue problem](./Chapter4/S4.2.html) * [4.3. Orthogonal systems](./Chapter4/S4.3.html) * [4.4. Ortogonal systems and Fourier series](./Chapter4/S4.4.html) * [4.5. Other Fourier series](./Chapter4/S4.5.html) * [Appendix 4.A. Calculation of negative eigenvalues in Robin problem](./Chapter4/S4.A.html) * [Appendix 4.B. Multidimensional Fourier series](./Chapter4/S4.B.html) * [Appendix 4.C. Harmonic Oscillator](./Chapter4/S4.C.html) ###[Chapter 5. Fourier transform](#chapter-5) * [5.1. Fourier transform, Fourier integral](./Chapter5/S5.1.html) * [Appendix 5.1.A. Justification](./Chapter5/S5.1.A.html) * [Appendix 5.1.B. Discussion: pointwise convergence of Fourier integrals and series](./Chapter5/S5.1.A.html) * [5.2. Properties of Fourier transform](./Chapter5/S5.2.html) * [5.3. Applications of Fourier transform to PDEs](./Chapter5/S5.3.html) ###[Chapter 6. Separation of variables](#chapter-6) * [6.1. Separation of variables for heat equation](./Chapter6/S6.1.html) * [6.2. Separation of variables: Misc equations](./Chapter6/S6.2.html) * [6.3. Laplace operator in different coordinates](./Chapter6/S6.3.html) * [6.4. Laplace operator in the disk: separation of variables](./Chapter6/S6.4.html) * [6.5. Laplace operator in the disk. II](./Chapter6/S6.5.1.html) * [Appendix 6.A. Linear second order ODEs](./Chapter6/S6.A.html) ###[Chapter 7. Laplace equation](#chapter-7) * [7.1. General properties of Laplace equation](./Chapter7/S7.1.html) * [7.2. Potential theory and around](./Chapter7/S7.2.html) * [7.3. Green function](./Chapter7/S7.3.html) ###[Chapter 8. Separation of variables](#chapter-8) * [8.1. Separation of variable in spherical coordinates](./Chapter8/S8.1.html) * [8.2. Separation of variable in polar and cylindrical coordinates](./Chapter8/S8.2.html) * [8.3. Helmholtz equation in the cylinder](./Chapter8/S8.3.html) * [8.4. Laplace equation in the cylinder](./Chapter8/S8.2.3.html) * [Appendix 8.A. Separation of variable in elliptic and parabolic coordinates](./Chapter8/S8.A.html) ###[Chapter 9. Wave equation](#chapter-9) * [9.1. Wave equation: energy method](./Chapter9/S9.1,html) * [9.2. Wave equation: energy method](./Chapter9/S9.2.html) ###[Chapter 10. Variational methods](#chapter-10) * [10.1. Functionals, extremums and variations](./Chapter10/S10.1.html) * [10. Functionals, extremums and variations. More](./Chapter10/S10.2.html) * [10.3. Variational methods in physics](./Chapter10/S10.3.html) * [10.P. Problems](./Chapter10/S10.P.html) ###[Chapter 11. Distributions and weak solutions](#chapter-11) * [11.1. Distributions](./Chapter11/S11.1.html) * [11.2. Distributions: more](./Chapter11/S11.2.html) * [11.3. Applications of distributions](./Chapter11/S11.3.html) * [11.4. Weak solutions](./Chapter11/S11.4.html) ###[Chapter 12. Non-linear equations](#chapter-12) * [12.1 Burgers equation](./Chapter12/S12.1.html) ###[Chapter 13. Eigenvalues and eigenfunctions](#chapter-13) * [13.1. Variational theory](./Chapter13/S13.1.html) * [13.2. Asymptotic distribution of eigenvalues](./Chapter13/S13.2.html) * [13.3. Properties of eigenfunctions](./Chapter13/S13.3.html) * [13.4. Definitions and classification](./Chapter13/S13.4.html) * [13.5. Continuous spectrum and scattering](./Chapter13/S13.5.html) ###[Chapter 14. Miscellaneous](#chapter-14) * [14.1. Conservation laws](./Chapter14/S14.1.html) * [14.2. Maxwell equations](./Chapter14/S14.2.html) * [14.3. Some quantum mechanical operators](./Chapter14/S14.3.html) ###[General appendices](#chapter-A) * [A.1. Field theory](./ChapterA/SA.1.html) * [A.2. Landau's notations ($O$, $o$, $\asymp$, $\sim$)](./ChapterA/SA.2.html)