## APM346 Partial Differential Equations

### Fall of 2015; Section L5101 – Lecture notes and Home assignments

Here current lecture notes will appear weekly. They will be based on Online textbook. So notes will be just references to sections of this textbook plus brief comments. Currently there are only placeholders. Usually each weekend (Sat or Sun) three new notes and a home assignment will be released covering the coming week and over next weekend (Fri or Sat) these notes will be finalized.

#### Week 1 (September 14—18)

1. Introduction: PDE Motivations and Context (Chapter 1)
2. Introduction: PDE Motivations and Context (continued; Chapter 1)
3. First order PDEs (Section 2.1)
4. #### Week 2 (September 21—25)

5. First order PDEs (Section 2.1 (end))
6. Homogeneous 1D Wave equation (Section 2.3)
7. 1D Wave equation reloaded: characteristic coordinates and more
(Section 2.4 and Section 2.5)
8. #### Week 3 (Sept. 28—Oct. 2)

9. 1D Wave equation: IBVP (Section 2.6)
10. 1D Wave equation: survey
11. Wave equation: Energy integral (Section 2.7)
12. #### Week 4 (October 5—9)

13. 1D Heat equation: method of self-similar solutions (Section 3.1)
14. Heat equation (Misc.) (Section 3.2)
15. Heat equation (Misc.) -- end
16. #### Week 5 (October 12—16)

17. 1D wave equation Separation of variables (the first blood) (Section 4.1)
18. Eigenvalue problems (examples) (Section 4.2)
19. Ortogonal systems (Section 4.3)
20. #### Week 6 (October 19—23)

21. Ortogonal systems (Section 4.3 (end))
22. Ortogonal systems and Fourier series
(Section 4.4)
23. Other Fourier series
(Section 4.5)

#### Week 7 (October 26—30)

1. Fourier transform, Fourier integral
(Section 5.1)
2. Properties of Fourier transform
(Section 5.2)
3. Applications of Fourier transform to PDEs
(Section 5.3)
4. #### Week 8 (November 2—6)

5. Separation of variables for heat equation
(Section 6.1)
Separation of variables: Misc equations
(Section 6.2)
6. Laplacian in polar and spherical coordinates
(Section 6.3)
Laplacian: separation of variables in polar coordinates (Section 6.4)
7. Laplace operator in the disk. II
(Section 6.5)
8. #### Week 9 (November 9—13)

9. General properties of Laplace equation
(Section 7.1)
10. Potential theory and around
(Section 7.2)
Green function (Section 7.3)
11. Separation of variable in spherical coordinates
(Section 8.1)
The rest of Chapter 8 skipped.
12. #### Week 10 (November 16—20)

13. Wave equation in 3D and 2D
(Section 9.1)
14. Wave equation in 3D and 2D (end)
15. Wave equation: energy method
(Section 9.2)
• Home Assignment 10
Problems to Chapter 10 (all)
• Term Test 2. Covers Chapters 4--7.
Solving HA 6—9 is a good preparation
16. #### Week 11 (November 23—27)

17. Functionals, extremums and variations
(Section 10.1)
18. Functionals, extremums and variations .II
(Section 10.2)
19. Variational methods in physics (Section 10.3)
20. #### Week 12 (Nov. 30—Dec. 04)

21. Distributions (Section 11.1)
22. Distributions: more (Section 11.2)
23. Applications of distributions (Section 11.3)
Weak solutions (Section 11.4)
• Quiz 7
 Review lecture (TBA) Special office hours (TBA) Final exam Solving HA 1—10 is a good preparation