## APM346 Partial Differential Equations

### Fall of 2016; 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 12—16)

1. Introduction: PDE Motivations and Context (Section 1.1)
2. Introduction: Initial and Boundary Value Problems (Section 1.2)
3. First order PDEs (Section 2.1)
4. #### Week 2 (September 19—23)

5. First order PDEs (Section 2.1 (end))
6. Homogeneous 1D Wave equation (Section 2.3)
7. Homogeneous 1D Wave equation--end (Section 2.3)
8. #### Week 3 (Sept. 26—30)

9. 1D Wave equation reloaded: characteristic coordinates and more
(Section 2.4 and Section 2.5)
10. 1D Wave equation: IBVP (Section 2.6)
11. Wave equation: Energy integral (Section 2.7)
12. #### Week 4 (October 3—7)

13. 1D Heat equation: method of self-similar solutions (Section 3.1)
14. Heat equation (Misc.) (Section 3.2)
15. Heat equation (Misc.) -- end
• Tutorial 4
• Home Assignment 4
Problems to Section 2.6 (1--5)
• Quiz 2 (Thu, Oct. 6, Lecture): covers Home Assignment 3
16. #### Week 5 (October 10—14)

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 17—21)

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 24—28)

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 (Oct. 31—Nov. 4)

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 7—11)

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)
12. Notes below are subject to modifications

#### Week 10 (November 14—18)

13. Separation of variable in spherical coordinates (Section 8.1 (end))
14. Wave equation in 3D and 2D
(Section 9.1)
15. Wave equation in 3D and 2D (end);
Wave equation: energy method
(Section 9.2)
16. #### Week 11 (November 21—26)

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)
• Tutorial 12
• Bonus Quiz: covers Home Assignment 11
 Review lecture (TBA) Special office hours (TBA) Final exam Solving HA 1—10 is a good preparation