Ma 650. Fall 2008.
 
Instructor: Pavel Dubovski
Classes: Mondays 6:15-8:45PM B 124
 
Office hours: MW 2-3 K226 or by appointment.
 

Textbooks:

[1] Tyn Myint-U, L.Debnath “Linear Partial Differential Equations for Scientists and Engineers”, 4^th edition. Birkhauser, 2007. ISBN 0817643931.

[2] P.V.O’Neil “Beginning Partial Differential Equation”, 2d edition. Wiley, 2008.

ISBN 9780470133903.

Syllabus:

1. Review of Ordinary Differential Equations:

- separable equations

- linear equations with constant coefficients

- Green function for boundary value problems of 2d order (section 8.11)

- eigenvalue problems (section 8.1)

2. Well posedness, linearity and superposition:

Read: sections 1.2-1.5

Problems # 1, 2, 5, 6, 7, 11, 23 (section 1.6)

3. 1^st order PDEs: characteristics method

Read sections 2.1, 2.2, 2.4, 2.5.

Solve # 3, 4, 5, 9, 12, 14, 17, 20, 23, 24 (section 2.8).

4. 1^st order PDEs, other methods.

Solve # 25, 26, 27, 28, 29, 31 (section 2.8)

5. Mathematical models: vibrating string, heat conduction, diffusion.

Read sections 3.1, 3.2, 3.5, 3.6

Solve # 1, 2, 5, 9, 14 (section 3.9)

6. Wave equation for infinite string.

Read sections 5.3, 5.4

Solve # 1, 2, 7, 8, 9, 10, 35, 36.

7. Fourier series and separation of variables. Eigenvalue problems

Read sections 6.3, 6.4, 6.5, 6.6, 6.7, 7.2, 7.3, 7.5, 7.7, 7.8

Solve # 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 17, 18, 21-35 (section 7.9)

8. Special functions: Bessel functions, Legendre functions.

Read sections 8.6, 8.8, 8.9.

Solve # 1, 6, 7, 11, 13, 14

9. Elliptic problems. Harmonic functions.

Read sections 9.2, 9.4, 9.5, 9.6, 9.7, 9.9.

Solve # 3, 7, 8, 9, 11, 12, 13, 15, 22, 23, 24, 33, 35 (section 9.10)

10. 2 and 3-dimensional boundary value problems

Read sections 10.2, 10.3, 10.4, 10.5, 10.6, 10.7, 10.8, 10.9, 10.12.

Solve # 1, 2, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 19, 25, 26, 27, 28, 29 (section 10.13).

11. Method of Green functions. Dirac delta-function.

Read 11.2, 11.3, 11.4, 11.5, 11.6, 11.7, 11.8.

Solve # 3, 7, 8, 9, 10, 11, 12, 13, (section 11.11)

12. Fourier transform and applications to heat distribution

Read sections 12.2, 12.3, 12.4, 12.5

Solve # 11, 12, 16, 17, 18 (section 12.18).

13. Nonlinear 1^st order equations. Burgers equation, dispersive waves.

Read sections 13.2, 13.3.

14. Other mathematical models.

Traffic flow, Black-Scholes model in finance, flood waves, Burgetrs equation and shock waves, Korteweg-de Vries equation, Solitons, Nonlinear Schrodinger equation, Solitary waves.

Read sections 13.6, 13.7, 13.9, 13.10, 13.11, 13.12

Solve # 1, 2, 3, 4, 5, 10, 11, 12, 14 (section 13.14).