# Partial Differential Equations and Applications

An engineer thinks that his equations are an approximation to reality.
A physicist thinks reality is an approximation to his equations.
A mathematician doesn't care.

 Instructor: Andrej V. Cherkaev, professor Department of Mathematics University of Utah Salt Lake City, UT 84112 Email: cherk@math.utah.edu Tel : +1 801 - 581 6822 Fax : +1 801 - 581 4148

Text:
Analytical and Computational methods of Advanced Engineering Mathematics,  Grant B. Gustafson and Calvin H. Wilcox Springer Verlag, 1998.
Also recommended: Advanced engineering mathematics, Erwin Kreyszig, 5th ed., New York : Wiley, c1983.

Description:
Partial Differential Equations (PDE) describe processes in continua, such as waves, diffusion, equilibria, etc. The course gives the introduction to the PDE and discusses simplest means to solve them. Elementary theory of Fourier series and Fourier transforms is discussed as well.

Prerequisites:
Mathematics 2210 (or 1260) and 2250.
High school trigonometry!

Tentative scheme of the course

 Chapter 7 (7.1 - 7.4) Derivation of differential equations. Uniqueness. Types of PDE: Hyperbolic (waves), Parabolic (diffusion), Elliptic (equilibria) 2 ++ lectures Chapter 8 (except 8.6) Fourier series and orthogonality, Fourier integrals, Sturm-Liuville theory: eigenvalues and eigenfunctions 11 lectures Chapter 9 (9.1 - 9.5) Methods of solution of PDE: Separation of variables and Laplace transform. Examples. 11 lectures

Internet sources
There are many interesting sites on the Internet, related to the course. Alta Vista returns 40,000 pages related to Partial Differential Equations, many of them tutorial. Please let me know if you have found interesting sites on the Internet. Some sites are listed below:

1. PDE Tutorial,
2. Fourier Series Tutorial ,
3. M- 353 Maple Tutorial (look for Gustafson's 353 Labs /w Maple Notes),
4. Bibliography

Pictures for the course
Heat transfer problem , Vibration of a string

### Course work

Computer projects
Two projects are assigned that use Maple, Mathematica, or Matlab. Because the Engineering College uses primary Maple in its instruction, and because of support available in the math. department, I advice using Maple unless your preference is strong for another language. There are those who swear by Matlab.

Tutoring
The Math. depaertment offers free tutoring, computer lab access, etc. in the Undergraduate Math Center Click here for details Homework assignments

 Set Based on: Problems Due day 1. 7.2, 8.1, 8.2 7.2: ## 1, 2, 3, 4 8.1: ## 1, 4, 8, 9, 10, 13, 14, 16 8.2: ## 2, 3, 4 January 18 2. 8.3, 8.4, 8.5 8.3: ## 5, 7, 9, 12 (use Maple, make 3d graphs of solutions) 8.4: ## 2, 5, 9, 11 8.5: ## 1, 2, 6 February 2 3. 8.8, 8.10 8.9: ## 1, 5, 6, 9 8.10: ## 1, 3 February 9 4. 8.11- 8.14 8.12, ## 2, 4, 7 8.13: # 5 8.14: # 1 March 5 5. 7.1, 9.2 7.1, # 5 9.2: ##1, 3, 6,13, 15 March 5. 7.4, 9.3 7.4, # 4 9.3 ##1, 6, 11 March 6. 9.3, 9.4, 9.6, 9.7 9.3, # 26 9.4, # 1 9.6, # 1 9.7 ## 3, 5 Last week of classes

Extra Credit Problems.

You may earn up to 10 points to the total score by solving the following three problems

Solve the problem, plot the graph of the solution using Maple

1. Find the steady-state distribution of the temperature T(x,y) inside the rectangular domain
0 < x < 1, 0 < y < 4,
if the boundary conditions are
u(x,0)= x, u(x, 4)= x^2, u(0, y)= 0, u(1, y)=1.
2. Find the steady-state distribution of the electrical current j(r, t ) in an infinite conducting plane outside the circular hole ( r > 1). The hole is isolated: j.n =0 on the boundary of the hole. The current at infinity is uniform and is directed along the OX axis.
3. Find the distribution of the temperature  T(x,y, t) inside the rectangular domain
0 < x < 1, 0 < y < 1,
if the boundary conditions are
u(x, 0, t)= x, u(x, 1, t)= 0, u(0, y, t)= 0, u(1, y, t)=1-y
and the initial conditions are: u(x, y, 0)=0.
The constant of conductivity is equal o one.
Plot the graph of u(x, y, 0.05),  u(x, y, 0.5),  u(x, y, 5).

You also may want to look on the last year assignments prepared by Peter Brinkman.

### Policies

The American with Disabilities Act requires that reasonable accommodations be provided for students with physical, sensory, cognitive, systemic, learning, and psychiatric disabilities. Please contact me at the beginning of the semester to discuss any such accommodations for the course.