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. Mathematical folklore
About the course
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:
-
PDE Tutorial,
-
Fourier Series Tutorial ,
-
M- 353 Maple Tutorial
(look for Gustafson's 353 Labs /w Maple Notes),
- 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
-
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.
-
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.
-
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
ADA statement:
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.
Grading:
The final grade will be determined by two one-hour midterm exams (20% each), one two-hours final exam (40%),
and homework (20%). Homework assignments will consist of traditional homework
problems as well as computer assignments.
Homework assignments and due dates will
be posted on the course homepage.
There will be no makeup exams, and
late homework will not be accepted.
Student's teams:
You may do your homework in groups consisting of no more than four students,
and submit one set of solutions per group. You should identify your group at
the second week of the semester and, as a rule, stick to it for the rest of the course.
Ideally, all members of a group should do the homework on their own, and
then the group should meet in order to discuss the solutions as well as
the topics covered in class. You will see that explaining your solutions
to the other members of your group is more challenging than just solving
a problems. This activity also prepares students to clearly formulate mathematical ideas and convincingly communicate them.
Acknowledgement:
While preparing this page, the instructor used the webpage of Peter Brinkman, advice of David Eyre, and various internet sources.
To the Homepage
,
To the Teaching Page
