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## __ MATH 443 - PARTIAL DIFFERENTIAL EQUATIONS __

## (1999-2000, Spring 2002, 2004-2005) Fall 2009-2010

**Math 443 : **__ Partial Differential Equations__. A Third year course.

** Text book **:
** Ian Sneddon **, Elements of Partial Differential Equations,
McGraw-Hill International Editions (Mathematics Series), 1985

** Other Books: **
**R. Dennemeyer **, Introduction to Partial Differential Equations
and Boundary Value Problems., McGraw-Hill, New York , 1986.

** Content **:
First Order equations, Cauchy-Kovalevsky theorem. Method of characteristics.,
Lagrange-Charpit method, Pfaff systems. Second order equations, classification,
Stokes' and Green's theorems, wave equation, Euler-Darboux equations,
Riemann Green function, heat equation, Green's function, Laplace and
Poisson's equations, Riesz' method.
Please see also the "Course Syllabus" below
## __ Course Schedule __

Tuesday 08.40-10.30 , SAZ-19
Thursday 10.40-12.30, SAZ-19

** Exams **

see previous exams

** (20%)** First Midterm Exam pdf file
: ** October 20 (10.40-12.30 SAZ 19)**
** (20%)**
Second Midterm Exam pdf file
: ** November 24 (10.40-12.30 SAZ 19)**
** (40%) ** Final Exam pdf file
: ** December 29, 2005 , 09.00-12.00 (SAZ 04) **

** (20%) ** Attendance, homeworks .

Solutions of the first midterm exam first
midterm ps file

Solutions of the second midterm exam second
midterm pdf file

Solutions of the final exam final exam pdf file

## Subjects Covered So far

**(new) 1. September 12 **

Curves and Surfaces in space (R^3)

Homework Set I

** (new) 2. September 19 **

Methods of solutions of dx/P=dy/Q=dz/R
Orthogonal Families Curves
Pfaffian Differential forms and Pfaffian Differential Equations

**(new) 3. September 26 **

Solutions of Pfaffian Differential Equations in Three Variables
First Order Partial Differential Equations

**(new) 4. Ocober 03 **

First Order Partial Differential Equations
Integral Surfaces passing through a given curve
Nonlinear Partial Differential Equations of first order

Homework Set II

**(new) 5. October 10 **

Nonlinear PDEs of First Order
Partial Differential Equations of Second Order

**(new) 6. October 17 **

Partial Differential Equations with Constant Coefficients
Partial Differential Equations with Variable Coefficients

Homework Set III

First Midterm Exam

**(new) 7. October 24 **

Partial Differential Equations with Variable Coefficients
Characteristic Curves of Second Order Equations

**(new) 8. November 7 **

The Solution of Linear Hyperbolic Partial Differential Equations
(Riemann's Method)
Characteristics of Equations in Three or more Variables
Separation of Variables
The Method of Integral Transform

Homework Set IV

**(new) 9. November 14 **

The Laplace Equation
Laplace equation in spherical, cylindrical and Cartesian coordinates
Some simple solutions
Equpotential surfaces

**(new) 10. November 21 **

The Laplace Equation
Seperation of variables
Boundary value problems
Kelvin's Theorem

Second Midterm Exam

** (new) 11. November 29 **
The Laplace Equation
The method of Green's function (in two and three dimensions)
Some Dirichlet and Neuman problems
The wave equation

Homework Set V

** (new) 12. December 5 **

The wave equation
D'Alembert's Solution
String with one fixed end
String with fixed ends
Initial Condtions on a curve (Riemann's Method)

** (new) 13. December 12 **

Membranes
Infinite and finite membranes
Uniqueness problems

** 14. December 19 **

Diffusion Equation
The fundemantal solution of the heat equation
Solution of the heat equation with several initial
and boundary problems,

** 15. December 26 **

Maximum and minumum principles of the heat equation
Uniqueness of the solutions of the heat equation
Similarity solutions of the PDEs F(x,y,u, ux,uy,...)=0

** end of the semester **

## __ Course Syllabus __

1.** Feb.07** __ Chapter 1. Section 1.1. Surfaces in three
dimensions __** Solve exercises at the end of the section on
page 7.**

2. ** Feb.14 **Pfaffian systems and their solutions __ Chapter 1.
Sections 2,3,4,5 completed. __** Solve all exercises.**
Sections 7 and 8 will not be done.
**Solve the Miscellaneous Problems at the end of the Chapter.**
.

3.** Feb.21 ** First order partial differential equations (2.1 - 2.6)
** Sections 2.1-2.6 completed ** . You are responsable from all
exercies at the end of each Section.

4.** Mar.14 ** Nonlinear partial differential equations of
first order
** (2.7-2.11) ** . Solve the Miscellaneous Problems (M.Ex.1)

5. ** Mar. 21 **Second order partial differential equations
** (3.1-3.5)**.

6. ** Mar. 28 ** Characteristic curves and characteristic equations :
Solution of the Cauchy problem: Existence and uniquenss of solutions
when the data given on (a) Non characteristic and (b) characteristic curves.
** All sections except section 11 **.

7.** April 8 ** __ L aplace's equation __ (4.1-4.5).

8.** April 15 ** __ Green's function for Laplace's equation __
** All sections of Ch 4. except Section 9.** (M. Ex.2)

9. ** April 28 **The wave equation ** (5.1-5.5)**.
__ Solve all exercises
__

10. ** May 7.** General solutions and Green's function for the wave equation
** (5.5-5.7)**. __ solve all exercises __

11. ** May 13 ** The diffusion (Heat) equation **(6.1-6.4)**.

12. ** May. 21 ** The use of integral transform and Green's function for the heat equations ** (6.5--6.6)**. __ Solve all exercises at the end of
each Sections __

13. May.01 **Symmetries of partial differential equations**.

14. May.08 **Group invariant solutions of differential equations**.

15. May.15 **Review **

For exercises please see pdeexercises

For midterms please see pdemidterms

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### Last update January 2002

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