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## __ MATH 443 - PARTIAL DIFFERENTIAL EQUATIONS __

## (1999-2000, Spring 2002, Fall of 2004-2005,2009-2011 )

# FALL 2017

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

** Text books **:

1. ** Ian Sneddon **, Elements of Partial Differential Equations,
McGraw-Hill International Editions (Mathematics Series), 1985

and

2. ** Richard Haberman **:Applied Partial Differential Equations: with Fourier
Series and Boundary Value Problems (Fourth Edition), Pearson Education (2004)

** Other Books: **

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

** Content of the Course **:
First Order equations. Method of characteristics.,
Lagrange-Charpit method, Pfaff systems. Linear Partial Differential Equations
with Constant Coefficients.Second order equations, classification,
The Method of Seperation of Variable, heat equation,the wave equation,
Green's function, Laplace and Poisson's equations.

modelling ,
Fourier Series

Please see also the "Course Syllabus" below
## __ Course Schedule __

Monday 08.40-10.30 , SA-Z19
Wednesday 10.40-12.30, SA-Z19

** Exams **

see previous exams

** (30%)** First Midterm Exam pdf file
(2005) ,
(2009) ,(2010),
(2011)
: ** November 1**
** (30%)**
Second Midterm Exam pdf file (2005) , (2009) ,
(2010),
(2011), (2017)
: ** December 6 **
** (40%) ** Final Exam pdf file
(2005) , (2009),
(2010),
(2011), (2017), Solution(2017)
: ** January 02, 2018, **

** MAKEUP EXAM: January **

Makeup 1
(2009),(2010)

Makeup 2 (2009)

Makeup 2 solutions (2009)

Solutions of the first midterm exam first
midterm ps file (2005) , (2009) ,
(2010), (2011)

Solutions of the second midterm exam second
midterm pdf file (2005) ,
(2009) , (2010),
(2011), (2017)

Solutions of the final exam final exam pdf
file (2005) , (2009), (2010),
(2011)

Some Exercies: Study problems

## Subjects Covered So far

** 1. September 18 **

Curves and Surfaces in space (R^3)

Chapter 1 (of Sneddon). Sections 1 and 2 including the Problems (pages 1-10)

Homework Set I

** 2. September 25 **

Methods of solutions of dx/P=dy/Q=dz/R
Applications of the system of equations dx/P=dy/Q=dz/R to dynamical
systems in R^3

(this application is not from Sneddon, see the lecture notes).
Applications of the system of equations dx/P=dy/Q=dz/R to Orthogonal
Families of Curves on a surface.
Pfaffian Differential forms and Pfaffian Differential Equations.

Chapter 1 (of Sneddon). Sections 3,4 and 5 (pages 10-26) including the
Problems (pages 15,18, and 26).

** 3. October 2 **

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

Chapter 1 (of Sneddon), Section 6 (pages 26-33) and Problems in page 33, and
Miscellaneous Problems in pages 42-43.

Chapter 2 (of Sneddon), Sections 1-4
(pages 44-55) and Problems in page 55.

** 4. Ocober 9 **

First Order Partial Differential Equations
Integral Surfaces Passing Through a Given Curve
Cauchy's Method of Characterics.

Chapter 2 (of Sneddon), sections 5,6,8 and Problems in the pages 55,57,59,66.

Homework Set II

** 5. October 16 **

Nonlinear PDEs of First Order
(Method of Envelopes) Nonlinear PFEs of First Order (section 2.7)
Partial Differential Equations of Second Order

Chapter 2 Sections (of Sneddon) 7,8,9,10,11. Problems at the end of each
section

** 6. October 23 **

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

Homework Set III

** 7. October 30 **

Last week's (6th week) subjects are not included the First midterm.

First Midterm Exam (2005),
(2010)

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

** 8. November 6 **

Characteristic Curves of Second Order Equations
Initial Value
Problems, Cauchy Problem .

After this week we shall use the book

"Applied Partial Differential Equations: with Fourier
Series and Boundary Value Problems"

by ** Richard Haberman ** (Fourth Edition), Pearson Education (2004)

Homework Set IV

** 9. November 13 **

The Heat Equation (Chapter 2)
The Method of Separation of Variables
An Initial and Boundary Value Problem With Zero Temperatures at the End
Points (2.3)
Justification of the solution (Convergence problem)

** 10. November 20 **

Solutions of the heat equations with different boundary conditions: Heat
Conduction in a Rod With Insulated Ends (2.4.1)
Heat Conduction in a Thin Circular Ring (2.4.2)
Laplace's Equation (2.5)
The Laplace Equation inside a rectangle (2.5.1)
The Laplace Equation for a circular disk (2.5.2)

** 11. November 27 **

Some properties of Laplace's equation (2.5.4)
Fourier Series (Chapter 3)

Homework Set V

** 12. December 4 **

Second Midterm
Exam , (2010)

The wave equation (Chapter 4)
Well Posed Problems
Vibrating String with fixed end Points

** 13. December 11 **

Vibrating String with fixed end Points
String with infinite length (D'Alembert's solution)
Regular Sturm-Liouville Problems

** 14. December 18 **

Properties of the Regular Sturm-Liouville Problems (Chapter 5)
The Rayleigh Quotient
Boundary Conditions of the third kind

** end of the semester **

## __ Course Syllabus __

1.** Sept 18** __ Chapter 1. Section 1.1. Surfaces in three
dimensions __** Solve exercises at the end of the section on
page 7.**

2. ** Sept 25 **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 September 2017

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