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## MATH544 APPLIED MATHEMATICS II

** Math544: Applied Mathematics II (Spring 2004,2006,2007,2008,2010,2012,
2016),
(Spring 2018) **

** Text Books **:

1. J. David Logan , "Applied Mathematics" ,
John Willey and Sons, Inc, New York , 1997 (Second Edition)

2. Roland B Guenter and John W. Lee, "Partial Differential Equations
of Mathematical Physics and Integral Equations"", Prentice Hall, 1988.

3. E.T. Copson, "partial Differential Equations", Cambridge University
Press, 1975.

** Other Books **

1. P. Dennery and A. Krzywicki, "Mathematics for Physicists",
Harper and Row, 1967.

2. F. B. Hildebrand, " Methods of Applied Mathematics", second edition,
Prentice Hall.

3. Sadri Hassan, "Mathematical Physics: A Modern Introduction to its
Foundations", Springer Verlag, New York, 1999.

4. WE Boyce and RC Di Prima, "Elementary DEs and BVPs" Sixth Edition.
In particular Chapter 9.

5. Richard Courant and David Hilbert, "Methods of Mathematical Physics",
Vol 1,2
2004 WILEY-VCH, Weinheim.

## Course Schedule

Tuesday 15:40-17:30 SA-Z02

Friday 13.40-15.30 SA-Z02

**[%30] First Midterm ** pdf file (2007) ,
(2008) ,
(2010) ,
(2012) ,
(2013) , ,
(2014) ,
,
(2016)
(solution) ( March 13)

** [%30] Second Midterm ** , pdf file (2007) ,
(2008) ,
(2010) ,
(2012) ,
(2013) ,
(2014) ,
Solution (2016) ( April 24)

** [%40] Final Exam ** pdf file (2007),
(2008) ,
(2010) ,
(2012) ,
(2013) ,
(2014) ,(2016) ,
Solution (2016)
: May xx, 2018

**[%00] Homework (Presentations) **

For exam results please see math544exams

**Exams will be closed book. You will be responsable from the lecture Notes
**

notes, D. Logan and assigned exercies here

** Lectures **

Lecture 1

Lecture 2

Lecture 3

Lecture 4

Lecture 5

Lecture 6

Lecture 7

** Assigned Exercises **

** 1 ** Latex File ,
DVI File ,
PS. File For Ch.1

** 2 ** Latex File ,
DVI File ,
PS. File For Ch.2

** 3 ** Latex File ,
DVI File ,
PS. File For Ch.2

** 4 ** Latex File ,
DVI File ,
PS. File For Ch.2

** 5 ** Latex File ,
DVI File ,
PS. File For Ch.3

** 6 ** Latex File ,
DVI File ,
PS. File For Ch.3

** 7 ** Latex File ,
DVI File ,
PS. File For Ch.4

** 8 ** Latex File ,
DVI File ,
PS. File For Ch.4

** 9 ** Latex File ,
DVI File ,
PS. File For Ch.5

## Subjects Covered

1.** February 6 **
Functionals
Extrimizing Functionals and varitional derivative
The necessary condition
Euler Lagrange Equations
First Integrals
Natural Boundary Conditions
Higher Derivative cases

Lecture 1

homework 1

2.** February 13 **

More dependent variables
Constrained systems
Discontinuous Lagrange functions
Variable endpoints
More independent variables
Examples
More general cases
Classification

Assigned exercises, set 1 (pdf file)

FIRST PROJECT (March 5, 2013)

3. ** February 20 **

Second Order Half-Linear PDEs
(local) Classification
Existence and Uniqueness of Solutions
Examples
Classification of HL equations with n-independent variables

Assigned exercises, set 2 (pdf file)

homework 2

4.** February 27 **

The Wave equation, well posed problems
Initial and Boundary Value problems
Inhomogeneous Wave equation
Hyperbolic Equations in two space dimensions

Assigned exercises set 3 (pdf file)

Lecture 2

5. ** March 5 **

Existence and uniqueness of linear hyperbolic PDEs of dergree two
The Riemann Method

Assigned exercises set 4 (pdf file)

homework 3

6. ** March 12 **
The Riemann Method
Parabolic equations (The heat equation)
The Maximium Principle (for the heat equation) and Consequences

Lecture 3

First Midterm Exam , solution
(2013)

Assigned exercises, set 5

7. ** March 19 **

The Green's Function for the Heat Equation
Heat Flow in an infinite Rod

Lecture 4

8. ** March 26 **

The Laplace Equation in Various Dimensions and Some Exact Solutions
Existence and Uniqueness of Some BV Problems
The Maximum Principle (for the Laplace equation)

Assigned exercises, set 6

homework 5

9. ** April 2 **

The Green's Function for Various Dirichlet's Problems
Sturm Liouville Problems
Green's Functions

Lecture 5

10.** April 9 **

Neumann Series
Homogeneous Fredholm equation

Assigned exercises, set 7 (Ch.4)

Assigned exercises, set 9

Assigned exercises, set 8

Lecture 6

11. ** April 16 **

Hilbert Schmidt Theory
Singular Integral Equations

** Second Midterm Exam **

homework 6

homework 6 (solution)

12.** April 23 **

An example and stability in two dimensional dynamical systens
Stability and Bifurcations (one dimensional case)

13. ** April 30 **

Stability and Bifurcations (one dimensional case)

Assigned exercises, set 10

14. ** May 7 **
Stability and Bifurcations (one dimensional case)

Lecture 7

## Contents

** Ch.1. Calculus of Variations (Logan and Hildebrand) **

Variational Problems
Necessary Conditions for Extrema
The simplest problem
Generalizations
Isoprimetric Problems

** Ch.2. Partial Differential Equation Models (Gunther-Lee, Copson and
Logan)**

Classification of 2nd order Partial Differential Equations
(G-L and Copson)

(first midterm exam was up to this section)
The Cauchy-Kowalewsky theorem (Copson)
Linear, quasi-linear, half-linear equations (Copson)
Initial and BV problems for the wave equation (GL)
Some existence and uniqueness theorems (G-L)
First order hyperbolic systems
The Riemann Method (G-L and Copson)

** Ch.3. Exact solutions, uniqueness, maximum-minimum theorems (G-L and DK)
**

Hyperbolic type of (the wave) equation
Parabolic type of (the heat) equation
Elliptic type of (the Laplace) equation
Green's function techniques

** Ch.4. Integral Equations and Green's Functions (G-L and Hildebrand) **

CH.4 Latex File ,
dvi file ,
ps file

Sturm-Liouville Problems
Green's Functions
Neumann Series
Examples
Hilbert-Schmidt Theory
Singular Integral Equations

(second midterm was up to this section)

** Ch.5. Stability and Bifurcation (Logan) and (Boyce and DiPrima , Ch. 9)**

CH.5 Latex File ,
dvi file ,
ps.file

Intuitive Ideas
One-Dimensional Problems
Two-Dimensional Porblems
Hydrodynamics Stability

## Course Syllabus of Math544

01. Febr. 05- Calculus of variations

02. Febr. 12- Calculus of Variations

03. Febr. 19- Calculus of Variations

04. Febr. 26- Partial Differential Equations

05. Marc. 05- Partial Differential Equations

06. Marc. 12- Partial Differential Equations

07. Marc. 19- Partial Differential Equations

08. Marc. 26- Partial Differential Equations

09. Aprl. 02- Integral equations and Green's Functions

10. Aprl. 09- Integral equations and Green's Functions

11. Aprl. 16- Integral Equations and Green's Functions

12. Aprl. 23- Integral Equations and Green's Functions

13. Aprl. 30- Integral Equations and Green's Functions

14. May. 07- Stability and Bifurcations

15. May. 14- Stability and Bifurcations
## Last update May 2012

#

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