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MATH544 APPLIED MATHEMATICS II
Spring 2019
Math544: Applied Mathematics II (Spring 2004,2006,2007,2008,2010,2012,
2016-2018,2019),
(Spring 2020)
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
Wednesday 13:40-15:30 SA-Z02
Friday 15.40-17.30 SA-Z02
[%30] First Midterm pdf file (2007) ,
(2008) ,
(2010) ,
(2012) ,
(2013) , ,
(2014) ,
,
(2016)
(solution) , (2019)
(solution) ( March 18)
[%30] Second Midterm , pdf file (2007) ,
(2008) ,
(2010) ,
(2012) ,
(2013) ,
(2014) ,
Solution (2016) , , (2019)
(solution) ( April 29)
[%40] Final Exam pdf file (2007),
(2008) ,
(2010) ,
(2012) ,
(2013) ,
(2014) ,(2016) ,
Solution (2016) , solution
(2019)
: May 21, 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: Calculus of Variations
Lecture 2:Classifications of Partial
Differential Equations
Lecture 3: Hyperbolic Type of Equations
Lecture 4: Parabolic Type of Equations
Lecture 5: Elliptic Type of Equations
Lecture 6: Integral Equations and Green's Function
Lecture 7: Stability and Bifurcations
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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