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MATH 543 APPLIED MATHEMATICS I
(2000-2019)
2022 Spring Semester
Math543: Applied Mathematics I
Text 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. J. David Logan, "Applied Mathematics", John Willey and Sons, Inc.,
New York, 1997 (Second Edition).
5. Haaser and Sullivan , "Real Analaysis", The University Series in
Undergraduate Mathematics.
6. Richard Courant and David Hilbert, "Methods of Mathematical Physics", Vol 1,
2004 WILEY-VCH, Weinheim.
7. W. E. Boyce and R. C. DiPrima, " Elementary Differential Equations
and Boundary Value Problems",
Sixth Edition, John Wiley and Sons, Inc.
some
subjects in applied mathematics
Course Schedule
Wednesday 10:30-12:20 (SA-Z19)
Friday 15:30-17:20 (SA-Z19)
Exams
For previous exams see prev.exams
1) (30%) First Midterm Exam 2007: pdf file
, 2008 ,
2010 ,
2011 ,
2012 , 2013 ,
2014 ,
2015 , 2017 ,
solution ,
2018 ,
solution 2018 ,
solution 2019
, March , 2022
(0%) Second Midterm Exam 2006: pdf file
, 2008 ,
2011 , 2012
( solution ),
2013
( solution ),
2014 ,
( solution ),
2015
( solution ),
2017
( solution ),
,
solution 2018 ,
solution 2019
,
2) (40%) Final Exam 2007: pdf
file ,
2008 , 2010 ,
2011 ,
2012 ,2013 -Solution ,
2015 -Solution ,
2016-Solution ,
2017-Solution ,
2018-Solution XCy
2018 Final makup exam ,
solution 2019
, May, 2022
3) 30% Homework(At least six homework assignments)
For exam results please see math543exams
In the exams you will be responsable from DK, Logan, Lecture notes
and the assigned exercises below
Lectures
1. Lecture 0 (some prelimineries)--[10 pages]
2. Lectures 1 and 2-- [Function Spaces]
3. Lectures 3 and 4-- [Hilbert Spaces,
Bases and Orthoganal Polynomials]
4. Lectures 5 and 6-- [Genralization of the Weirtstrass Theorem and
Classical Orthogonal Polynomials]
5. Lectures 7 and 8-- [Second Order
ODEs and Green's Function Method]
6. Lectures 9 and 10-- [Series
solutions of DEs, Frobenius Method and
Fuchsian DEs]
7. Lecture 10-- [Confluent Hypergeometric
Equation is included]
8. Lecture 11 [Regular Perturbation Method]
9. Lecture 12 [Singular Perturbation Method]
Contents of Applied Mathematics I
For preparation read the last Chapter of Haaser and Sullivann
"Real Analysis"
Assigned Exercises
1. set 1 pdf file ,
ps file
2. set 2 pdf file ,
ps file
3. set 3 pdf file ,
ps file
4. set 4 pdf file ,
ps file
1st exam pdf file
5. set 5 dvi file ,
ps file
6. set 6 dvi file ,
ps file
7. set 7 dvi file ,
ps file
2nd exam with solutions pdf file
8. set 8 dvi file ,
ps file
9. set 9 pdf file
10. set 10 pdf file
Final exam (pdf file)
Subjects covered
1. February 3
Funcion Space, completeness, square integrable functions.
Chapter III of Dennery, Chapter 5 (Hilbert Spaces) of Sadri Hassan
assigned exercises, set 1 pdf file ,
Homework Set 1 pdf file ,
2. February 7
Orthogonal set of vectors and the Bessel inequality.
Basis and the Parseval's relation
Weierstrass theorem
assigned exercises, set 2, pdf file ,
3. February 14
Classification of orthogonal polynomials
Classical orthogonal polynomials
Homework Set 2 pdf file ,
assigned exercises, set 3 pdf file
Spherical Harmonics in D
Dimensions
Generating function of the Legendre
Polynomials , Other generating functions
4. February 21
Trigonometric series
Generalized functions
assigned exercises, set 4 pdf file
5. February 28
Fourier transform of Generalized Functions
Second order differential equations: Fundamental solutions,
method of variation of constants.
6. March 7
Generalized Greens identiy: Adjoint operators and adjoint bounday
conditions.
(2007)
7. March 14
The Method of Green's Function
The Method of Green's Function (applications)
First Midterm Exam pdf file (2006) ,
Homework Set 3 pdf file ,
assignaed exercises, set 5 pdf file
8. March 21
The Sturm Liouville Problem
Classification of singular points
9. March 28
The Frobenius method: The series solutions of linear DEs
assigned exercises, set 6 pdf file ,
(Assigned exercises : Solve also the problems of the Sections
5.4-5.8 of Boyce and DiPrima)
10. April 4
The Frobenius method: The series solutions of linear DEs
(Assigned exercises : Solve also the problems of the Sections
5.4-5.8 of Boyce and DiPrima)
Second Midterm Exam pdf file
11. April 11
Fuchsian Differential Equations
The Hypergeometric Function
assigned exercises, set 7 pdf file
Solutions of DEs by Integral Representations
Integral Representations of Hypergeometric Functions
assigned exercises, set 8 pdf file
Homework Set IV: From set 7 Problems 12,13 and from set 8 Problems 2,3
(Due December 28)
12. April 18
Regular Perturbations
Poincare-Lindstedt Method
Assigned exercises, set 9 pdf file
13. April 25
Poincare-Lindstedt Method
Singular Perturbations
Boundary Layer Problems
Second Midterm Exam pdf file
14. May 2
WKB Approximation
Asymtotic Expansion of Integrals
Calculus of Variations
15. May 9
Calculus of Variations
Necessary Condition
Euler-Lagrange Equations
Lagrange functions depending on higher derivatives
Null Lagrange functions
Lagrange function of given DE
Lagrange function with several dependent variables
Assigned exercises, set 10 pdf file
end of the semester
16. May 16
Lagrangeans with Several Functions
Isoperimetric Problems
Lagrange Function of a given DE
Subjects to be covered
Ch.1. Function Space, Orthogonal Polynomials and Fourier Analysis
(Dennery and Krzywicki)
Space of Continuous Functions
Expansion in Orthoganal Functions
The Classical Orthogonal Polynomials
Trigonometric Series
Generalized Functions
Linear Operators in Infinite Dimensional Space
First Midterm
Ch.2. Ordinary Differential Equations (Dennery and Krzywicki)
Second Order Differential Equations
Generalized Green's Identity
Green's Functions
The Sturm-Liouville Problem
Series Solution of Linear Differential Equations
The Hypergeometric Function
Second Midterm
Ch.3. Perturbation Methods (Logan)
Regular Peturbations
Singular Perturbation
Boundary Layer Analysis
Applications
The WKB Approximation
Asymptotic Expansions of Integrals
Final Exam
Ch.4. Calculus of Variations (Logan and Hildebrand)
Variational Problems
Necessary Conditions for Extrema
The simplest problem
Generalizations
Isoperimetric Problems
Course Syllabus of Math543
01. Sept. 24 - Function Space
02. Oct. 01 - Function Space
03. Oct. 08 - Function Space
04. Oct. 15 - Ordinary Differential Equations
05. Oct. 22 - Ordinary Differential Equations
06. Oct. 29 - Ordinar Differential Equations
07. Nov. 05 - Ordinary Differential Equations
08. Nov. 12 - Perturbation Methods
09. Nov. 19 - Perturbation Methods
10. Nov. 26 - Perturbation Methods
11. Dec. 03 - Perturbation Methods
12. Dec. 10 - Calculus of Variations
13. Dec. 17 - Calculus of Variations
14. Dec. 21 - Calculus of Variations
Last update May 2017
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