Introduction To The Foundations Of Applied Mathematics Homework

MATH 543 APPLIED MATHEMATICS I

(2000-2017)

2017 Fall 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

Tuesday 08:40-10:30 (SA-Z02)
Thursday 10:40-12:30 (SA-Z02)

Exams

For previous exams see prev.exams

(30%) First Midterm Exam 2007: pdf file , 2008 , 2010 , 2011 , 2012 , 2013 , 2014 , 2015 , 2017 , solution October 26, 2017
(30%) Second Midterm Exam 2006: pdf file , 2008 , 2011 , 2012 ( solution ), 2013 ( solution ), 2014 , ( solution ), 2015 ( solution ), 2017 ( solution ), November 30 , 2017
(40%) Final Exam 2007: pdf file , 2008 , 2010 , 2011 , 2012 ,2013 -Solution , 2015 -Solution , 2016-Solution , 2017-Solution , January xx, 2017


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. September 17
  • 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. September 24
  • Orthogonal set of vectors and the Bessel inequality.
  • Basis and the Parseval's relation
  • Weierstrass theorem
    assigned exercises, set 2, pdf file ,
    3. October 1
  • 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. October 8
  • Trigonometric series
  • Generalized functions
    assigned exercises, set 4 pdf file
    5. October 15
  • Fourier transform of Generalized Functions
  • Second order differential equations: Fundamental solutions, method of variation of constants.
    6. October 22
  • Generalized Greens identiy: Adjoint operators and adjoint bounday conditions.
    (2007)
    7. November 29
  • 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. November 5
  • The Sturm Liouville Problem
  • Classification of singular points
    9. November 12
  • 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. November 19
  • 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. November 26
  • 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. December 3
  • Regular Perturbations
  • Poincare-Lindstedt Method
    Assigned exercises, set 9 pdf file
    13. December 10
  • Poincare-Lindstedt Method
  • Singular Perturbations
  • Boundary Layer Problems
    Second Midterm Exam pdf file
    14. December 17
  • WKB Approximation
  • Asymtotic Expansion of Integrals
  • Calculus of Variations
    15. December 24
  • 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. December 28
  • 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


    End of Applied1's Home Page



  • Math 451      Homework Assignments       Spring, 2018
     

    (Subject to change: check web page for each week’s assignment)

    Office Hours: Tuesdays 2:30-3:00 p.m., and by appointment.

    Book: Mark H. Holmes, Introduction to the Foundations of Applied Mathematics.  Springer, 2009
    Available as an ebook from the NCSU library.

    Homework handed in should be written carefully.

    HOMEWORK

    HW #

    Due date

    Problems 


    1/11
      Read sections 6.1 - 6.4

    1/18

      Chap. 6: problems 6.1 - 6.5.  Read sections  6.5-6.7. Hand in problems 6.2abcd, 6.5af.

    2

    1/25

      Read 6.6--6.8. Problems: 6.10, 6.11
    3
    2/1
      Read sections 8.1 to 8.3. Problems,page 394: 8.1, 8.2 (a-d) 
      4

    2/8

      Read sections 8.4 to 8.6.   Problems, page 395:  8.6, 8.7.

    2/15

    No homework due; get started on Chapter 9 reading sections 9.1, 9.2

    5
    2/22


    6

    3/1

    TEST 1old test for practice
    7
    3/5-9
     SPRING BREAK

    3/15

      Read 8.11. Hand in at least 4 parts of the following: 8.14, 8.21(a), 8.22 - 8.24.

    8

    3/22

     Read  Chapter 9:

    9

    4/5

     

    10

    4/12

      Chapter 9, problems 9.4, 9.5, 9.6

     

    4/15

     


                    Final Exam: Thursday, May 3rd,  1:00 - 4:00.

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