MATH544's home page

MATH544 APPLIED MATHEMATICS II


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

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


Monday 10:40-12:30 SB-Z10
Thursday 08.40-10.30 SB-Z10

[%30] First Midterm pdf file (2007) , (2008) , (2010) , (2012) , (2013) , , (2014) , , (2016) (solution) ( March 14)
[%30] Second Midterm , pdf file (2007) , (2008) , (2010) , (2012) , (2013) , (2014) , Solution (2016) ( April 25)
[%40] Final Exam pdf file (2007), (2008) , (2010) , (2012) , (2013) , (2014) ,(2016) , Solution (2016) : May 5, 2016 (S-AZ04) 15:30-17:30
[%00] Homework (Presentations)

For exam results please see math544exams

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


    End of Applied2's Home Page