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     Math 560/561  -  Applied Math I/II *    

Math 560



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Textbook Principles of Applied Mathematics (1rst or 2nd ed.), James Keener (not required but highly recommended)
Instructor Mark Pernarowski
Grading Your grade will be based strictly on homework assignments.  The raw score for each assignment will vary. If P is the percentage of the sum of all the raw scores then your letter grade will be determined from:

A A- B+ B B- C+ C C- D F
90-100 87-89 84-86 80-83 77-79 74-76 70-73 67-69 60-66 0-59
 
Assignments must be written in a clear and logical fashion.
   

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Homework Homework 1 :Inner product spaces, diagonalization, Fredholm applications, Least Sq. Due: Thursday, Sept 22
  Homework 2: Operator boundedness, eigenfunctions, adjoint calculations Due: Thursday, Oct 13
  Homework 3: Operators, Fredholm alternative and Distributions Due: Thursday, Nov 3-7
  Homework 4: Green's functions and Sturm Liouville Due: Nov 17-22
  Homework5:  Euler Lagrange Eqns, Natural BC, Extrema, Geodesics Due: Thursday, Feb 2
 
Homework 6: Isoperimetric problems, Transversaiity, Convexity, Several independent
                    variables, Lagrangian Mechanics.
Due: Thursday, Feb 23
 
Homework 7: Regular perturbation approximations and intro to asymptotics
Due: Thursday, Mar 30
 
Homework_8: Singular Boundary Value Problems
Due: Thursday, Apr 20

 

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Notes Inner product spaces, bases, coordinates, Gram Schmidt, Least Squares  
  Eigenvalues, self adjoint, diagonalization, maximum principle  
  Subspaces, Projections and the Fredholm Alternative for Matrices  
  Least Squares and Normal Equations  
  Hilbert Spaces and Operators  
  Functional 101, Adjoints and Fredholm Alternative  
  Operator Inverses  
  Distributions                         (useful old notes here)  
  Greens functions  
  Sturm Liouville Theory            
  Sturm Liouville Applications  
  Calculus of Variations - Introduction  
  Calculus of Variations - Theory, issues, NBC, L(x,y,y') and L(x,y,y',y'')  
  Calculus of Variations - Multi-variable L(x,y1,y1',y2,y2',...) introduction  
  Calculus of Variations - Geodesics and Isoperimetric Problems and Transversality  
  Calculus of Variations - Convexity, Sufficiency, Existence and Convexity overview  
  Calculus of Variations - Several independent variables.  
  Calculus of Variations - Least Actions and Hamiltonian Mechanics   
  Perturbation Theory - Intro  
  Perturbation Theory - Regular         (Implicit Function Theorem)  
  Perturbation Theory - Asymptotics  
  Perturbation Theory - Asymptotic root approximations  
  Perturbation Theory - Matching Theory and Concepts  
  Perturbation Methods - BVP  
  Perturbation Methods - Oscillations and Multiple Scales  MS-systems  
  Perturbation Methods - Adiabatic Invariants  
  Perturbation Methods - Averaging (big)  
  Perturbation Methods - Kuzmak-Luke Strongly nonlinear osc.  


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Supplemental Operator Theory Summary Sheet  and Compact Operators  
  Boundary Value Problems - 2nd order  
  Integral Equation examples  
  Functional Analysis - Bounded Operators  
  Functional Analysis - Fredholm Alternative  
  Functional Analysis - Mixed with Integral equations  
     
     
     
     
     
     
     
     
     
     
     
     
Learning
Outcomes:

Upon completion of the course students will have an understanding of the following:Inner product spaces: Bases, coordinates, Gram Schmidt orthogonalization, Least squares approximations Matrices: adjoints, diagonalization, maximum principle, projections, Least squares solutions, singular value decomposition, normal equations and the Fredholm alternative Introductory theory of Hilbert spaces including an overview of the properties of bounded, unbounded, and compact operators and their adjoints. Completeness and spectral properties of self adjoint operators. Fredholm alternative and applications. Operator inverses. Introduction to distributions on test function spaces and their properties. Green's functions and Sturm Liouville theory for Boundary Value Problems. Calculus of Variations introduction: functionals, admissible sets, natural boundary conditions for Lagrangians of single and several dependent variables. Euler Lagrange equation derivations Intermediate Calculus of Variation topics: geodesics, isoperimetric problems, transversality, Lagrangians with several independent variables, convexity and sufficiency conditions for minima. Lagrangian and Hamiltonian mechanics and the principle of Least action. Regular perturbation theory for algebraic and differential systems. Implicit function Theorem. Asymptotic expansions and series with applications to root approximation. Matching Theory for asymptotic expansions, overlap domains, intermediate variables. Matched asymptotic expansions for boundary value problems. Perturbation techniques for oscillatory systems: Multiple scales approaches with different temporal straining, the method of averaging and adiabatic invariants.