Math 450/451 Applied Mathematics
Instructor 

Mark Pernarowski 
Textbook 

Applied Mathematics (3rd ed) 


J. David Logan 
Office Hours 

Schedule (Wil 2236) 
Phone 

9945356 
Classroom 

Rob Hall 307 


MWF 10:0010:50am 








Grading: The course % for each of M450 and M451 is determined by:
Midterm M 100 Final F 100 Homework HW 200 _______________________________ 400 % = (M+F+HW)/4
The final is not comprehensive.
The Midterm and Final are take home and due at the date indicated below.
Midterm: TBA (takehome)
Final: TBA (takehome)
Homework due dates and exam dates will be announced in class and posted here at a later date. Exam content will also be announced in class.


Syllabus: Material for the M450/M451 sequence will be selected from:
Class Notes ODE Review Chapter 1 Dimensional Analysis Chapter 2 Perturbation Methods Chapter 3 Calculus of Variations
Chapter 4 Eigenvalue Problems, Green's Functions Chapter 6 Partial Differential Equations Chapter 7 Wave Phenomena Chapter 8 Models of Continua
Homework: Assigned homework and some of their solutions will be posted below as the course develops.
Homework scores will vary depending on their length and difficulty. The raw scores will be summed, and converted into a % to yield the 200 points in the final grade.

Classnotes for M450/M451:
 Ordinary Differential Equation review (Text 1.3,2.2)
 First Order Linear Equations
 First Order Nonlinear Equations
 Second Order Constant Coefficient
 Systems of Equations (second order only) and some additional Review Problems We may at some point also need a review of Laplace Transform techniques Other than your previous text, and class notes you may want to consider looking at the review drafted by Paul Dawkins at Lamar University, TX
2. Dimensional Analysis (Text Section 1.11.2)
 Dimensional Analysis Introductory examples. and Similarity Solns. Here's a unit summary sheet.
 Dimensional Analysis Theory
 Scaling in differential equations (nondimensionalization)
 Perturbation Theory (Text Chapter 3)
 Introductory Problems
 Regular Perturbation Problems  algebraic
 Regular Perturbation Problems  Systems and Integrals
 Regular Perturbation Problems  ordinary differential equation
 Regular Perturbation Problems  nonlinear oscillations
 Asymptotics
 Singular Perturbation Theory  algebraic
 Singular Perturbation Theory  BVP intro
 Singular Perturbation Theory  BVP Matching
 Singular Perturbation Theory  BVP worked examples
 Singular Perturbation Theory  BVP Matching failure
 Singular Perturbation Theory  IVP Oscillator example, Chemical Model example
 Calculus of Variations (Text Chapter 4)
 Calculus of Variations  Introduction
 Calculus of Variation  Theory
 EulerLagrange equations  Necessary conditions for minima
 Natural Boundary Conditions
 Higher Dimensional Problems
 SpringPendulum example
 Geodesics
 Isoperimetric Constraints  example
 Eigenvalue Problems, Integral Equations and Green's Functions (Text Chapter 5)
 Function expansions in L2[a,b] using orthogonal sets: generalized fourier series
 Regular Sturm Liouville problems and related eigenfunction expansions
 Fredholm Integral Equations
 Green's Functions
 Distributions
 Partial Differential Equations (Text Chapter 6)
 Introductory examples, definitions, concepts.
 Multivariate Calculus Overview
 Conservation Laws and Constituitive Relations
 Diffusion as Random Walks
 Series Solutions for PDE's
 Method of Characterisitics  introduction
Homework and Exams for M450: Below are the homework assignments and takehome exams. In the "Sample" column I will occasionally post old versions of similar assignments and their solutions. These should help augment your notes with more worked examples.

Class 
Due Date 
Content 
Solutions 





HW 1 
M450 
Wed. Sept. 17 
ODE Review 

HW 2 
M450 
Mond., Oct 2 
Dimensional Analysis 

HW 3 
M450 
Wed. Oct 18 
Regular Perturbations 






Midterm 

Frid. Oct 27 
Is a takehome exam. You may use, class notes, posted notes and the text.
You may not use electronic devices (including internet) nor may you
discuss the problems with other students. If you need clarification
about a problem, you may ask me.
 Dimensional Analysis  Buckingham Pi theorem
 Nondimensionalizing an ODE(s) system
 Regular expansions of f(x,epsilon)=0
 Regular expansion of a system
 Regular expansion of an IVP












HW 4 
M450


Poincare Lindstedt method, asymptotics, singular root approximations, Singular boundary value prob. 

HW 5 
M450 
Wed. Nov 29 
Nonlinear BVP, functional First Variations, Simple Euler Lagrange equations. 






Final 
M450 
Frid, Dec 8

Is on HW 45 (TAKEHOME GIVEN OUT FRI DEC 4 IN CLASS1 






HW 6 
M451

Frid, Jan 26 
Problems with Natural Boundary Conditions (NBC) for Lagrangians L=L(x,y,y') and L=L(x,y,y',y'') tex


HW_7 
M451 
Frid, Feb 9 
Isoperimetric problems, Geodesics and multiple dependent variables. 

HW_8 
M451 
Frid, Feb 23 
Fourier Series and Sturm Liouville Problems 

