Math 450/451 Applied Mathematics I-II
| Instructor |
Mark Pernarowski |
| Textbook |
Applied Mathematics (3rd ed) |
|
J. David Logan |
| Office Hours |
Schedule (Wil 2-236) |
| Phone |
994-5356 |
| Classroom |
Wil 1-115 |
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I have the final exams. If you really need them back soon email me.
Otherwise I will keep final exams for you to pick up in the Fall.
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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
and both the final and midterm
are take home exams.
Homework and exam due dates will
be announced in class and posted
here at a later date. Exam content
will be announced in class.
Midterm and Final due dates and
details will be announced in
class.
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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.
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Homework and Exams for M450/M451:
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Due Date
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Content |
Solutions |
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| Homework 1 |
Friday, Sept 18 |
M450 |
ODE Review |
HW_1_Solns |
| Homework 2 |
Friday, October 2 |
M450 |
Dimensional Analysis |
HW_2_Solns |
| Homework 3 |
Friday, October 16 |
M450 |
Regular Perturbation Problems |
HW_3_Solns |
| Midterm 1 |
Friday, October 23 |
M450 |
on HW1-HW3 |
Midterm 1 Solns |
| Homework 4 |
Friday, November 6 |
M450 |
Singular perturbations, Asymptotics |
HW_4_Solns |
| Homework 5 |
Friday, November 20 |
M450 |
Singular BVP, IVP |
HW_5_Solns |
| Homework 6 |
Friday, December 4 |
M450 |
Calculus of variations I |
HW_6_Solns |
| Final |
Wed, December 16 |
M450 |
on HW4-HW6 |
Final Solns |
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| Homework 7 |
Wed, January 27 |
M451 |
Calculus of variations II |
HW_7_Solns |
| Homework 8 |
Friday, Feb 12 |
M451 |
L2[a,b] and Sturm Liouville Theory |
HW_8_Solns |
| Homework 9 |
Wed, Feb 24 |
M451 |
Fredholm Integral Equations |
HW_9_Solns |
| Midterm 1 |
Monday, March 8 |
M451 |
on HW1-HW3 |
Midterm 1 Solns |
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(Errata: #2 should have lambda < 0) |
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| Homework 10 |
Monday, April 5 |
M451 |
Green's fns, Distributions, PDE I |
HW_10_Solns |
| Homework 11 |
Friday, April 23 |
M451 |
Random Walks, PDEs - series and transforms |
HW_11_Solns |
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Skip 1b) and use boundedness of u in 6) |
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| Final |
Thur, May 6 at 4pm |
M451 |
on HW10-11 only. |
Final_Solns |
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at my office Wil 2-236 |
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Classnotes:
- Ordinary Differential Equation (ODE) review (Text Section 1.3).
- First Order Linear Equations
- First Order Nonlinear Equations
- Second Order Constant Coefficient Equations
- Systems of Differential Equations (second order only)
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.
- Dimensional Analysis (Text Section 1.1-1.2)
- A summary of some physical quantities and their units (MKSA system)
- An introduction to dimensional analysis with examples
- General theory for unit free laws and the Pi-theorem
- Nondimensionalizing (scaling) models: Logistic, Chemostat
- Perturbation Theory (Text Chapter 2)
- Introductory examples illustrating issues
- Summary sheet for Taylor Series
- Regular Perturbation Theory/Examples for Algebraic problems
- Regular Perturbation Theory/Examples for Initial Value Problems
- Regular Perturbation Theory - Nonlinear Oscillations
- Asymptotic Expansions
- Singular Perturbation Theory/Algebraic problems
- Singular Perturbation Theory/Boundary Value Problems- Intro.
A casual introduction to boundayer layers and how one finds
approximations valid in (inner) and away from (outer) the layer. The
ideas of outer and inner approximations and matching are introduced and
applied to a single model problem. Some Maple code I used in developing
the example is included here.
- Examples of solved problems
- Matching Theory - notes on the theory behind matching and why it works
- A few examples of method failures - result from incorrect assumptions
- Singular Perturbation Theory/Initial Value Problems
- Strongly damped oscillator - nondimensionalization/solution/issues
- Chemical Reaction Kinetics - not covered in class
- Calculus of Variations (Text Chapter 3)
- Functionals, function spaces and functional minimization - introduction and examples
- Introductory theory for functional minimization: Normed spaces, Local minima definition, admissible variations, Gateaux variations, necessary conditions for minima
- Euler-Lagrange Equations
-- for finding extremal of functionals having
Lagrangian L(x,y,y'). These are necessary conditions. First
integrals for the E-L equations are discussed and examples given.
- Higher Dimensional Lagrangians
-- Euler Lagrange Equations for finding extrema of functionals
whose Lagrangian is higher dimensional, such as: L(x,y,y',y'') or
L(x,y1,y2,y1',y2')
- Natural Boundary Conditions: Derivable boundary conditions extrema of functionals must satisfy when none or too few are specified.
- Isoperimetric Problems:
Optimization of integral functionals J(y) subject to integral
functional constraints K(y)=constant. For a summary of more general
Lagrange Multiplier theory see here.
- Eigenvalue Problems, Integral Equations and Green's Functions (Text Chapter 4)
- Function expansions in L2[a,b] using orthogonal sets: generalized fourier series
- Sturm Liouville Problems (SLP): Operators, Eigenvalue Problems
- Fredholm Integral Equations
- Green's Functions for Sturm Liouville Operators
- Distributions - delta function, distributional solutions of ODE's
- Partial Differential Equations (Text Chapter 6)
- Introductory examples, definitions and concepts
- Multivariate Calculus Overview
- Conservation Laws and Constituitive Relations
- Random Walks, Diffusion and Fokker-Planck Equations
- Initial and Boundary Value Problems: Uniqueness and Well-Posed problems
- Series Solutions: Laplace, Heat, Wave Equations (bounded spatial domains)
- Laplace Transform Techniques (Transform Table)
- Scalar Reaction Diffusion Systems: Equilibria and their stability
- A Reaction Diffusion System: Chemotactic driven instability
- Method of characteristics: Introduction for linear scalar 1rst order equations
- Travelling Waves: Burger's equation and phase plane description
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