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## Math 703 – Methods of Applied Mathematics I – Fall 2018

Lecture room: 1333 Sterling hall
Lecture time: Mon-Wed-Fri 1:20-2:10pm

Office hours: Monday and Friday 2:10-3:00pm

Lecturer:
Office: 405 Van Vleck
E-mail: craciun at math dot wisc dot edu

Main textbook:

Introduction to Applied Mathematics, by Gilbert Strang.

Other recommended textbooks:

Advanced Mathematical Methods for Scientists and Engineers, by C. M. Bender and S. A. Orszag.

Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering, by Steven Strogatz.

Course Content: The course introduces methods to solve mathematical problems that arise in areas of application such as physics, engineering, chemistry, biology, and statistics.

Prerequisites: Math 319 (ODEs), Math 321 (Vector and complex analysis), Math 322 (Sturm-Liouville, Fourier Series, intro to PDEs), Math 340 (Linear Algebra) or equivalent. In addition, Math 521-522 (Mathematical Analysis), and Math 623 (Complex Analysis) are strongly recommended.

Grading: Homework and class participation: 10%, Midterm exam: 40%, Final exam: 50%.  Additional credit may be obtained for work on research projects.

MATLAB help

Several MATLAB tutorials are available here.

A free online textbook “Numerical Computing with MATLAB” by Cleve Moler is available here.

Review of Linear Algebra

If you need a detailed review of Linear Algebra you can find Gilbert Strang’s video lectures here.

Homework 1

1. Have a look at these other online resources for linear algebra:

2. Try to solve the following review problems from the Strang textbook: 1.5.1, 1.5.3, 1.5.7, 1.5.9, 1.5.11, 1.5.13.

Solutions to Homework 1 will not be collected, but these problems give you an idea of topics that you should already know a lot about.

Homework 2

Solve and hand in your solutions for the following problems from the Strang textbook:

1.2.6, 1.2.9, 1.2.10, 1.2.11, 1.3.1, 1.3.5, 1.3.7, 1.3.8, 1.3.9, 1.3.11.

Due date/time: Wednesday Sept 19,  just before class.

Homework 2 solutions are here.

Homework 3

Watch the videos on “Essence of linear algebra” by 3Blue1Brown here:

Homework 4

Solve and hand in your solutions for the following problems from the Strang textbook:

1.4.1, 1.4.2. 1.4.5, 1.4.6, 1.4.7, 1.4.8, 1.4.9,

1.5.3, 1.5.5, 1.5.6, 1.5.12, 1.5.15, 1.5.16, 1.5.18, 1.5.19, 1.5.20.

Due date/time: Wednesday Oct 3,  just before class.

Homework 4 solutions are here.

Homework 5

Study all the examples in sections 1.2, 1.3, 1.4, 1.5, 1.6 in the Strang textbook.

Homework 6

Solve and hand in your solutions for the following problems from the Strang textbook:

2.1.2, 2.1.3, 2.1.4, 2.1.6, 2.1.7, 2.1.9, 2.1.11, 2.1.12, 2.1.13.

Due date/time: Wednesday Oct 24,  just before class.

Homework 6 solutions are here.

Homework 7

Study all the examples in sections 2.1 and 2.2 in the Strang textbook.

Homework 8

Solve and hand in your solutions for the following problems from the Strang textbook:

2.2.1, 2.2.2, 2.2.3(b), 2.2.4, 2.2.5, 2.2.7, 2.2.8, 2.2.11, 2.2.15, 2.2.16.

Due date/time: Friday Nov. 2,  just before class.

Homework 8 solutions are here.

MIDTERM EXAM: Friday Nov. 9, during regular class time.

Homework 9

Study all the examples in sections 4.1 and 4.3 in the Strang textbook.

Homework 10

Solve and hand in your solutions for the following problems from the Strang textbook:

4.1.1, 4.1.2, 4.1.3, 4.1.4, 4.1.5, 4.1.6, 4.3.1, 4.3.2, 4.3.3, 4.3.4, 4.3.5, 4.3.6, 4.3.7.

Due date/time: Wednesday Nov. 28,  just before class.

Homework 10 solutions are here.

Homework 11

Study all the examples in sections 5.2, 6.2, 6.3, 6.4, 6.5 in the Strogatz textbook.

Homework 12

Solve and hand in your solutions for the following problems:

1. Use "pplane" or any other software to plot solutions of 2D linear dynamical systems that show (i) a saddle point, (ii) a stable spiral, (iii) a stable node, and (iv) a center.

2. Is it possible for a 2D linear dynamical system to have more than one equilibrium? Explain your answer.

3. Use "pplane" or any other software to plot solutions of 2D dynamical systems that has two equilibria, one stable and one unstable.

4. Use "pplane" or any other software to plot solutions of 2D dynamical systems that has a limit cycle.

Due date/time: Wednesday Dec. 12,  just before class.

Suggestions for Class Project topics (a good Class Project will replace the Final Exam; groups of 1-4 students may work together on a Class Project; if you choose a topic please send me an email to let me know):

1.    Choose your own topic: something that you are interested in and has connection to applied math. Please send me an email to let me know what that topic is.

2.    Choose a section from the Strang textbook, or from the Strogatz textbook (or from some other book) that we have not covered in class, and write a report on that topic, including maybe solutions to homework problems from that section, maybe also applications and examples.

3.    Mathematics in the news: find some news stories that have connections to mathematics, and explain those connections.

4.    Specific applications of mathematics in your field of interest: engineering, physics, chemistry, biology, etc.

5.    Mathematical models of the spread of infectious diseases.

6.    Mathematical models in “game theory” and relationships to Linear Algebra.

7.    The diffusion equation.

8.    The wave equation.

9.    Transport equations.

10.Solutions of linear dynamical systems in three or more dimensions.

11.The fast Fourier transform.

12.The use of Fourier transforms in signal processing.

13.The use of Fourier transforms in image processing.

14.Dynamical system models in chemistry or biochemistry.

15.Dynamical system models in population dynamics (for example: competitive exclusion).

In general, please also think about including some use of software (you don’t have to, but it would be great if you can do so).  For example, if your project has any differential equations, you can include numerical simulations using the software “pplane”. Otherwise, you can use Matlab, Mathematica, or any software you prefer.

Submit your Class Project by email before the end of the day on Dec 20, 2018.

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Alternative assignment for the final exam (you don’t have to do this if you are preparing a Class Project):

Choose four
of the following seven problems to solve. Submit your solutions by email by Dec 20, 2018.

1. Solve problem 3.3.1 (a)-(d) on pages 81-82 of the Strogatz textbook.

2.  Solve problem 3.3.2 (a)-(c) on page 82 of the Strogatz textbook.

3. Solve problem 3.5.6 (a)-(e) on pages 85 of the Strogatz textbook.

4. Solve problem 3.7.5 (a)-(e) on pages 90-91 of the Strogatz textbook.

5. Solve problem 6.1.14 on pages 180-181 of the Strogatz textbook.

6. Solve problem 6.4.7 (a)-(c) on page 185 of the Strogatz textbook.

7. Solve problem 6.5.19 (a)-(d) on page 189 of the Strogatz textbook.

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