**Lecture room: **1333 Sterling hall

**Lecture time: **Mon-Wed-Fri 1:20-2:10pm

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

**Lecturer: **Gheorghe Craciun

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:*

Linear Algebra
Done Wrong, by Sergei Treil

http://en.wikipedia.org/wiki/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.*

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**Homework 3**

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

https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab

**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.*

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**Homework 5**

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

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*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.*

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**Homework 7**

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

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*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.*

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**Homework 9**

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

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*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.*

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*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.*

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*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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