Methods of Applied Math (Math 558) Offered in Fall 2014

Course:  Math 558 -- Methods of Applied Mathematics

Instructor:  Prof. Jared Bronski, Dept. of Mathematics

Course Overview:  "Applied Mathematics" is a diverse field encompassing modeling, applications, numerics, as well as techniques drawn from many areas of mathematics, sciences and engineering. In this course we will cover some a few of the most important mathematical techniques used in modern applied mathematics, as well as introducing a some of the interesting current problems in the eld. We will cover both analytical techniques as well as computational techniques. This course is aimed both at graduate students interested in research in applied mathematics as well as graduate students in applied elds interested in learning more mathematical techniques. The course will work from a number of texts (see below) as well as some current research papers. The content, particularly the applications, will be tailored to the interests of the class.

Syllabus:

1. Nondimensionalization, scaling analysis and simple modeling.  10 Lectures. Material will include Chapters 0-3 of [1] and Chapters 1-2 of [2]

2. Regular asymptotics: Integrals, roots, and eigenvalues.  12 Lectures: Selected Chapters of [3], Chapters 5-7 of [4].

3. Singular asymptotics: Multiple times-scales, Modulations, Boundary Layers and WKB analysis.  12 Lectures: Selected Chapters of [3], Chapter 9 of [4]

4. Coherent Structures and Stability  Applications: Stability of solitons and traveling waves. Synchronization in networks.  8 Lectures. Material will be drawn from a variety of papers.

Partial Bibliography

1. G.I. Barenblatt, \Scaling, Similarity and intermediate Asymptotics"

2. Mark H. Holmes, \Introduction to the foundations of Applied Math"

3. C. Bender and S. Orszag, \Advanced Mathematical Methods for Scientists and Engineers"

4. C.W. Gardiner \Handbook of Stochastic Methods for Phsyics, Chemistry and the natural sciences."