MATH 404
CLN , TTh 1:00-2:30, EH 4088

# Intermediate (Applied) Differential Equations

(Prerequisites: MATH 216, 256, 286, or 316)

## Instructor:

Dr. Patrick Nelson
East Hall 3071
tel: 763-3408
fax: 763-0937
pwn@math.lsa.umich.edu
office hours: Tues 3-4:30, Wed 9:30 - 11:00, and by appt.

 Course description Textbook Syllabus Lecture Notes Objectives Schedule

## Course Description

There are three main objectives to this class. First, we will present and explain mathematical methods for obtaining approximate analytical solutions to differential equations that cannot be solved exactly. The material in this section will mostly come from Bender and Orszag and we will present it in a very introductory manner. Second, we will introduce and explain the theories of dynamical systems, specifically bifurcation and chaos, phase portraits, linear stability analysis, and local and global behavior of both linear and non-linear differential equations. Third, we will use the computer via matlab and maple to visualize what we are learning and to gain a better understanding of the dynamics. Learning the material will be done through homework that will require both pen and paper analysis and computation support. I would expect all students to have had an introductory course in differential equations and some familiarity with matlab. There will be a special session or two on matlab for students who do not have familiarity with this computation method.

## Textbooks

1. Advanced Mathematical Methods for Scientists and Engineers (required), Carl M. Bender and Steven A. Orszag, McGraw-Hill, 1978,. This book presents the methods of asymptotics and perturbation theory for obtaining approximate analytical solutions to 'real' differential equations that arise in physics and engineering. These equations are usually not solvable in closed form and numerical methods may not converge to useful solutions. The aim is to teach the insights that are most useful in approaching new problems and it avoids the special methods and tricks that work only for particular problems. We will use this book in an introductory manner where some of the more advanced topics will not be covered.
2. Non-linear Systems (recommended), P.G. Drazin, Cambridge texts in Applied Mathematics,1994.
3. Nonlinear Oscillations, Dynamical systems, and Bifurcations of Vector Fields (recommended), J. Guckenheimer and P. Holmes, Springer

## Mathematical concepts to be covered

1. Ordinary differential equations
2. Approximate solutions of linear differential equations
3. Approximate solutions of non-linear differential equations
4. Stability analysis, bifurcations, and limit cycles
5. Phase portraits
6. Perturbation theory
7. Boundary layer theory
8. Bessel, parabolic cylinder, and Airy functions
9. Numerical analysis with Matlab

 Homework 1 404Hmwk1.ps 404Hmwk1.pdf Homework 2 404Hmwk2.ps 404Hmwk2.pdf Homework 3 404Hmwk3.ps 404Hmwk3.pdf Homework 4 404Hmwk4.ps 404Hmwk4.pdf Homework 5 404Hmwk5.ps 404Hmwk5.pdf

## Learning Objectives and Instructor Expectations

The objective of this course (as worded in Bender and Orszag) is to help young and established scientists and engineers to build the skill necessary to analyze equations that they encounter in the real world. Asymptotic and perturbation analysis are some of the most useful and powerful, as well as beautiful, methods for finding approximate solutions to equations. Combining these techniques with those of dynamical systems and computation provide the student with a powerful tool for analyzing most ordinary differential equations.