Introduction to Applied Mathematics

Math 7382
Louisiana State University
Fall Semester, 2022

Prof. Stephen Shipman

Place: Lockett 381
Time: 10:30-11:50 Tuesday and Thursday

Office: Room 314 of Lockett Hall
Telephone: 225/578-1674
Email: shipman@lsu.edu
Office Hours: Tuesday 9:00-10:30 and Thursday 9:00-10:30; also by appointment.

Course Description

The purpose of this course is to introduce students interested in applied mathematics to the mathematical (differential) equations of physics and the basic techniques that are used. It does not take the place of courses in PDEs or real analysis or Fourier analysis. Instead, it teaches how a breadth of mathematical techniques and subjects bear, often simultaneously, upon the study of applied problems. Material on specific equations of physics, such as those of electromagnetics, fluids, acoustics, quantum mechanics, and other areas, will supplement the main textbook. Topics will come from the following list. Some will be dealt with in more detail than others, but I hope to at least introduce the significance of each.

• Equations
  • Maxwell equations of electromagnetism
  • Navier-Stokes equations of fluids
  • Schrödinger equations of quantum mechanics
  • Many others
• Concepts
  • Elliptic/Parabolic/Hyperbolic classification
  • Boundary-value and initial-value problems
  • Well-posedness of solutions
  • Conservation Laws
  • Maximum principles
  • Linear vs. nonlinear equations
• Techniques
  • Method of characteristics
  • Weak formulations of PDEs in function spaces
  • Distributional solutions
  • Fourier analysis
  • Operator theory
  • Green functions
  • Energy methods

Textbook

Prerequisite

Literature

There is a bibliography of relevant works below.

Assignments and Evaluation

Evaluation will be based on your solutions to problems assigned throughout the semester and a final exam. Assigned problem sets are listed below. The final exam will be worth 20% of the grade.

Students may discuss problems with each other and other people (including me, of course) and consult other literature; in fact students are encouraged to search the literature and discuss ideas. However, all work that is turned in must ultimately be that of the submitter alone. If a student receives aid on an assigned problem from discussions with people or other sources, he or she must begin from scratch in writing the solution so that the result is the product of his or her own understanding alone.

Lecture Notes

Problem Sets

Grading scale (required by LSU)

A+: at least 95%	A: at least 90%	A-: at least 88%
B+: at least 85%	B: at least 80%	B-: at least 78%
C+: at least 75%	C: at least 70%	C-: at least 68%
D+: at least 65%	D: at least 60%	D-: at least 50%
F: less than 50%

Bibliography

1. Michel Cessenat, Mathematical Methods in Electromagnetics: Linear Theory and Applications, World Scientific 1996
2. Excerpts on compact operators:
   Naylor and Sell
   Kress
   Reed and Simon
   Riesz and Sz.-Nagy
   Akhiezer and Glazman, Chapter V; see the link below.
3. Michael Reed and Barry Simon, Methods of Modern Mathematics: Vol. I Functional Analysis, Vol. II Fourier Analysis and Self-Adjointness, Vol. III Scattering Theory, Vol. IV Analysis of Operators , excerpts from Vol. I (spectral theorem), Vol. II (self-adjoint extensions), Academic Press, 1980.
4. Gerald B. Folland, Introduction to Partial Differential Equations, Second Edition, Princeton University Press, 1995. Excerpt: Chapter 0.
5. J. Billingham and A. C. King, Wave Motion, Cambridge Texts in Appl. Math., Camb. Univ. Press, 2000. Excerpts: Chapter 7, KdV images.
6. A. J. Chorin and J. E. Marsden, A Mathematical Introduction to Fluid Mechanics, Texts in Applied Mathematics, 4. Springer.
7. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955. Excerpt: Dependence on initial conditions.
8. Lawrence C. Evans, Partial Differential Equations, AMS Graduate Studies in Math., Vol. 19, 1998.
9. Fritz John, Partial Differential Equations, Springer-Verlag, Appl. Math. Sci. Vol. 1, Fourth Ed., 1982.
10. Carl M. Bender and Steven A. Orszag, Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation Theory, Springer-Verlag, 1999. Excerpt: Lapace Integrals.
11. Richard Haberman, Applied Partial Differential Equations, Fourth Edition, Pearson Prentice Hall, 2004. Excerpts: Separation of Variables, Waves and the Korteweg-deVries Equation.
12. N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space, Dover, 1993, replication of the edition of F. Ungar Publishing, NY, 1961, 1963.