Department of Mathematics
Math 188. Topics in Applied Mathematics Fall
2017
Time and Place: MWF 10:00 am - 10:55
am Millikan 2393.
Instructor: Dr.
Adolfo J. Rumbos
Office: Andrew
2287.
Phone/e-mail: ext. 18713 /
Office Hours: TuTh 9:00 am - 10:00 am or by appointment
Texts: Calculus of Variations
(with applications to physics and
Engineering by Robert Weinstock (
Introduction to
Partial Differential Equations
and Hilbert Space
Methods by
Karl E. Gustafson (
Prerequisites: Multivariable Calculus, Linear Algebra
and Differential Equations (Math 102 at
Course
Description. The topic for this year is Variational Methods and Optimization. This course is an introduction to the calculus of variations and
the variational approach in the theory of
differential equations. The calculus of variations is a subject as old as the
Calculus of Newton and Leibniz. It arose
out of the necessity of looking at physical problems in which an optimal
solution is sought; e.g., which configurations of molecules, or paths of
particles, will minimize a physical quantity like the energy or the action? Problems like these are known as variational problems.
Since its beginnings, the calculus of variations has been intimately
connected with the theory of differential equations; in particular, the theory
of boundary value problems. Sometimes a variational problem leads to a differential equation that
can be solved, and this gives the desired optimal solution. On the other hand, variational
methods can be successfully used to find solutions of otherwise intractable
problems in nonlinear partial differential equations. This interplay between the theory of partial
differential equations and the calculus of variations will be one of the major
themes in the course.
Course Requirements.
Reading assignments will be given according to the attached (tentative)
schedule. Problem sets will be assigned
and collected on an alternate basis.
Students are strongly encouraged to work on every assigned problem. Students will also be expected to give a
written and oral formal presentation on some topic of their interest related to
the course material or some suggested one (see attached list of special topics).
Grading Policy. Grades will be based on the homework, two
examinations (see attached schedule), plus a written and oral
presentation. The grades will be
computed as follows:
Homework 20%
Two
exams 50%
Paper
and presentation 30%
Math 188 Fall
2017
Tentative Schedule of Topics,
Presentations and Examinations
Date Topic
W Aug. 30 Soap films and minimal surfaces.
F Sep. 1 Variational problems
M Sep. 4 Variational
problems (continued)
W Sep. 6 Normed
linear spaces
F Sep.
8 Continuous functionals on normed linear spaces
M Sep.
11 Indirect Methods
W Sep. 13 Gateaux derivatives and the first variation
F Sep.
15 The Euler-Lagrange
equations
M Sep. 18 The Euler-Lagrange equations (continued).
W Sep.
20 Examples
F Sep.
22 Problems
M Sep. 25 Convex
functionals.
W Sep.
27 Convex functionals (continued).
F Sep.
29 Minimization of convex
functions
M Oct. 2 Convex minimization theorem
W Oct.
4 Examples
F Oct.
6 Problems
M Oct. 9 Review
W Oct.
11 Exam 1
F Oct. 13 Problems
M Oct.
16 Fall Recess!
W Oct.
18 Direct methods
F Oct.
20 Isoperimetric problems
M Oct. 23 The
variational approach
W Oct.
25 Hilbert space methods
F Oct.
27 The Dirichlet principle
M Oct. 30 Solving
the Dirichlet problem
W Nov.
1 Solving the Dirichlet problem (continued)
F Nov.
3 Problems
Date Topic
M Nov. 6 Eigenvalues of the Laplacian
W Nov.
8 Sturm-Lioville problems
F Nov. 10 Problems
M Nov.
13 Examples
W Nov. 15 Examples
F Nov. 17 Problems
M Nov.
20 Review
W Nov. 22 Exam
2
F Nov. 24 Thanksgiving Recess!
M Nov.
27 Special Topic
W Nov. 29 Special Topic
F Dec. 1 Special
Topic
M Dec. 4 Special Topic
W Dec.
6 Special Topic