2023 Students Enrolled in or before 2015 School of Science Mathematics
Advanced courses in Analysis B
- Academic unit or major
- Mathematics
- Instructor(s)
- Kai Koike
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Mon (M-102(H115))
- Class
- -
- Course Code
- ZUA.C332
- Number of credits
- 100
- Course offered
- 2023
- Offered quarter
- 2Q
- Syllabus updated
- Jul 8, 2025
- Language
- English
Syllabus
Course overview and goals
This lecture is an introduction to the mathematical analysis of nonlinear partial differential equations (NLPDEs). NLPDEs are abundant in nature and each of them form a unique landscape. However, there are certain principles underlying many theories. And in this lecture, through analyzing specific NLPDEs, we will learn these ideas.
Note that this lecture is a continuation of Advanced topics in Analysis A1 (MTH.C405).
Course description and aims
To understand important methods and ideas in the mathematical analysis of nonlinear partial differential equations.
Keywords
Nonlinear partial differential equations, Direct method in the calculus of variations, Method of characteristics
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This is a standard lecture course. Problems for reports are given occasionally.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Calculus of variations: The Euler−Lagrange equation | To be able to explain the relation between calculus of variations and the Euler−Lagrange equation. |
| Class 2 | Direct method in the calculus of variations (1) | To be able to apply the direct method to show the existence of solutions to the Euler−Lagrange equation. |
| Class 3 | Direct method in the calculus of variations (2) | Same as above. |
| Class 4 | Inviscid Burgers' equation: Method of characteristics | To be able to use the method of characteristics to solve first-order differential equations. |
| Class 5 | Weak solutions to inviscid Burgers' equation: Shock waves and rarefaction waves | To be able to explain what shock waves and rarefaction waves are. |
| Class 6 | Weak solutions to inviscid Burgers' equation: Entropy condition | To understand the role of the entropy condition in the uniqueness of weak solutions to inviscid Burgers' equation. |
| Class 7 | Epilogue | Other nonlinear PDEs are briefly touched upon to indicate directions for further study. |
| Class 8 | Other topics | Details will be provided in the class. |
Study advice (preparation and review)
To enhance effective learning, by referring to textbooks and other course materials, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class contents afterwards (including assignments) for each class.
Textbook(s)
None required.
Reference books, course materials, etc.
[1] Lawrence C. Evans, Partial Differential Equations, American Mathematical Society, 2010
[2] A. Matsumura and K. Nishihara, Global-in-time Solutions of Nonlinear Differential Equations, Nihon-hyoron-sha, 2004 (Japanese)
[3] Stanley Farlow, Partial Differential Equations Scientists Engineers, Dover Publications, 1993
Evaluation methods and criteria
Evaluation is based on attendance and assignments.
Related courses
- MTH.C405 ： Advanced topics in Analysis A1
Prerequisites
Students are required to take Advanced topics in Analysis A1 (MTH.C405).