| Academic Year |
2025Year |
School/Graduate School |
Graduate School of Advanced Science and Engineering (Master's Course) Division of Advanced Science and Engineering Mathematics Program |
| Lecture Code |
WSA71002 |
Subject Classification |
Specialized Education |
| Subject Name |
数学特別講義（反応拡散方程式の基礎理論と伝播現象） |
Subject Name
（Katakana） |
スウガクトクベツコウギ |
Subject Name in
English |
Special Lectures in Mathematics |
| Instructor |
To be announced.,TAKIMOTO KAZUHIRO |
Instructor
(Katakana) |
タントウキョウインミテイ,タキモト カズヒロ |
| Campus |
Higashi-Hiroshima |
Semester/Term |
1st-Year, Second Semester, 3Term |
| Days, Periods, and Classrooms |
(3T) Inte |
| Lesson Style |
Lecture |
Lesson Style
(More Details) |
Face-to-face |
| |
| Credits |
1.0 |
Class Hours/Week |
|
Language of Instruction |
B
:
Japanese／English |
| Course Level |
6
:
Graduate Advanced
|
| Course Area（Area） |
25
:
Science and Technology |
| Course Area（Discipline） |
01
:
Mathematics/Statistics |
| Eligible Students |
Graduate course students |
| Keywords |
Reaction-diffusion system, Dynamical system, Asymptotic behavior, Stability, Traveling wave, Spreading |
| Special Subject for Teacher Education |
|
Special Subject |
|
Class Status
within Educational
Program
(Applicable only to targeted subjects for undergraduate students) | |
|---|
Criterion referenced
Evaluation
(Applicable only to targeted subjects for undergraduate students) | |
|---|
Class Objectives
/Class Outline |
This is an introductory lecture on nonlinear systems of parabolic partial differential equations, specifically reaction-diffusion equations.
First, we will understand the classification of reaction-diffusion systems and the comparison principles related to cooperative and competitive systems. As an application, we will demonstrate the stability of steady-state solutions based on the structure of order-preserving dynamical systems. Next, we will learn the stability analysis of steady-state solutions using the linearized stability principle. To prepare for this, we will study the eigenfunction expansion of the Laplacian, which will help us understand the phenomenon known as Turing instability. Furthermore, we will explore fundamental concepts related to traveling waves and spreading phenomena of the initial value problem for reaction-diffusion equations. |
| Class Schedule |
1. Derivation of the diffusion equation, maximum principle for a single parabolic equation, comparison principles for cooperative and competitive systems
2. Stability of steady-state solutions in reaction-diffusion systems, classification of reaction-diffusion systems
3. Dynamical systems and phase diagrams, limit sets, Lyapunov functions, linear stability principle
4. Eigenfunction expansion, stability of steady-state solutions in nonlinear parabolic equations, stability of steady-state solutions in reaction-diffusion systems and Turing instability
5. Traveling waves in a single parabolic equation, spreading speed and asymptotic behavior |
Text/Reference
Books,etc. |
References (lecture notes) are handed out at every class. |
PC or AV used in
Class,etc. |
Text |
| (More Details) |
|
| Learning techniques to be incorporated |
Post-class Report |
Suggestions on
Preparation and
Review |
Reading the distributed materials in advance or reviewing them afterward will be efficient. Questions about the lecture content are highly encouraged. |
| Requirements |
|
| Grading Method |
Evaluation will be based on the assessment of submitted reports. |
| Practical Experience |
|
| Summary of Practical Experience and Class Contents based on it |
|
| Message |
|
| Other |
|
Please fill in the class improvement questionnaire which is carried out on all classes.
Instructors will reflect on your feedback and utilize the information for improving their teaching. |