Theory of Partial Differential Equations

Course title

6M008700

	YAMAZAWA Hiroshi
	HIROSE Sampei

Course content

In this class, the first half will be given by Yamazawa and the second half by Hirose.
The first half of the class will cover introductory topics on Painlevé equations, one of the most important differential equations in the study of differential equations. Bäcklund transform, symmetric form, and τ (tau) function will be explained.
The second half of the class will focus on curves, explaining variational methods. These are related to elastic curves and interfacial phenomena, and are also important in applications. In addition, discretization, which is important for simulation, will be discussed.

Purpose of class

The purpose of the first half of the class is to understand the fundamentals of the Painlevé equation.
The second half of the class aims to understand the basic properties of variation of curves.

Goals and objectives

To be able to understand the definition of the Bäcklund transform and the Bäcklund transform by the symmetric form of the Painlevé equation.
To be able to understand the relationship between the τ (tau) function and the Painlevé equation, and the Bäcklund transform of the τ function.
To be able to understand the basic properties of curve and its curvature
To be able to understand the basic properties of variational methods.
To be able to understand the relationship between variation of curves and differential equations.

Language

Japanese

Class schedule

	Class schedule	HW assignments (Including preparation and review of the class.)	Amount of Time Required
1．	Bäcklund transform 1	Review of class notes.	190minutes
2．	Bäcklund transform 2	Review of class notes.	190minutes
3．	Symmetric form 1	Review of class notes.	190minutes
4．	Symmetric form 2	Review of class notes.	190minutes
5．	Tau function 1	Review of class notes.	190minutes
6．	Tau function 2	Review of class notes.	190minutes
7．	Tau function 3	Review of class notes.	190minutes
8．	Curve and its curvature 1	Review undergraduate mathematics (calculus, linear algebra) before class. Review the notes after class.	190minutes
9．	Curve and its curvature 2	Review the notes after class.	190minutes
10．	Fundamentals of variational methods 1	Review the notes after class.	190minutes
11．	Fundamentals of variational methods 2	Review the notes after class.	190minutes
12．	Variation of curves and differential equations 1	Review the notes after class.	190minutes
13．	Variation of curves and differential equations 2	Review the notes after class.	190minutes
14．	Discretization	Review the notes after class.	190minutes
Total.	-	-	2660minutes

Relationship between 'Goals and Objectives' and 'Course Outcomes'

	Reports	Total.
1.	25%	25%
2.	25%	25%
3.	17%	17%
4.	17%	17%
5.	16%	16%
Total.	100%	-

Evaluation method and criteria

Exercises (approximately 7 points each) will be assigned during each class. The exercises should be calculations and proofs that can be done in the same way as in class.

Feedback on exams, assignments, etc.

ways of feedback	specific contents about "Other"
The Others	状況に応じてフィードバックを行う

Textbooks and reference materials

Reference books and bibliography will be introduced in class as needed.

Prerequisites

Review of undergraduate mathematics (calculus, linear algebra).

Office hours and How to contact professors for questions

30 minutes after class

Regionally-oriented

Non-regionally-oriented course

Development of social and professional independence

Course that cultivates an ability for utilizing knowledge

Active-learning course

N/A

Course by professor with work experience

Work experience	Work experience and relevance to the course content if applicable
N/A

Education related SDGs:the Sustainable Development Goals

9.INDUSTRY, INNOVATION AND INFRASTRUCTURE

Last modified : Sat Sep 09 05:54:55 JST 2023