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.
In the second half of the course, we will explain the basic principles of the calculus of variations and discuss advanced topics such as its discretization and variational problems of curves. The discretization of the calculus of variations is important for simulations, while variational problems of curves are relevant to elastic curves and interfacial phenomena, making them significant in applications.

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 principles of the calculus of variations.

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 understand the fundamental properties of the calculus of variations and to be able to perform calculations.
To understand variational problems of curves and be able to perform calculations.

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

	Reports	Total.
1.	25%	25%
2.	25%	25%
3.	36%	36%
4.	14%	14%
Total.	100%	-

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．	Fundamentals of the Calculus of Variations 1	Review undergraduate mathematics (calculus, linear algebra) before class. Review the notes after class.	190minutes
9．	Fundamentals of the Calculus of Variations 2	Review the notes after class.	190minutes
10．	Fundamentals of the Calculus of Variations 3	Review the notes after class.	190minutes
11．	Fundamentals of the Calculus of Variations 4	Review the notes after class.	190minutes
12．	Discretization of the Calculus of Variations	Review the notes after class.	190minutes
13．	Curves and Curvature Variational Problems of Curves 1	Review the notes after class.	190minutes
14．	Variational Problems of Curves 2	Review the notes after class.	190minutes
Total.	-	-	2660minutes

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	N/A

Education related SDGs:the Sustainable Development Goals

9.INDUSTRY, INNOVATION AND INFRASTRUCTURE

Last modified : Wed Sep 24 04:04:12 JST 2025