Theory of Partial Differential Equations

Course title

6M008700

	YAMAZAWA Hiroshi
	HIROSE Sampei

Purpose of class

First Half: To understand the fundamental principles of q-analysis.
Second Half: To understand the basic properties of the Laplacian (Laplace operator) and its applications.

Course content

This course is co-taught, with Yamazawa leading the first half and Hirose leading the second half.

First Half (q-Differentiace Equations):
The course begins with an introduction to analysis using q-analogs. Topics include Jacobi's triple product and q-difference equations.

Second Half (The Laplacian):
This portion covers topics related to the Laplacian, one of the most critical operators in partial differential equations.
Specifically, we will discuss fundamental matters such as Laplace's equation and harmonic functions, boundary value problems for Poisson's equation, and eigenvalue problems of the Laplacian. Time permitting, we will also explore advanced topics such as the finite element method, discrete Laplacians, and geometry processing.

Goals and objectives

Understand the definitions of operations in q-analysis and perform basic calculations.
Understand methods for solving q-difference equations and solve fundamental problems.
Understand the properties of Laplace's equation and its solutions (harmonic functions).
Understand construction methods for solutions to boundary value problems of Poisson's equation.
Understand the properties of eigenfunctions and eigenvalues in Laplacian eigenvalue problems.
Understand the practical applications of the Laplacian.

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

	Reports	Total.
1.	25%	25%
2.	25%	25%
3.	14%	14%
4.	14%	14%
5.	14%	14%
6.	8%	8%
Total.	100%	-

Language

Japanese

Class schedule

	Class schedule	HW assignments (Including preparation and review of the class.)	Amount of Time Required
1．	q-Analysis and Taylor's Formula	Review course notes	190minutes
2．	Gauss's Binomial Theorem and Heine's Binomial Theorem	Review course notes	190minutes
3．	Jacobi's Triple Product	Review course notes	190minutes
4．	q-Analog Difference Equations and Their Solutions I	Review course notes	190minutes
5．	q-Analog Difference Equations and Their Solutions II	Review course notes	190minutes
6．	q-Analog Difference Equations and Their Solutions III	Review course notes	190minutes
7．	q-Analog Difference Equations and Their Solutions IV	Review course notes	190minutes
8．	Introduction to the Laplacian	Review undergraduate mathematics: Calculus, Linear Algebra, and Functional Analysis before class. Review the lecture content after class.	190minutes
9．	Laplace Equation and Harmonic Functions	Review the lecture content after class.	190minutes
10．	Boundary Value Problems of Poisson Equation I	Review the lecture content after class.	190minutes
11．	Boundary Value Problems of Poisson Equation II	Review the lecture content after class.	190minutes
12．	Eigenvalue Problems of the Laplacian I	Review the lecture content after class.	190minutes
13．	Eigenvalue Problems of the Laplacian II	Review the lecture content after class.	190minutes
14．	Applications of the Laplacian	Review the lecture content after class.	190minutes
Total.	-	-	2660minutes

Evaluation method and criteria

Grading is based on exercises assigned during each class (50 points for the first half, 50 points for the second half).
Criteria: Exercises will consist of calculations and proofs similar to those covered in class.
Passing Grade: A total score of 60 out of 100 is required to pass. A passing grade indicates that the student has understood approximately 60% of the course material.

Feedback on exams, assignments, etc.

ways of feedback	specific contents about "Other"
The Others	Feedback will be provided as appropriate, depending on the research progress and situation.

Textbooks and reference materials

Reference books and literature will be introduced as needed during the lectures.

Prerequisites

Students should review undergraduate-level mathematics, specifically: Calculus (Differential and Integral Calculus), Linear Algebra, and Functional Analysis.

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 Apr 29 10:36:24 JST 2026