Essential Mathematics for Global LeadersU[17S1007]

科目名
Course Title
Essential Mathematics for Global LeadersU[17S1007]
Essential Mathematics for Global LeadersU
科目区分・科目種 共通科目(前期課程) クラス 博士課程共通
CCBM   キャリアデザイン  
単位数 2.0単位 履修年次 13

担当教員 DAHAN Xavier
学期 後集中

授業の形態
講義,演習

教科書・参考文献
a) Dynamical Systems with applications using Mathematica (S. Lynch, Birkhauser 2007)
b) Introduction to Partial Differential Equations for Scientists and Engineers Using Mathematica (Kuzman Adzievski, Abul Hasan Siddiqi. Taylor & Francis, 2013)

評価方法・評価割合
その他=Short Report, Attendance

主題と目標
heme: Differential Equations (in the broader sense of dynamical systems) are the core topics in mathematical modeling.
数理モデルにおいて最も利用される「微分方程式」(広い意味で、力学系)
Objective: Through examples in Mathematica to understand: Mathematicaでの例を通じて以下のことを理解すること:
- what are Ordinary, Partial Differential Equations (ODE, PDE)
ODEとPDEは何か、どうやって扱うか。
- some methods of resolution: closed forms, Series Solutions, Fourier & Laplace Transforms…
代表的な解法:閉形式解、級数解、フーリエとラプラス変換など)
- how to use Mathematica to solve and visualize solutions.
Mathematicaを使って解き、解を視覚化する.

授業計画
The following topics will be introduced, (maybe not exactly in this order !). 以下の科目(必ずこの順番に沿うとかぎらない!)を紹介する。
All topics are illustrated with Examples and some small projects in Mathematica.
科目をすべて例もしくはMathematicaでの小プロジェクトを挙げて説明する。

0. Introduction to Mathematica. Mathematica入門

1. Continuous differential systems. 連続常微分方程式
1.1 Planar linear systems: phase portrait. Linear systems.
平面線形微分方程式:位相平面上で解軌道を見る。線形系。
1.2 Planar non-linear system: Stability, Linearization (Hartman theorem).
平面非線形微分方程式:線形化、安定性(Hartman theorem).
1.3 Planar non-linear systems II. Limit Cycles. Poincare theorem. 
平面非線形微分方程式 II: 極限円。ポアンカレ定理。
1.4 Three-dimensional non-linear systems: Notion of Chaos. Lorentz’s strange attractor.
   3次元非線形微分方程式:ローレズストレンジアトラクターとキャオスの概念。

2. Discrete differential systems (finite-difference systems). 差分方程式。
2.1 Linear recurrence relations. Leslie Model. 線形漸化式
2.2 Non-linear examples: logistic model. Logistic and Henon map. 非線型の例:ロジスティックモデル。

3. Partial Differential Equations. 偏微分方程式。
3.1 Introduction & classification. Boundary conditions. 序論、分類。境界条件。
3.2 Hyperbolic PDE: Wave equations, vibrating string. 双曲型偏微分方程式:波動方程式
3.3. Parabolic PDE: Heat equation, diffusion problem. 放物方偏微分方程式:熱方程式
3.4 Elliptic PDE: Laplace equations. 楕円型偏微分方程式:ラプラス方程式

学生へのメッセージ
“As part of the Essential course series, Essential Maths I (Statistics) and II (modeling ODE and PDE) are supposed to endow/increase capability to model concrete problems with mathematical equations. Essential Math II focuses on the use of a math software, and through its visualization functionality to learn/put into practice basic methods of resolution. Mathematical notions will be introduced formally, but main theorems will be stated in a concrete way. Most proofs will be omitted, in particular only basic notions of Calculus and of Linear Algebra are expected for this course.