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2020　Partial Differential Equations for Science and Engineering

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Academic unit or major
Undergraduate major in Transdisciplinary Science and Engineering
Instructor(s)
Kinouchi Tsuyoshi  Nakamura Takashi  Tsutsui Hiroaki  Nakamura Takashi 
Class Format
Lecture / Exercise    (ZOOM)
Media-enhanced courses
Day/Period(Room No.)
Mon7-8(W351)  Thr7-8(W351)  
Group
-
Course number
TSE.M202
Credits
2
Academic year
2020
Offered quarter
3Q
Syllabus updated
2020/9/18
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

This course focuses on the fundamental mathematics of the partial differential equation (PDE) and methods to solve the PDE. Topics include "description of physical problems with PDE", "features of PDE", "typical method to get an analytical solution of PDE" and "numerical method to solve PDE". In the classes concerning "numerical methods", computer practices are scheduled.
PDE plays an important role as a common language to describe and solve various physical problems. It must be useful to acquire the skill to describe physical problems with PDE and to understand physical phenomena using PDE.

Student learning outcomes

By the end of this course, students will be able to:
I. Describe typical physical problems in PDE.
II. Understand physical phenomena from PDE.
III. Solve various PDE by analytical method.
IV. Solve various PDE by numerical method.

Keywords

Partial differential equation (PDE), modeling of the physical phenomena, advection equation, wave equation, diffusion/heat equation, Poisson equation, analytical solution, numerical solution

Competencies that will be developed

✔ Specialist skills Intercultural skills Communication skills Critical thinking skills ✔ Practical and/or problem-solving skills

Class flow

In the first half part of this course, theoretical aspects of PDE are introduced. In the latter half part, numerical methods to solve PDE and computer practices are discussed.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Introduction of partial differential equation Review mathematical definition and laws of partial difference. Learn types or categories of partial differential equations.
Class 2 Modeling of flow phenomena with the hyperbolic PDE Learn the modeling of advection (transport) phenomena due to flow
Class 3 Analytical solution of the hyperbolic PDE (D'Alembert Solution) Learn the method of analytical solution of the hyperbolic PDE using the D'Alembert Solution
Class 4 Analytical solution of the hyperbolic PDE (Fourier Series) Learn the method of analytical solution of the hyperbolic PDE using the Fourier Series
Class 5 Modeling of diffusion phenomena with the parabolic PDE Learn the modeling of diffusion phenomena
Class 6 Analytical solution of the parabolic PDE Learn the method of analytical solution of the parabolic PDE
Class 7 Modeling of a steady state with the elliptic PDE and its analytical solution Learn physical modeling of an elliptic PDE
Class 8 Numerical solution of PDE Learn how to solve PDE using numerical methods
Class 9 Theory of numerical solution of PDE - Parabolic equations & elliptic equations equations Learn the numerical solution of parabolic PDE based on Finite Difference Method
Class 10 Practice of numerical solution of PDE - Parabolic equations & elliptic equations Implement a program to solve the parabolic and elliptic PDE
Class 11 Theory of numerical solution of PDE - Combined parabolic and hyperbolic equations Learn the numerical solution of a combined parabolic and hyperbolic PDE based on Finite Difference Method
Class 12 Practice of numerical solution of PDE - Combined parabolic and hyperbolic equations Implement a program to solve the combined parabolic and hyperbolic PDE
Class 13 Theory of numerical solution of PDE - Hyperbolic equations Learn the numerical solution of hyperbolic PDE using multiple approaches
Class 14 Practice of numerical solution of PDE - Hyperbolic equations Implement a program to solve the hyperbolic PDE

Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.

Textbook(s)

Advanced engineering mathematics, Erwin Kreiszig, John Wiley & Sons.
「キーポイント 偏微分方程式」
https://www.iwanami.co.jp/book/b260895.html

Reference books, course materials, etc.

Advanced engineering mathematics, Erwin Kreiszig, John Wiley & Sons. (English)
登坂宣好、大西和栄、偏微分方程式の数値シミュレーション、東京大学出版会 (Japanese)
越塚誠一、数値流体力学、培風館 (Japanese)
https://docs.python.org/3/ for python.

Assessment criteria and methods

Students' knowledge of "description of physical phenomena with PDE", "analytical solution of PDE", "numerical solution of PDE", and their ability to apply them to problems will be assessed. The first half part of class 1-7 is evaluated through a midterm exam and exercise problems, the later part of class 9-15 is assessed by an end-of-term report.

Related courses

  • TSE.M201 ： Ordinary Differential Equations and Physical Phenomena

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students must have successfully completed "Ordinary Differential Equations and Physical Phenomena" or have equivalent knowledge.

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