2018 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  Varquez Alvin Christopher 
Course component(s)
Lecture / Exercise
Day/Period(Room No.)
Mon7-8(S513)  Thr7-8(S513)  
Group
-
Course number
TSE.M202
Credits
2
Academic year
2018
Offered quarter
3Q
Syllabus updated
2018/9/14
Lecture notes updated
2018/11/15
Language used
English
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

Intercultural skills Communication skills Specialist 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 Definition and categories 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
Class 4 Analytical solution of the hyperbolic PDE (Fourier Series) Learn the method of analytical solution of the hyperbolic PDE
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 the physical modeling with the elliptic PDE
Class 8 Test level of understanding with exercise problems for the first part of the course - Solve exercise problems covering the contents of classes 1–7. Test level of understanding of classes 1–7.
Class 9 Numerical solution of PDE Learn how to solve PDE using numerical methods
Class 10 Theory of numerical solution of PDE - Parabolic equations Learn the numerical solution of parabolic PDE based on Finite Difference Method
Class 11 Practice of numerical solution of PDE - Parabolic equations Implement a program to solve the parabolic PDE
Class 12 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 13 Practice of numerical solution of PDE - Combined parabolic and hyperbolic equations Implement a program to solve the combined parabolic and hyperbolic PDE
Class 14 Theory of numerical solution of PDE - Hyperbolic equations Learn the numerical solution of hyperbolic PDE using multiple approaches
Class 15 Practice of numerical solution of PDE - Hyperbolic equations Implement a program to solve the hyperbolic PDE

Textbook(s)

None

Reference books, course materials, etc.

Advanced engineering mathematics, Erwin Kreiszig, John Wiley & Sons.
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)

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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