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.
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.
Partial differential equation (PDE), modeling of the physical phenomena, advection equation, wave equation, diffusion/heat equation, Poisson equation, analytical solution, numerical solution
Intercultural skills | Communication skills | ✔ Specialist skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
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 | |
---|---|---|
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 |
None
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)
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.
Students must have successfully completed "Ordinary Differential Equations and Physical Phenomena" or have equivalent knowledge.