| Course title | |||||
| 物理数学Ⅱおよび演習 [Fundamentals of Partial Differential Equations in Physics] | |||||
| Course category | technology speciality courses,ets. | Requirement | Credit | 2 | |
| Department | Year | 2～4 | Semester | Fall | |
| Course type | Fall | Course code | 022524 | ||
| Instructor(s) | |||||
| 池上 貴志 [IKEGAMI Takashi] | |||||
| Facility affiliation | Graduate School of Bio-Applications and Systems Engineering | Office | Email address | ||
| Course description |
|
Objectives: Basic subjects such as mechanics learned at the Faculty of Engineering, we can see many expressions using differential equations as a method of describing the phenomena. In the course of "physical mathematics I and practice," mainly vector analysis was learned. However, here, to change a viewpoint, we will use a method to stress a physical interpretation of mathematical formula, discussing the characteristic of the differential equations to describe physical phenomena. Especially, we will emphasize the characteristic of partial differential equations. Resume: As to the second order, linear differential equations, choosing wave and heat conduction equations as examples, which are basics in physics, analytical solving methods and numerical solving ones will be learned. Especially, classification of standard forms by the change of variables, separation method of variables as a powerful analytical solving method, approach to the inhomogeneous equation, and difference approximation, which is a numerical solving method as a final means etc. will be acquired. |
| Expected Learning |
| Physical mathematics is a base of the special field of study at the Faculty of Engineering. In this lecture, we aim at making students understand main partial differential equations describing physical phenomena. |
| Course schedule |
|
(Course of knowledge-based design) First Lecture - Introduction and ordinary/partial differential equations: Examples of partial differential equations and review of ordinary differential equation. Then, we will explain what partial differential equations are. Second Lecture - First order partial differential equation (I): As to first order partial differential equation, general solutions using such as total differential and Lagrange equations will be explained. Third Lecture - First order partial differential equation (II): The later part of the first order partial differential equation will be explained. Fourth Lecture - Classification of second order, linear partial differential equations: Classification of second, linear partial differential equations, hyperbolic, parabolic, and elliptic types, transformation to standard form using the change of variables will be explained. Fifth Lecture - Derivation of second order, linear partial differential equations in physics: Partial differential equations will be derived. Deriving wave and heat conduction equations from real physical phenomena, their characteristics will be investigated. Sixth Lecture - Variable separation method: Applying a variable transformation method to a couple of the basic second order, linear partial differential equations, transformation to a few ordinary differential equations will be learned. Seventh Lecture - Review and exercise of the first half lecture: Eighth Lecture - Two-dimensional wave equation: Wave equation as a partial differential equation will be solved by the use of separation of variables. Bessel's differential equation and expansion of orthogonal functions will be learned. In addition, we will make students understand superposition of solutions satisfying boundary conditions. Ninth Lecture - Laplace equation (I): As a steady state solution, Laplace equation will be treated. We will also investigate the characteristic of Laplace equation. Tenth Lecture - Laplace equation (II): The second half part of solution of Laplace equation will be given in our lecture. Eleventh Lecture - Eigenfunction expansion method: This method will be introduced in the case of inhomogeneous differential equation, and also variation of parameters will be learned. Twelfth Lecture - Fourier transform: This transform and its solving method will be explained briefly. Thirteenth Lecture - Special functions: The functions such as Delta, Gamma, and Bessel functions frequently appear in treating partial differential equations will be reviewed. Fourteenth Lecture - Supplemental lecture: Fifteenth Lecture - Review and exercise of the second half lecture: |
| Prerequisites |
| It is better to understand the contents of basic subjects such as related mathematics and physics etc. |
| Required Text(s) and Materials |
|
(Course of knowledge-based design) Tetsuya Kawamura, "Key point: partial differential equation," Iwanami Shoten. |
| References |
| Assessment/Grading |
|
Ten times of quiz and reports in class (33%) Midterm examination (33%) Final examination (34%) |
| Message from instructor(s) |
| Although it is better to remember mathematical formulae and rules etc., doing your utmost to clear their physical images is desirable, not sticking them. It is enough that to know the place in books to find the formulae etc. |
| Course keywords |
| Differential equation, separation of variables, expansion of eigen function, finite difference, special function. |
| Office hours |
| Just after the lecture or appointment determination at that time. |
| Remarks 1 |
| Remarks 2 |
| Related URL |
| Lecture Language |
| Japanese |
| Language Subject |
| Last update |
| 3/2/2018 1:30:55 PM |