Course title
V0320600,Z3320500
Theory of Functional Equations 1

 takeuchi shingo Click to show questionnaire result at 2018
Course description
I will explain the theory of linear ordinary differential equations with variable coefficients. In "Differential Equations" we dealt with the case of constant coefficients, but the theory becomes more general and more applicable if the equations are allowed to have variable coefficients. First, I will describe the uniform convergence and the series expansion of functions, and show the existence and uniqueness theorem of initial value problems. After that, we understand an important role played by fundamental solutions for seeking general solutions of (in)homogeneous differential equations. Next, I will introduce the method of series expansion to obtain the fundamental solutions. Finally, we will discuss the Bessel differential equation, which is important in applications. The students should note that this class is more theoretic than "Differential Equations" because this class is the "theory" of functional equations.
Purpose of class
Differential equations are necessary to model phenomena on the basis of laws in various sciences. In particular, they are often reduced to linear ordinary differential equations for simplicity. This class is the sequel period for "Differential Equations" and the students will learn the theory of linear ordinary differential equations if they take this class after that class. In this area, we frequently use differential/integral calculus, but the theory is in linear algebra. Therefore, the students can learn how to harmonize differential/integral calculus and linear algebra, not to mention differential equations.
Goals and objectives
  1. To understand the importance of the existence and uniqueness theorem.
  2. To understand the structure of set of solutions of linear ordinary differential equations with variable coefficients.
  3. To understand the method of series expansion.
Language
Japanese
Class schedule

 Class schedule HW assignments (Including preparation and review of the class.) Amount of Time Required
1. Preparation (1) what is the theory of differential equations Review "Differential Equations" in advance. 190minutes
2. Preparation (2) uniform convergence, series expansion Review "Mathematics I" and "Fundamental Analysis" in advance. 190minutes
3. Existece and uniqueness for initial value problem Review "Differential Equations" in advance. 190minutes
4. General solutions of homogeneous differential equations (1) Wronskian Review the case of constant coefficients in advance. 190minutes
5. General solutions of homogeneous differential equations (2) fundamental solutions Understand with "Linear Spaces" in advance. 190minutes
6. General solutions of homogeneous differential equations (3) examples Review Sessions 3 and 4 in advance. 190minutes
7. General solutions of inhomogeneous differential equations: variation of constants Review the case of constant coeffients in advance. 190minutes
8. Mid-term exam Review Sessions 1-7 in advance. 190minutes
9. The method of series expansion (1) regular point Review Session 2 in advance. 190minutes
10. The method of series expansion (2) Legendre's differential equation Review Session 9 in advance. 190minutes
11. The method of series expansion (3) regular singular point Review Sessions 9-10 in advance. 190minutes
12. The method of series expansion (4) Bessel's function Review Session 11 in advance. 190minutes
13. The method of matrix Review Session 12 in advance. 190minutes
14. Final exam Review Sessions 9-13 in advance. 190minutes
Total. - - 2660minutes
Relationship between 'Goals and Objectives' and 'Course Outcomes'

 Mid-term exam Final exam Total.
1. 25% 25% 50%
2. 25% 25%
3. 25% 25%
Total. 50% 50% -
Evaluation method and criteria
Mied-term exam and Final exam.
Textbooks and reference materials
No textbook is necessary, but as references, I will recommand the textbook which you have used in the first grade or Chapter 1 in Takasi Kusano, "KYOUKAICHIMONDAI NYUUMON", Asakura Shoten.
Prerequisites
"Differential Equations", you have learned in the first grade.
Office hours and How to contact professors for questions
  • Lunchtime on every Tuesday.
Relation to the environment
Environment-related course ( 5%)
Regionally-oriented
Non-regionally-oriented course
Development of social and professional independence
  • Non-social and professional independence development course
Active-learning course
More than one class is interactive
Last modified : Wed Oct 17 07:33:01 JST 2018