Differential Equations
| Numbering Code | U-SCI00 33146 LJ55 | Year/Term | 2022 ・ First semester | |
|---|---|---|---|---|
| Number of Credits | 2 | Course Type | Lecture | |
| Target Year | 3rd year students or above | Target Student | ||
| Language | Japanese | Day/Period | Thu.2 | |
| Instructor name | SAKAJOU TAKAYUKI (Graduate School of Science Professor) | |||
| Outline and Purpose of the Course | This course covers advanced topics related to ordinary differential equations. They are important mathematical objects that appear in various fields of mathematical science as model equations describing time evolution of natural phenomena. They are also utilized when we solve partial differential equations. Students will learn some of the topics listed in the "Course Plan and Lecture Content" section by focusing on theoretical handling based on functional and complex analysis methods, with appropriate application examples. | |||
| Course Goals |
* Understand the advanced topics from the basic theory of ordinary differential equations * Learn how to construct solutions to differential equations and boundary value problems of Sturm-Liouville type ordinary differential equations * Learn how to deal with ordinary differential equations in the complex plane mathematically |
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| Schedule and Contents |
We put emphasis on the theoretical aspects of ordinary differential equations as listed below, but some applications will be shown as necessary. The course consists of 15 lectures (including feedback) in total. The lecturer will decide the order of lectures for each item and sub-items according to the students' understanding. (a) Existence and uniqueness of solutions for the initial value problem of ordinary differential equations [4-5 lectures] * The space of continuous functions on a compact metric space (Banach space) * Ascoli Arzela's theorem * Construction of a solution using the contraction mapping theorem * Existence of solutions in the space of continuous functions (b) Boundary value problem of second-order linear differential equations (Sturm-Liouville theory) [4-5 lectures] * Self-adjoint operator * Green's function, and compactness of the integral operator defined by the Green's function * Sturm-Liouville eigenvalue problem * Completeness of the sequence of eigen-functions (c) Definite singular point differential equation [4-5 lectures] * Power series expansion * Regular singular point * Frobenius method * Special functions that appear in mathematical physics (Bessel function and Legendre function) * Fuchsian equation |
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| Course Requirements | Prerequisites: Calculus A/B (lecture and practicum), Linear Algebra A/B (lecture and practicum), Calculus Continuation II, Function Theory, and Sets and Phases. | |||
| Study outside of Class (preparation and review) | As the course progresses, we will give appropriate instructions on what to review so that the students can fully understand lecture content. | |||
| Textbooks | Textbooks/References | No particular textbooks will be specified. | ||
| References, etc. |
* Koji Kasahara, 微分方程式の基礎 (数理科学ライブラリー), Asakura Shoten [Basics of Differential Equations, Mathematical Science Library], ISBN: 978-4254114157. * J?rgen Jost, ポストモダン解析学 原著第3版, Maruzen Publishing [Postmodern Analysis (3rd ed.)]. * K?saku Yosida, 積分方程式論, Iwanami Shoten [Lectures on Differential and Integral Equations]. |
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