Introduction to Nonlinear Analysis
| Numbering Code | U-SCI00 22108 LJ55 | Year/Term | 2022 ・ Second semester | |
|---|---|---|---|---|
| Number of Credits | 2 | Course Type | Lecture | |
| Target Year | 2nd year students or above | Target Student | ||
| Language | Japanese | Day/Period | Thu.2 | |
| Instructor name | ISHIMOTO KENTA (Research Institute for Mathematical Sciences Associate Professor) | |||
| Outline and Purpose of the Course |
When attempting to understand various phenomena, it is often helpful to extract specific parameters of particular interest among the numerous factors involved and describe their relationship mathematically. This method, called "mathematical modeling," has produced many of the differential equations available to us. Generally, derived differential equations are nonlinear, and there is no general theory to obtain their exact solutions; thus, analyzing them requires understanding the features of their models well. In this course, we will discuss concrete phenomena beginning with introductory content regarding the derivation of mathematical models for these phenomena and nonlinear analysis methods for the obtained equations. For each target phenomenon, the course will proceed as follows: (1) Derivation of the model (2) Mathematical theories (definitions, theories, applications thereof, etc.) required for the analysis of the model This process will deepen our understanding of both the phenomena and associated mathematical concepts. When we introduce numerical results and incorporate problems requiring numerical calculations, we will encourage the use of calculation devices. |
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| Course Goals | Students will learn techniques used to construct mathematical models, how they help us understand various phenomena, related concepts, and the mathematical techniques necessary for their analysis. You will do this by learning about the mathematical structure behind the derivation of mathematical models for specific phenomena and the techniques necessary for their analysis. Further, you will learn how to apply this knowledge when analyzing specific problems. | |||
| Schedule and Contents |
We will cover six items from mathematical models used to describe the following phenomena. There will be 15 lectures (including feedback) on this material. The items covered in the lectures might differ slightly from the content listed here at the lecturer's discretion. (a) Models in electrical mechanics (1-2 lectures): Equation of motion of simple pendulum: Energy conservation law, correspondence between solution and pendulum motion (equilibrium point, periodic orbit, homoclinic orbit) (b) Engineering models (about 2 lectures): Van der Pol equation describing the state of a simple electric circuit: existence and uniqueness of periodic solutions (c) Models that appear in biology (about 2 lectures): Logistic equations and Lotka-Volterra equations derived from problems in population ecology: equilibrium point stability vs. linearization stability, phase plane analysis (d) Meteorological models (about 2 to 3 lectures): Lorenz equations derived from the thermal convection problem: bifurcation of the equilibrium point, the concept of chaos (e) Models that appear in nonlinear waves (about 2-3 lectures): The Korteweg-De Vries (KdV) equation and the Burgers equation describing nonlinear waves: soliton solution, shock wave solution, traveling wave solution, Cole-Hopf transformation (f) Model for fluid phenomena (about 3-4 lectures): ・Two‐dimensional incompressible Euler equations describing the flow around an airfoil: functional theory, harmonic equation, potential flow, wing theory ・Two‐dimensional Stokes equations describing the motion of microorganisms in fluids: functional theory, multiple harmonic equations, and motion of microorganisms (g) Variation model (around 3-4 lectures): Optimization problems such as minimal surfaces: Euler-Lagrange equation, necessary and sufficient conditions for extreme value, constrained minimization (h) Models of pattern formation in biology and materials science (about 3-4 lectures): Random walks and diffusion, Fourier-series solutions, nonlinear reaction-diffusion equation, stability of the steady-state solution |
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| Course Requirements | The lecturers will assume that students have a working knowledge of the concepts and techniques covered in the following classes: Calculus A/B; Linear Algebra A/B, and Calculus II (Faculty of Science, First Semester). Taking Function Theory of a Complex Variable will also help students' understanding. | |||
| Study outside of Class (preparation and review) | We encourage students to review the lecture content. | |||
| Textbooks | Textbooks/References | No textbook will be used | ||
| References, etc. | Introduced in classes | |||
