The main topic of this course is nonlinear differential equation. Other topics cover existence and uniqueness of initial value problem, continuous dependence on parameters and theory of finite dimensional dynamical system. This course is a continuation of Differential Equations I.
Differential equations are fundamental notions appearing in all fields of mathematics. Space of solutions have algebraic structure, existence theorems of solutions give various geometric and analytic objects of great interests. This course is an entry to these paths.
The main topic of this course is a basic theory and its applications of ordinary differential equations of one unknown variable. Ordinary differential equations describe various natural phenomena and physical laws, thus, method of solving equations and its theory are important mathematically as well as for applications. Method of solving ordinary differential equation and theory for analyzing qualitative properties of solutions are to be discussed. Applications to science and engineering will be also discussed.
nonlinear ordinary differential equation, existence and uniqueness of solution, stability
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
Standard course lecture.
|Course schedule||Required learning|
|Class 1||nonlinear ordinary differential equation||Details will be provided during each class session|
|Class 2||existence and uniqueness of solution||Details will be provided during each class session|
|Class 3||dependence of solution on initial value and parameter||Details will be provided during each class session|
|Class 4||vector field and its flow||Details will be provided during each class session|
|Class 5||stability||Details will be provided during each class session|
|Class 6||phase space||Details will be provided during each class session|
|Class 7||Hamilton flow, gradient flow||Details will be provided during each class session|
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Introduction to differential equations, Koji Kasahara, Asakura Shoten (Japanese)
Earl A. Coddington and Norman Levinson, ``Theory of ordinary differential equations'', McGraw-Hill (1955)
Evaluation based on mid-term and final exam. Details will be provided in the class.
Students are expected to have passed Differential Equations I.