2020/2021

Research Seminar "Ordinary Differential Equations"

Type: Optional course (faculty)

Delivered by: Faculty of Mathematics

Where: Faculty of Mathematics

When: 3, 4 module

Instructors: Takashi Takebe

Language: English

ECTS credits: 6

Contact hours: 72

Course Syllabus

Abstract

Ordinary differential equations are ubiquitous in sciences. They are convenient tools for describing scientific laws. Of course they are important in various branches in mathematics. We discuss their basic examples, solving methods, fundamental properties and several topics in applications (for example, integrable systems, dynamical systems, etc.).

Learning Objectives

• Examples of ordinary differential equations.
• Local construction of solutions of initial value problems
• Extension of local solutions
• Ordinary differential equations with boundary conditions
• Solving ODE by quadrature
• Linear ordinary differential equations
• Phase space and vector fields
• Liouville integrability and symmetries
• Dynamical systems: basics.

Expected Learning Outcomes

• Acquaintance with examples of ODE and their solutions. Understanding of necessity of general theories.
• Understanding of existence theorems and their differences.
• Understanding of boundary value problems, their equivalence with integral equations, integral operators and resolvents.
• Ability of solving simple ODE by quadrature.
• Understanding of general theory of linear ODE. Ability of solving linear ODE with constant coefficients (homogeneous/inhomogeneous).
• Acquaintance with formalisms (Lagrangian/Hamiltonian) of mechanics, variational problems and canonical transformations.
• Acquaintance with notion of vector fields, dynamical systems. Understanding of theorems on behaviours of orbits around fixed points.

Course Contents

• Local construction of solutions of initial value problems
  Initial value problems in analytic category (Cauchy's theorem), in Lipschitz continuous case (Picard's theorem) and in continuous case (Peano's theorem). Extensions of local solutions. Dependence on parameters.
• Examples of ordinary differential equations
  Simple examples of ODE and their solving methods. Motivation to general theories.
• Ordinary differential equations with boundary conditions.
  Sturm-Liouville problem. Dirichlet and Neumann conditions. Integral operator. Resolvent and Green function.
• Solving ODE by quadrature
  Method of separation of variables. Exact differential equation. Examples (d'Alembert's equation, Clairaut's equation).
• Linear ordinary differential equations
  Solution space and superposition principle. Linear independence and Wronskian. Homogeneous equation with constant coefficients. Inhomogeneous equation. Method of variation of constants.
• Dynamical systems: basics.
  Dynamical systems. Vector fields and one-parameter transformation groups. Fixed points and asymptotic behaviour of orbits around them.
• Analytical mechanics: basics
  Lagrangian formalism. Principle of least action. Hamiltonian formalism. Phase space. Canonical transformations. Liouville integrability.

Assessment Elements

• homeworks
• short tests

Interim Assessment

• Interim assessment (4 module)
  0.5 * homeworks + 0.5 * short tests

Bibliography

Recommended Core Bibliography

• A course in ordinary differential equations, Swift, R. J., 2007
• Gorain, G. C. (2014). Introductory Course on Differential Equations. New Delhi: Alpha Science Internation Limited. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=1878058
• Ordinary differential equations : introduction and qualitative theory, Cronin, J., 2008

Recommended Additional Bibliography

• llyashenko, Y., & Yakovenko, S. (2008). Lectures on Analytic Differential Equations. Providence, Rhode Island: AMS. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=971274
• Theory and examples of ordinary differential equations, Lin, C.- Y., 2011