This course enhances mathematical knowledge of the ordinary differential equations and the complex function theory through lectures and exercises.
The ordinary differential equations and the complex function theory are required to analyze the subjects in wide mechanical engineering fields and this course covers their fundamentals.
By the end of the course, students will be able to:
1) Explain basic problems of the ordinary differential equations.
2) Solve basic problems of the ordinary differential equations.
3) Explain basic problems of the complex function theory.
4) Solve basic problems of the complex function theory.
Ordinary differential equations, complex function theory
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
Students are given exercise problems and solutions of the problems are reviewed.
Course schedule | Required learning | |
---|---|---|
Class 1 | 1st order ordinary differential equations, n-th order ordinary differential equations | Understand elementary solutions, particular solutions, general solutions and Wronski determinant. |
Class 2 | Differentiation and integral in the complex plane, basics of second-order partial differential equation | Derivation of Cauchy-Riemann equation, relation to elliptic second-order partial differential equation. |
Class 3 | n-th order ordinary differential equations, simultaneous ordinary differential equations | Understand differential operator, variation of parameters and eigenvalue. |
Class 4 | Integral in the complex plane, integral theorem of Cauchy, Taylor and Laurent series expansion | Set-up of the integral path, derivation of the series expansion. |
Class 5 | Solving by series, nonlinear ordinary differential equations | Understand solving by series, perturbation method and Bessel function. |
Class 6 | Residue theorem, Jordan's lemma, Bromwitch integral path | Examples of integrals using the residue theorem, application to Laplace transform. |
Class 7 | Fourier series, Summary of the ordinary differential equations | Understand Fourier series. Summary. |
Class 8 | Multivalued function, Riemann surface, Summary of the complex function theory | Determination of the branches on the Riemann surface. Summary. |
Kentaro Yano and Shigeru Ishihara: Basic Analysis, Shokabo Co., Ltd, .ISBN: 978-4-7853-1079-0. (Japanese)
Ryuichi Watabe, Hiroshi Miyazaki, Shizuo Endo: Complex Function, Baifuukan (1980)
Osamu Takenouchi: Differential Equations and Their Application, Saiensu-sha Co., Ltd. Publishers, ISBN: 4-7819-1060-2. (Japanese)
Students' knowledge of ordinary differential equations and complex function theory will be assessed.
Exam 50%, exercise problems 50%.
Students must have knowledge of the ordinary differential equations and the complex function theory.