The course teaches the fundamentals of mathematical analysis of partial differential equations modeling various phenomena.
Students learn the derivation of the partial differential equations and the method of the Fourier series.
Students will be able to apply them to various problems.
By completing this course, students will be able to;
1) derive the partial differential equations as the models of various phenomena.
2) understand the theory of Fourier series and solve the partial differential equations.
3) understand properties of the solutions by using the character of each equations.
partial differential equations, heat equation, wave equation, Laplace equation, Fourier series
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
The lecture is devoted to the fundamentals to the derivation and the resolution of partial differential equations. In order to cultivate
a better understanding, some exercises are given.
Course schedule | Required learning | |
---|---|---|
Class 1 | Some examples of partial differential equaitons | Understand the contents in the lecture. |
Class 2 | Derivation of the heat equation | Understand the contents in the lecture. |
Class 3 | Maximum principles and their applications | Understand the contents in the lecture. |
Class 4 | Fourier’s method and separation of variables | Understand the contents in the lecture. |
Class 5 | Theory of Fourier series | Understand the contents in the lecture. |
Class 6 | Hilbert spaces and complete orthonormal system | Understand the contents in the lecture. |
Class 7 | Completeness of Fourier series | Understand the contents in the lecture. |
Class 8 | Solving heat equations via Fourier series | Understand the contents in the lecture. |
Class 9 | Derivation of the wave equation | Understand the contents in the lecture. |
Class 10 | Energy conservation law and its applications | Understand the contents in the lecture. |
Class 11 | d'Alembert 's solution for the wave equation | Understand the contents in the lecture. |
Class 12 | Solving the wave equation via Fourier series | Understand the contents in the lecture. |
Class 13 | Derivation of the Laplace equation | Understand the contents in the lecture. |
Class 14 | Solving the Laplace equation | Understand the contents in the lecture. |
Class 15 | Mean value theorem and its applications | Understand the contents in the lecture. |
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.
Partial differential equations (Science library), Yoshio Kato (Japanese)
Unspecified
Learning achievement is evaluated by a final exam and so on.
No prerequisites. Students should understand the fundamentals of calculus.