2021 | Mathematics for Systems and Control B

Undergraduate
Graduate

2021　Mathematics for Systems and Control B

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Academic unit or major: Undergraduate major in Systems and Control Engineering

Instructor(s): Kosaka Hidenori

Class Format: Lecture / Exercise

Media-enhanced courses

Day/Period(Room No.): Mon5-7(W331) Thr5-7(W331)

Group: -

Course number: SCE.A202

Credits: 3

Academic year: 2021

Offered quarter: 3Q

Syllabus updated: 2021/3/19

Lecture notes updated: -

Language used: Japanese

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Syllabus

Course description and aims

This course focuses on the solving methods of partial differential equations. Firstly, the classification of partial differential equations and their derivation are presented. Secondly, the separation variables, eigenfunction expansion, Fourier transform, and Laplace transform are presented as the solving methods of parabolic partial differential equation. By combining lectures and exercises, the course enables students to understand and acquire the fundamentals of mathematical tools widely applicable to analysis of physical phenomena and design of control systems.

Partial differential equations are used for mathematical expression of various physical phenomena such as electromagnetic field, thermo-flow field, and also for design of control systems. In this course, the solving methods of partial differential equations are explained from the viewpoint of applying them to the engineering problems rather than mathematical rigorousness. The 1-D heat transfer in a rod is taken as an example of solving the partial equation. Students will understand and acquire the fundamentals on solving methods of parabolic partial differential equations for the application of them to engineering issues.

Student learning outcomes

By the end of this course, students will be able to:
1) Explain the basic classification and properties of partial differential equations.
2) Apply separation variables to solve simple parabolic partial differential equations.
3) Apply eigenfunction expansion to solve simple parabolic partial differential equations.
4) Apply Fourier transform to solve simple parabolic partial differential equations.
5) Apply Laplace transform to solve simple parabolic partial differential equations.

Keywords

parabolic partial differential equation, separation variables, eigenfunction expansion, Fourier transform, Laplace transform

Competencies that will be developed

✔ Specialist skills

Intercultural skills

Communication skills

Critical thinking skills

✔ Practical and/or problem-solving skills

Class flow

At the beginning of each class, solutions to exercise problems that were assigned during the previous class are reviewed. Towards the end of class, students are given exercise problems related to the lecture given that day to solve. To prepare for class, students should read the course schedule section and check what topics will be covered. Required learning should be completed outside of the classroom for preparation and review purposes.

Course schedule/Required learning

	Course schedule	Required learning
Class 1	Classification and properties of partial differential equations	Understand the properties of parabolic, hyperbolic, and elliptic partial differential equations
Class 2	Derivation of partial differential equations (Heat equation)	Understand the expression of physical phenomena by partial differential equations
Class 3	Boundary and initial conditions	Understand the properties of boundary and initial conditions
Class 4	Separation variables method	Apply separation variables method for solving parabolic partial differential equation
Class 5	Nonhomogeneous boundary conditions	Transformation of nonhomogeneous boundary condition to homogeneous one
Class 6	Eigenvalue	Understand Sturm-Liouville problem
Class 7	Transformation of partial differential equation	Understand the transform of partial differential equation to use the separation variables method
Class 8	Nonhomogeneous partial differential equation	Apply eigenfunction expansion method for solving parabolic partial differential equation
Class 9	Integral transform	Understand sine and cosine transforms of partial differential equation
Class 10	Fourier series and Fourier transform	Understand the properties of Fourier series and transform
Class 11	Fourier transform	Apply Fourier transform for solving parabolic partial differential equation
Class 12	Laplace transform	Apply Laplace transform for solving parabolic partial differential equation
Class 13	Duhamel’s principle	Understand Duhamel’s principle
Class 14	Convection-diffusion equation	Understand the use of moving coordinates to reduce the convection effect

Out-of-Class Study Time (Preparation and Review)

To enhance effective learning, students are encouraged to spend a certain length of time outside of class on preparation and review (including for assignments), as specified by the Tokyo Institute of Technology Rules on Undergraduate Learning (東京工業大学学修規程) and the Tokyo Institute of Technology Rules on Graduate Learning (東京工業大学大学院学修規程), for each class.
They should do so by referring to textbooks and other course material.

Textbook(s)

Materials will be provided if they are required.

Reference books, course materials, etc.

Reference book: Farlow, Stanley, Partial differential equations for Scientists and Engineers, Dover Publications, Inc.(1996)

Assessment criteria and methods

Students' knowledge on solving partial differential equations will be assessed by the reports on the exercises.

Related courses

Calculus I
Calculus II
Mathematics for Systems and Control A
Fundamentals of System Science
System Modeling

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students are expected to have successfully completed both Calculus I and Calculus II or have equivalent knowledge.

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