### 2019　Mathematics for Systems and Control B

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Undergraduate major in Systems and Control Engineering
Instructor(s)
Kosaka Hidenori
Course component(s)
Lecture / Exercise
Day/Period(Room No.)
Tue2-4(S516)  Fri2-4(S516)
Group
-
Course number
SCE.A202
Credits
3
2019
Offered quarter
3Q
Syllabus updated
2019/3/18
Lecture notes updated
-
Language used
Japanese
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### 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

 Intercultural skills Communication skills ✔ Specialist 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 Understand the expression of physical phenomena by partial differential equations
Class 3 Heat equation Understand the derivation of energy conservation equation in a rod
Class 4 Boundary and initial conditions Understand the properties of boundary and initial conditions
Class 5 Separation variables method Apply separation variables method for solving parabolic partial differential equation
Class 6 Nonhomogeneous boundary conditions Transformation of nonhomogeneous boundary condition to homogeneous one
Class 7 Eigenvalue Understand Sturm-Liouville problem
Class 8 Transformation of partial differential equation Understand the transform of partial differential equation to use the separation variables method
Class 9 Nonhomogeneous partial differential equation Apply eigenfunction expansion method for solving parabolic partial differential equation
Class 10 Integral transform Understand sine and cosine transforms of partial differential equation
Class 11 Fourier series and Fourier transform Understand the properties of Fourier series and transform
Class 12 Fourier transform Apply Fourier transform for solving parabolic partial differential equation
Class 13 Laplace transform Apply Laplace transform for solving parabolic partial differential equation
Class 14 Duhamel’s principle Understand Duhamel’s principle
Class 15 Convection-diffusion equation Understand the use of moving coordinates to reduce the convection effect

### 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 final exams.

### 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.