Course title
A00070002
Applied Analysis

 SAWA Takekazu
Course description
Fourier series, Fourier transform and Inverse Fourier transform are important techniques to solve partial differential equations such as equation of heat conduction.
Laplace transform and Inverse Laplace transform are the effective techniques to solve various differential equations.
In this course, fundamentals and applications of these techniques are lectured.
In addition, basics of complex function theory are explained to apply them to the mechanical problems
Purpose of class
The aim of this course is to acquire the basics of Fourier transform and Laplace transform, and to learn how to apply these techniques to solve various problems in the mechanical engineering. In addition, basics of complex function theory will be learned to apply them to the mechanical problems
Goals and objectives
  1. Students can solve Fourier series of Even functions and Odd functions．
  2. Students can solve Fourier series of trigonometric and exponential functions．
  3. Students can solve partial differential equations using Fourier transform and Inverse Fourier transform．
  4. Students understand the basics of the complex function theory, and can associate with the visualized figures in complex space.
  5. Students acquire definitions and characteristics of Laplace transform and Inverse Laplace transform．
  6. Students can solve differential equations in problems of mechanical engineering by applying Laplace transform.
Relationship between 'Goals and Objectives' and 'Course Outcomes'

 Intermediate test Term-end exam Total.
1. 10% 10%
2. 10% 10%
3. 10% 10%
4. 10% 10%
5. 30% 30%
6. 30% 30%
Total. 30% 70% -
Evaluation method and criteria
Final grade will be calculated according to mid-term examination (30%) and term-end examination (70%).
Mid-term examination includes issues 1 to 6 and partly issue 7.
Term-end examination includes issues 7 to 15.
Language
Japanese
Class schedule

 Class schedule HW assignments (Including preparation and review of the class.) Amount of Time Required
1. What is the Fourier series ? What is the Fourier transform？
Euler's coefficients(1)
　　Even function, Odd functions
　　Orthogonalitys of trigonometric functions
　　How to solve Euler's coefficients 		Preparation and review using reference materials 190minutes
2. Euler's coefficients(2)
　　An example: Saw-tooth wave
　　Half-range expansion 		Preparation and review using reference materials 190minutes
3. Description of fourier series in the exponential functions
　　Definition of exponential functions, Euler's formulae
　　Description of fourier series in the trigonometric
　　and exponential functions 		Preparation and review using reference materials 190minutes
4. Fourier series to Fourier transform
　　Change of the independent variables
　　Replacement to the continuous functions 		Preparation and review using reference materials 190minutes
5. Fourier transform and Inverse Fourier transform (1)
　　Single square wave
　　Exponential function 		Preparation and review using reference materials 190minutes
6. Fourier transform and Inverse Fourier transform (2)
　　Step function
　　Delta function
　　Trigonometric function

Characteristics of Fourier transform(1)
　　Linearity, transitivity, and similarity 		Preparation and review using reference materials 190minutes
7. Characteristics of Fourier transform(2)
　　Fourier transform of differentials and integrations

Parseval's equality and Gaussian function
　　Parseval's equality
　　Gaussian function 		Preparation and review using reference materials 190minutes
8. Mid-term examination and review Preparation and review using reference materials 190minutes
9. Application of Fourier transform on partial differential equations
　　Equation of heat conduction
　　Application of Fourier transform on the equation of heat conduction 		Preparation and review using reference materials 190minutes
10. Basics of complex functions (1)
　　Complex functions and their operations
　　Regular function
　　Cauchy-Riemann's equation 		Preparation and review using reference materials 190minutes
11. Basics of complex functions (2)
　　Visualization of complex functions
　　Conformal mapping
　　Application to mechanical problems 		Preparation and review using reference materials 190minutes
12. Laplace transform (1)
　　Definitions of Laplace transform and Inverse Laplace transform
　　Characteristics of Laplace transform 		Preparation and review using reference materials 190minutes
13. Laplace transform (2)
　　Laplace transform of a typical function
　　Application of Laplace transform 		Preparation and review using reference materials 190minutes
14. Laplace transform (3)
　　How to solve differential equations using Laplace transform ?
　　Applications on the problems in mechanical engineering

Term-end examination and the summary 		Preparation and review using reference materials 190minutes
Total. - - 2660minutes
Feedback on exams, assignments, etc.
ways of feedback specific contents about "Other"
Feedback in the class
Textbooks and reference materials
Reference materials will be provided before each lecture (in the file folder of network system).
Prerequisites
It is desireable 'Elementary Mathematics (Analysis)', 'Differential and Integral Calculus and Exercise' and 'Basic Analysis' are completed before this lecture.
Office hours and How to contact professors for questions
  • Tuesday 11: 00-12: 00. It is desirable to notify the visit in advance.
    Accept questions during the class and any time by e-mail.
Regionally-oriented
Non-regionally-oriented course
Development of social and professional independence
  • Non-social and professional independence development course
Active-learning course
N/A
Course by professor with work experience
Work experience Work experience and relevance to the course content if applicable
Applicable Describe how to solve problems with showing examples of practical applications on practical experience .
Education related SDGs:the Sustainable Development Goals
  • 4.QUALITY EDUCATION
  • 9.INDUSTRY, INNOVATION AND INFRASTRUCTURE
Last modified : Sat Mar 08 04:18:42 JST 2025