2020 Partial Differential Equations

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Academic unit or major
Undergraduate major in Mechanical Engineering
Okuno Yoshihiro  Aono Yuko 
Class Format
Lecture    (ZOOM)
Media-enhanced courses
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Course description and aims

This course focuses on the Partial differential equations and Laplace transform. Topics include 1st order partial differential equations, 2nd order partial differential equations, Laplace transforms, the properties of Laplace transforms, and solving differential equations with the Laplace transform. By combining lectures and exercises, the course enables students to understand and acquire the fundamentals of mathematical tools widely applicable to linear systems in Engineering.
Mathematical approaches, such as solutions of partial differential equations and Laplace transform, taught in this course are not only useful in analyzing problems in mechanical engineering, but are applicable to those in the various field of engineering. Students will experience the satisfaction of solving practical problems by using their mathematical knowledge acquired through this course.
1. Partial derivatives and partial differential equations.
2. Solution of partial differential equations.
3. Solution by using Laplace transform.
4. Integral equations.

Student learning outcomes

By the end of this course, students will be able to:
1. Explain partial derivatives and partial differential equations.
2. Derive the partial differential equations and their solutions.
3. Explain the Laplace transform and apply to solve differential equations.
4. Explain series solutions of integral equations.


partial derivative, partial differential equations, integral equations, Laplace transform, fundamental solutions

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 class, solutions for exercise problems assigned in the previous class are explained. At the end of the class, exercise problems related to the lecture that day are given to the students.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Fundamentals of partial derivatives and partial differential equations Explain and obtain partial derivatives.
Class 2 First order linear partial differential equations Solve the first order partial differential equations.
Class 3 Second order linear partial differential equations Explain the second order partial differential equations.
Class 4 Solution of second order linear partial differential equations Derive fundamental solutions of the second order partial differential equations.
Class 5 Definition and properties of Laplace transform Transform the differential equations by using Laplace transform.
Class 6 Solution of differential equations by using Laplace inverse transform Solve differential equations by using Laplace inverse transform.
Class 7 Integral equations Explain series solutions of integral equations.

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

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.


None required.

Reference books, course materials, etc.

Handouts will be distributed at the beginning of the class when necessary.

Assessment criteria and methods

Understanding of partial differential equations, fundamental solutions, Laplace transform, series solutions, and the ability to apply them to engineering problems will be assessed.
Exams 80%, exercise problems 20%.

Related courses

  • MEC.B211 : Ordinary Differential Equations
  • MEC.F201 : Fundamentals of Fluid Mechanics
  • MEC.D231 : Fundamentals of Analytical Dynamics (Mechanical Engineering)

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

Students must have successfully completed Ordinary Differential Equations (MEC.B211.A) or have equivalent knowledge.

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