2016 Partial Differential Equations

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Academic unit or major
Undergraduate major in Mechanical Engineering
Takahara Hiroki  Oshima Shuzo  Yamazaki Takahisa 
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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.

Student learning outcomes

By the end of this course, students will be able to:
1) Explain partial derivatives and partial differential equations.
2) Explain the partial differential equations in Engineering, and properties of fundamental solutions with their applications.
3) Explain the fundamentals of the Laplace transform.
4) Apply Laplace transforms to solve differential equations.


partial difference, total differential equations, fundamental equations, integral equations, Laplace transform, fundamental solutions

Competencies that will be developed

Intercultural skills Communication skills Specialist skills Critical thinking skills Practical and/or problem-solving skills
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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.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Fundamentals of partial differential equations, definition of partial derivatives. Write partial derivatives, solve the derivative of functions.
Class 2 First order linear partial differential equations Solve the first order partial differential equations.
Class 3 Second order linear partial differential equations Explain governing equations in engineering.
Class 4 Solution of second order linear partial differential equations . Explain fundamental solutions of governing equations.
Class 5 Integral equations Explain solution by series.
Class 6 Definition and properties of Laplace transform Transform the differential equations and integral equations using Laplace transform.
Class 7 Solution of differential equations using Laplace inverse transform. Solve differential equations by using Laplace inverse transform
Class 8 Summary


None required.

Reference books, course materials, etc.

Handouts will be distributed at beginning of class when necessary.

Assessment criteria and methods

Students' knowledge of partial differential equations, fundamental solutions, solve by series, Laplace transform, and their ability to apply them to 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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