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.
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.
parabolic partial differential equation, separation variables, eigenfunction expansion, Fourier transform, Laplace transform
Intercultural skills | Communication skills | ✔ Specialist skills | Critical thinking skills | ✔ Practical and/or problem-solving skills |
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 | |
---|---|---|
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 |
Materials will be provided if they are required.
Reference book: Farlow, Stanley, Partial differential equations for Scientists and Engineers, Dover Publications, Inc.(1996)
Students' knowledge on solving partial differential equations will be assessed by final exams.
Students are expected to have successfully completed both Calculus I and Calculus II or have equivalent knowledge.