Volume: 25 lectures @ 90 minutes
Syllabus: Download Course Description (PDF)
Textbooks: Material from several sources will be used, including:
Y. Pinchover and J. Rubinstein, An Introduction to Partial Differential Equations, Cambridge University Press 2005.
Link to publisher's description,
I. Stakgold and M. Holst, Green’s Functions and Boundary Value Problems, 3rd Edition, Wiley 2011, Free download from within SJTU network
E. Zauderer, Partial Differential Equations of Applied Mathematics, 3rd Edition, Wiley 2011,
Free download from within SJTU network
WT Ang, A Beginner’s Course in Boundary Element Methods, Universal Publishers, 2007
Prerequisites: Graduate standing or permission of instructor
Background and Goals: This two-part introduction to advanced mathematical methods aims to provide mathematical foundations to graduate students majoring in both ME and ECE. The topics covered are closely aligned with those of parallel graduate engineering courses.
The present course focuses on the theory of partial differential equations.
Key Words: Point sources, distributions, classical, weak and fundamental solutions to differential equations, causal fundamental solutions, formal adjoint differential operator, conjunct, Green's formula, Green functions for boundary value problems for ODEs, modified Green functions for ODEs, solution formulas for ODEs and PDEs, Green functions for PDEs, eigenfunction expansions, method of images, conformal mappings, boundary element method.
Detailed Content: The course consists of three parts, each dealing with certain mathematical techniques useful for solving differential equations. Examples from mechanical as well as electrical engineering will be used throughout.
Our initial motivation is the desire to understand the treatment of point sources. Starting from the Dirac delta function as a formal symbol to denote a point source, we begin a formal treatment of generalized functions (distributions), including principal value integrals, notions of convergence and delta families, the distributional Fourier transform and solutions of distributional equations.
The second part of the course applies the theory of distributions to ordinary differential equations (ODEs). Strong, weak and distributional solutions are introduced and general solution formulas obtained. The main focus is then on obtaining Green's functions for boundary value problems (BVPs) for ODEs, leading to a brief discussion of solvability and modified Green's functions for ODEs.
The final third of the course extends the ODE methods to PDEs. Green's formulas for boundary value problems of the first, second and third kind are derived. Subsequently, methods for finding Green's functions are explored, including that of fukll and partial eigenfunction expansions, the method of images and (if time permits) conformal mappings. Finally, a short introduction to the use of Green's functions for the Laplace equation in the boundary element method (BEM) is presented.
Overview: The following textbooks are referenced below:
[SH] I. Stakgold and M. Holst, Green’s Functions and Boundary Value Problems
[Z] E. Zauderer, Partial Differential Equations of Applied Mathematics
[A] WT. Ang, A Beginner’s Course in Boundary Element Methods
Lecture | Lecture Subject | Textbook | Video Files |
---|---|---|---|
1 | Introduction | [SH] 1.1 - 1.2 | 1-6 |
2 | Point Sources and Green Functions | [SH] 1.1 - 1.2 | 1-6 |
3 | Distributions | [SH] 2.1 | 7-9 |
4 | Operations on Distributions | [SH] 2.1 | 10-12 |
5 | Families of Distributions | [SH] 2.2 | 13-14 |
6 | The Fourier Transform | [SH] 2.4 | 15-17 |
7 | The Fourier Transform for Tempered Distributions | [SH] 2.4 | 18-19 |
8 | First Midterm Exam | [SH] 2.1-2.4 | 7-19 |
9 | Differential Operators and Types of Solutions to Differential Equations | [SH] 2.4 | 18-19 |
10 | Review of Initial Value Problems for ODEs | [SH] 2.5, 3.1 | 22-25 |
11 | Second-Order Boundary Value Problems for ODEs | [SH] 3.2 | 26-28 |
12 | Second-Order Boundary Value Problems for ODEs | [SH] 3.2 | 26-28 |
13 | Adjoint Boundary Value Problems and Higher Order Equations | [SH] 3.2-3.4 | 29-31 |
14 | Solvability Conditions | [SH] 3.5 | 32-34 |
15 | Modified Green Functions | [SH] 3.5 | 32-34 |
16 | Second Midterm Exam | [SH] 3.1-3.5 | 20-34 |
17 | Second Order PDEs and BVPs | [Z] 7.1, 7.3 | 35-39 |
18 | Second Order PDEs and BVPs | [Z] 7.1, 7.3 | 35-39 |
19 | Eigenfunction Expansions | [Z] 7.3 | 40-41 |
20 | Partial Eigenfunction Expansions | [Z] 7.3 | 42 |
21 | Partial Eigenfunction Expansions | [Z] 7.3 | 42 |
22 | The Method of Images | [Z] 7.5 | 43-44 |
23 | The Method of Images | [Z] 7.5 | 45-46 |
24 | The Boundary Element Method | [A] | 47-48 |
25 | Final Exam | [Z] 7.1, 7.3, 7.5, [A] | 35-48 |
Video Lecture Slides (PDF):
Worksheets (2017 version; PDF files):