课程介绍

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 full 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.

课程大纲

学习要求

Students are expected to have taken courses in single- and multi-variable calculus and in ordinary differential equations.
Some knowledge of classical solution methods of PDEs (e.g., separation of variables) will be helpful.

考核标准

线上成绩构成比例为：
[课件浏览]： 40%
[客观练习]： 60%
（暂定，以教学团队最终成绩公布为准）

教材教参

Stakgold, I. and Holst, M.: Green’s Functions and Boundary Value Problems, 3rd Ed., Wiley 2011.
http://onlinelibrary.wiley.com/book/10.1002/9780470906538

E. Zauderer, Partial Differential Equations of Applied Mathematics, 2nd Ed., Wiley 1989,
http://onlinelibrary.wiley.com/book/10.1002/9781118033302

For some background on PDEs: Y. Pinchover and J. Rubinstein, An Introduction to Partial Differential Equations, Cambridge University Press 2005,
http://ebooks.cambridge.org/ebook.jsf?bid=CBO9780511801228