The course revolves around solving differential equations through methods inspired by the treatment of point sources (charges, masses, forces, etc.). 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.
We will then apply 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 part 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 and the method of images. To round off the topic, a short introduction to the use of Green's functions for the Laplace equation in the boundary element method (BEM) is presented.
Offering School：UM-SJTU Joint Institute
Course Level：Postgraduate Level
Course Component：Lecture
Time Zone：Beijing Time（UTC+8）
Language of Instruction：English
Course Introduction Video：English