University of Michigan - Shanghai Jiao Tong University Joint Institute

# Vv 557: Methods of Applied Mathematics II

## General information

Course description: 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 inte- grals, 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 per- mits) 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.
Credit hours: 3 Credits.
Required/Elective course: Elective.
Pre-requisites: Vv 256  Applied Calculus IV, Vv 286 Honors Mathematics IV, or consent of instructor
Terms offered: Summer
Cognizant faculty: Horst Hohberger.
Potential instructors: Horst Hohberger.
Textbook/Required material:
- Y. Pinchover and J. Rubinstein, An Introduction to Partial Differential Equations, Cambridge University Press 2005 (for review only)
- I. Stakgold and M. Holst, Green’s Functions and Boundary Value Problems, 3rd Edition, Wiley 2011
- E. Zauderer, Partial Differential Equations of Applied Mathematics, 2nd Ed., Wiley 1989
Syllabus