### Methods of Applied Math II

##### back to homepage ##### Project year: 2015 #### Horst Hohberger Horst Hohberger, Horst Hohberger, associate teaching professor, receivedhis Ph. D. in mathematical physics at the University of Potsdam, Germany, in 2006.He joined the University of Michigan—Shanghai Jiao Tong University Joint Institute in 2007, where he developed the mathematics curriculum. Over the past 11 years he has been teaching numerous courses, covering topics such as calculus/analysis, differential equations, discrete mathematics (number theory, combinatorics, algorithms), probability and statistics, function alanalysis and many more. For his teaching, he has received numerous awards, funding and prizes from Shanghai JiaoTong University and the Shanghai City Government.

#### Objectives

This is a graduate level course on solving differential equations using Greens functions, touching also on concepts such as non-regular solutions, physical point sources, generalized functions and the Fourier transform. A solid background in calculus and the classical theory of ordinary differential equations is assumed while some experience with partial differential equations is helpful. Students will (1)Gain an understanding of the formal mathematical definition of the “delta function”and how to manipulate it and generalized functions, (2) learn how to construct Green functions and solve partial and ordinary differential equations which involve piece-wise continuous as well as“ point source” inhomogeneities, (3) understand the mathematical background behind the numerical boundary element method.

#### Contents

Point sources and physical ideas (non-smooth solutions to differential equations, different approaches to the “delta function”, heuristic approximation of solutions to the inhomogeneous equation,example of an eigenfunction expansion); Distributions (locally integrable and smooth, compactly supported functions; regular and singular distributions, operations on distributions, families of distributions and convergence, tempered distributions, continuity and properties of the Fourier transform of tempered distributions, application to the heat equation); Differential equations (classical, weak and fundamental solutions, causal fundamental solutions, formal adjoint equation, conjunct, Lagrange’s and Greens formula); Green functions for boundary value problems for ODEs (initial value problems, separated boundary conditions, mixed conditions, construction of Green functions, higher order equations, adjoint problem, solv ability criteria, modified Green functions, solution formulas via Green functions); Elliptic, hyperbolic and parabolic PDEs (operator types, Dirichlet, Neumann and Robin conditions, adjoint problems, Green and Lagrange formula, solution formula); Green functions for PDEs (eigenfunction expansions, method of images); Boundary element method (basic idea only)

#### Textbooks and Reference Books #### Relatived Courses ##### Probability and Statistics ##### Calculus 