University of Michigan - Shanghai Jiao Tong University Joint Institute

Pages

Vv 454:Boundary Value Problems for Partial Differential Equations


General information

Prerequisites: Vv255 and Vv256; or Vv285 and Vv286; or permission of instructor
Background and Goals: The course is intended as an introduction to the classical theory of first- and second-order
quasilinear partial differential equations (PDEs) and boundary value problems. The main focus lies on gaining
familiarity with a range of solution methods. Rigorous proofs play only a subordinate role. A strong emphasis is
placed on applications from fluid dynamics, electrodynamics, the theory of heat transfer, the analysis of vibrations
and other fields of engineering.
Content: Conservation laws and the derivation of PDEs from physical models; quasilinear first-order PDEs and
the method of characteristics; Burgers’s equation and weak solutions; shock waves; the eikonal equation and other
nonlinear first-order PDEs; classification of quasilinear second-order PDEs and their transformation into normal
form; boundary value problems of various kinds; the wave equation on an infinite string and d’Alembert’s method;
the heat equation in a finite bar and its solution through separation of variables; Fourier-Euler series and their
convergence; spaces of weighted square-integrable functions and the problem of best approximation; Sturm-
Liouville boundary value problems; separation of variables for nonhomogeneous one-dimensional evolution
equations; problems on infinite and semi-infinite bars and the Fourier transform; dispersive solutions; analysis of
the telegraph equation; separation of variables in higher dimensions; Bessel functions and Legendre polynomials;
multipole expansions in electromagnetics; the Poisson equation and properties of harmonic functions.
Alternatives: None.
Course Profile
Syllabus