MATH 401 Study Guide - Midterm Guide: Boundary Value Problem, Legendre Polynomials
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Be sure that this examination has 3 pages. Consider axially symmetric di usion for u(r, z, t) in a nite cylinder of radius a > 0 and height h > 0 with insulating boundary conditions modeled by ut = urr + Determine an eigenfunction expansion representation for the time-dependent solution u(r, z, t), and also calculate the steady-state solution. Assume that > 0 and 0 are constants, and consider the following tra c ow model for the density (x, t) of cars given by. T + (2 ) x + = 0 , < x < , t > 0 , 2 + x2 . (i) first let = 0. Determine a parametric form for the solution (x, t). Plot qualitatively the characteristics in the (x, t) plane, and sketch the solution (x, t) versus x at di erent times. Determine the time tb as a function of when the solution rst becomes multi-valued. (ii) now let > 0.
