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Unformatted text preview: . t0 Copyright c 2000 SIAM Buy online from SIAM Buy from 610 Chapter 7 Eigenvalues and Eigenvectors Example 7.9.7 It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] Nondiagonalizable Mixing Problem. To make the point that even simple problems in nature can be nondiagonalizable, consider three V gallon tanks as shown in Figure 7.9.1 that are initially full of polluted water in which the ith tank contains ci lbs of a pollutant. In an attempt to flush the pollutant out, all spigots are opened at once allowing fresh water at the rate of r gal/sec to flow into the top of tank #3, while r gal/sec flow from its bottom into the top of tank #2, and so on. Fresh D E r gal/sec 3 T H r gal/sec 2 r gal/sec IG 1 R Y r gal/sec Figure 7.9.1 Problem: How many pounds of the pollutant are in each tank at any finite time t > 0 when instantaneous and continuous mixing occurs? P Solution: If ui (t) denotes the number of pounds of pollutant in tank i at time t > 0, then the concentration of pollutant in tank i at time t is ui (t)/V lbs/gal, so the model ui (t) = (lbs/sec) coming in − (lbs/sec) going out produces the nondiagonalizable system: ⎛ ⎛ ⎞⎛ ⎞ ⎞ ⎛⎞ u1 (t) −1 1 0 u1 (t) c1 ⎜ ⎟⎜ ⎟ r⎜ ⎟ u2 (t) ⎠ = ⎝ 0 −1 1 ⎠⎝ u2 (t) ⎠, or u = Au with u(0) = c = ⎝ c2 ⎠ . ⎝ V c3 u (t) u (t) 0 0 −1 3 O C 3 This setup is almost the same as that in Exercise 3.5.11 (p. 104). Notice that A is simply a scalar multiple of a single Jordan block J = −1 0 0 1 −1 0 0 1 −1 , so eAt is easily determined by replacing t by rt/V and λ by −1 in the second equation of (7.9.18) to produce ⎞ ⎛ 2 1 rt/V (rt/V ) /2 ⎟ ⎜ eAt = e(rt/V )J = e−rt/V ⎝ 0 1 rt/V ⎠. 0 Copyright c 2000 SIAM 0 1 Buy online from SIAM Buy from 7.9 Functions of Nondiagonalizable Matrice...
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This document was uploaded on 03/06/2014 for the course MA 5623 at City University of Hong Kong.

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