2 04 06 08 1 100 200 figure 742 copyright c 2000

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Unformatted text preview: g e−At v produces d e−At v = e−At v − e−At Av = 0, dt Copyright c 2000 SIAM so e−At v is constant for all t. Buy online from SIAM http://www.ec-securehost.com/SIAM/ot71.html Buy from AMAZON.com 542 Chapter 7 Eigenvalues and Eigenvectors http://www.amazon.com/exec/obidos/ASIN/0898714540 At t = 0 we have e−At v t=0 = e0 v(0) = Ic = c, and hence e−At v = c for all t. Multiply both sides of this equation by eAt and use (7.4.5) to conclude v = eAt c. Thus u = eAt c is the unique solution to u = Au with u(0) = c. Finally, notice that vi = Gi c ∈ N (A − λi I) is an eigenvector associated with λi , so that the solution to u = Au, u(0) = c, is It is illegal to print, duplicate, or distribute this material Please report violations to [email protected] u = eλ1 t v1 + eλ2 t v2 + · · · + eλk t vk , (7.4.6) and this solution is completely determined by the eigenpairs (λi , vi ). It turns out that u also can be expanded in terms of any complete set of independent eigenvectors—see Exercise 7.4.1. Let’s summarize what’s been said so far. Differential Equations T H D E If An×n is diagonalizable with σ (A) = {λ1 , λ2 , . . . , λk } , then the unique solution of u = Au, u(0) = c, is given by u = eAt c = eλ1 t v1 + eλ2 t v2 + · · · + eλk t vk (7.4.7) IG R in which vi is the eigenvector vi = Gi c, where Gi is the ith spectral projector. (See Exercise 7.4.1 for an alternate eigenexpansion.) Nonhomogeneous systems as well as the nondiagonalizable case are treated in Example 7.9.6 (p. 608). Y P Example 7.4.1 An Application to Diffusion. Important issues in medicine and biology involve the question of how drugs or chemical compounds move from one cell to another by means of diffusion through cell walls. Consider two cells, as depicted in Figure 7.4.1, which are both devoid of a particular compound. A unit amount of the compound is injected into the first cell at time t = 0, and as time proceeds the compound diffuses according to the following assumption. O C α β Cell 1 Cell 2 Figure 7.4.1 Copyright c 20...
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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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