Lecture 15: Three-tank Mixing and Lead Poisoning

Transcription

1 Lecure 15: Three-ak Miig ad Lead Poisoig Eigevalues ad eigevecors will be used o fid he soluio of a sysem for ukow fucios ha saisfy differeial equaios The ukow fucios will be wrie as a 1 colum vecor [ 1 () () ()] T, ad he differeial equaios will be rereseed by a mari A ' A + f where f ad (0) are give cosa 1 colum vecors Here we will assume ha A has disic real ad o-zero eigevalues Less resricive assumios o A require more advaced course work However, we will be able o give very ieresig alicaios o ime deede miig, lead oisoig ad hea rasfer (see homework roblem 5) The seady sae soluio requires ' 0 so ha 0 A + f We have assumed f is ideede of ime, ad A has o zero eigevalues so ha A has a iverse mari Ofe he seady sae soluio is referred o as a aricular soluio, which we will deoe by -A -1 f I order o form he homogeeous or ime deede ar of he soluio, we use he eigevecors, u j a o-zero 1 colum vecor, ad eigevalues, r j, so ha Au j r j u j The defiiio of eigevalues, r, ad eigevecors, u, was moivaed by he fac ha ue r mus saisfy ' A More geerally ay liear combiaio of he colum vecors will also be a soluio of ' A Oher roeries are summarized i Proosiio 8 We will use he followig oaio: Le Au r u where u is o zero 1 colum vecor j j j j U () u e u e u e is mari, r 1 r 1 c1 1 c r r Uc () ue 1 ue ue c ue c + ue c + + ue c r 1 r

2 Proosiio 8 Cosider ' A + f where f is a cosa 1 vecor Assume A has disic o-zero real eigevecors Defie U()c + where c is a cosa 1 vecor ad -Af The he followig are rue: 1 U()c + saisfies ' A + f A has a iverse mari 3 U() has a iverse mari 4 If c U(0) -1 ((0) ), he U()c + also saisfies (0) equals a give 1 colum vecor 5 Moreover, if all he eigevalues are egaive, he as goes o ifiiy () coverges o Proof of 1 Lefside ' ( U ( ) c + )' Righside A + f ( Uc ( ) )' + ( )' ( ue c + ue c + + ue c )' + 0 r 1 r ( ue c)' + ( ue c )' + + ( ue c )' r 1 r uc( e )' + uc ( e )' + + uc ( e )' r 1 r uce r+ uce r + + uce r r 1 r 1 A( Uc ( ) + ) + f AU ( ( ) c) + A + f r 1 r r 1 r 1 Aue c + ue c + + ue c + A A f + f r 1 r 1 ( ) ( ) ( Au ) e c + ( Au ) e c + + ( Au ) e c f + f ( ru ) e c + ( ru ) e c + + ( ru ) e c Proof of 3 This will have o wai for a more advaced course Alicaio o Three-ak Miig Cosider he roblem ha was formulaed i lecure 6 where 3 ad 1 he amou i he lef ak, y he amou i he ceer ak ad 3 z he amou i he righ ak

3 ' A+ f 1 10/4 4/ / 4 1 / /4 8/ (0) A f ad The Malab commad [u d] eig(a) will roduce wo 33 marices, ad colum j of u will be he eigevecor associaed wih he j diagoal comoe of d [ u d] Here U(0) u ad c U(0) -1 ((0) ) c U (0) ( 1 10 ) The reader should cosul he Malab demo ak3_imem for more deails, ad he creaio of he followig lo of he soluios / e( / )

4 Alicaio o Lead Poisoig See chaer 7 i he ODE e by Borrelli ad Colema A simlified model of lead oisoig i he huma body is somewha similar o he hree-ak miig roblem We will assume he lead is locaed i he blood ( 1 ()), issue ( ()) or boes ( 3 ()) The followig grahic deics he flow bewee hese hree regios I 1 Source k 31 1 k 1 1 Boes Blood Tissue k 13 3 k 1 k 01 1 k 0 Urie Hair, Nail ad Persiraio The flow raes k ij are assumed o be cosas from grou j o grou i These mus be deermied from measured daa, ad ossibly by usig a leas squares curve fi o muliliear fucios By usig he rae of chage of grou i is equal o he rae i mius he rae ou, we ca formulae he followig sysem of differeial equaios 1 ( k01+ k1+ k31) k1 k13 1 I1 k ( k + k ) k31 0 k Secific values of he cosas are give i he Malab demo lead3_imem The cosa k 01 ca be corolled by medicaios, ad he cosa I 1 ca be corolled by alerig he evirome For he values use i lead3_imem all he eigevalues were egaive, ad herefore, he soluio () mus coverge o he seady sae soluio [ ] The followig grah was geeraed for (0) [0 0 0] T where he verical ais is scaled by The wo curves for 1 () ad () are almos overlaig because of his scalig

5 / e( / ) / Homework 1 Use ak3_imem o eerime wih differe iiial coceraios Observe ha he soluios always coverge o he same seady sae soluio Use lead3_imem o eerime wih differe I 1 How does his affec he seady sae soluio? Will i remai he limi of ()? 3 Use lead3_imem o eerime wih differe medicaios by adjusig k 01 away from Prove ars 4 ad 5 i Proosiio 8 5 Cosider he hea diffusio i a rod i eamle 3 of lecure 14 The cosa c i he mari A has he form (K/(ρc ))/(h ) where K hermal coduciviy 001, ρ desiy 1, c secific hea 1, h segme size ¼ so ha c 016 Modify ak3_imem so ha his roblem is solved wih f [1 1 1] T ad (0) [ ] T You may wish o comare your code wih hea3_imem Eerime wih differe iiial emeraures ad observe ha () always coverges o he seady sae soluio