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FIGURE 96 AnMJile soveo tridiogonol to sysiern
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An M-file that solvesa tridiagonalsystemof equations listed in Fig 96 Note that the is algorithmdoesnot includepartialpivoting Although pivoting is sometimes required,most tridiagonalsystems routinely solvedin engineering and science not requirepivoting do Recall that the computationaleffort for Gauss elimination was proportional to n3 Because its sparseness, effort involved in solving tridiagonalsystems proportional of the is to n Consequently, algorithm in Fig 96 executes the much, much fasterthan Gausselimination,particularlyfor large systems
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Bockground Linear algebraicequationscan arise when modeling distributed systems For example,Fig97 shows a long, thin rod positionedbetweentwo walls that are held at constant temperaturesHeat flows through the rod as well as between the rod and the surrounding air For the steady-state case,a differential equation basedon heat conservation can be written for such a system as d2T ax-
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fo: 40
Is = 200
ti{it
,r=0
-- Lx;'7,,:20
FIGURE 97 two temperoture rod between wolls constont different of but A noninsuloted uniform positioned
The {inite-d'flerence representolion 66is5 employsf6ur inls1ie,
where I: temperature("C), x : distancealong the rod (m), h' : aheat transfercoefficient ("C) betweenthe rod and the surroundingair 1m-2), ztrdTo: the air temperature Given values for the parameters,forcing functions, and boundary conditions,calculus = can be usedto developan analyticalsolutionFor example,if h' :001, 4, : 20, T(0) 40, and (10) : 200, the solutionis t^ T :134523n0'tx - 534523e-0 + 20
(92'
In Although it provided a solutionhere,calculusdoesnot work for all suchproblems we such instances,numerical methods provide a valuable alternative In this case study, will use finite differencesto transformthis differentialequationinto a tridiagonalsystem deof linear algebraic equationswhich can be readily solved using the numerical methods scribed in this chapter into a setof linear algebraic equations by Solution Equation(924) canbe transformed the conceptualizing rod as consistingof a seriesof nodesFor example,the rod in Fig97 between nodesSincethe rod has a length of 10, the spacing is divided into six equispaced nodesis A,x :2 it Calculus was necessary solve Eq (920 because includes a secondderivative to provide a meansto transfom approximations As we learnedin Sec434,finite-difference algebraic form For example, the second derivative at each node canh derivatives into approximated as dzT d-t
Ti+t -ZTi *Ti-r
into where I designatesthe temperatureat node i This approximation can be substituted (92q to give Eq Ti+r-2Tr * Ti*t
n, -2
+h'(7"-4):0
95 CASE STUDY
continued
gives Collecting terms and substituting parameters the -Ti*r*204Ti4+r:08
(e26)
Thus, Eq (924)has beentransformed from a differentialequationinto an algebraicequation Equation(926) can now be appliedto eachof the interior nodes:
-To*204T1-4:08 - Tt * 20472 Z: : 08 - Tz 1204\ - Z+: 08 - Tt -t 20474 Is : 08 -
(927)
The values of the fixed end temperatures, : 40 and Ts:200, can be substituted Io and with four unknownsexpressed moved to the right-handsideThe resultsarefour equations in matrix form as
a; {i^l:{}i} ffi' 'it;]
>> A=12A4 -1 0 0 -r 204 -1 0 a -r 2 0,1 -L t) 0 -1,2A4); >> b=i408 08 08 20081';
>> T= (A\ b) ' T= 659698 93',i18a 'r245382 r5941 9a
(928)
So our original differentialequationhas been convertedinto an equivalentsystemof lineal algebraicequations Consequently, can usethe techniques we described this chapin ter to solve for the temperatures For example, using MATLAB
A plot can also be developed comparing these results with the analytical solution obtained with Eq (925),
L j' i t;
>> x=[0:2:101; >> xanal=f0:101; >> TT-0 (r) 13452 -j*exp (01*xanal) (-01*xanal)+20;
t-,rrj=1:(Ydl--); >> plot it ,'l , 'o' , ranal, Tartal )
-534523*exp
As in Fig 98, the numericalresultsare quite closeto thoseobtainedwith calculus
GAUSSELIMINATION
confinued
Analytical (linel and numerical (pointsl solutions
220 200
180 160
91 | the r (Fig 4
92 |
Chec equa
93 (
140 r- 120 100 80 60
(a) S
rl (b) c
(c) c 94 (
(a) C
FIGURE 98 versus rod A plot temperoture distonce of olong heoted Boih o onolyticol ondnumericol {line)
solutions disployed ore lpoints)
(b) u (d) s
(c) U tt
We In addition to being a linear system,notice that Eq (928) is also tridiagonal can use an efficient solutionschemelike the M-file in Fis 96 to obtain the solution:
>> e=10 -1 -1 -11; >> f=1204 204 204 204); >> g=i-1 -1 -1 01 >> r= [408 08 08 2008] ; >> TridiaS (e, f, q, r )
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