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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 48 "Diffusion and Reaction in
 a One Dimensional Slab" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "Our ex
ample involves the concentration in diffusion and reaction. The govern
ing ODE is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "ODE := D*Diff
(C,z,z)+r=0: ODE;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&%\"DG\"\"\"-
%%DiffG6%%\"CG%\"zGF,F'F'%\"rGF'\"\"!" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 6 "where " }{TEXT 257 1 "D" }{TEXT -1 31 " is the diffusion c
oefficient, " }{TEXT 258 1 "r" }{TEXT -1 26 " is the reaction rate and
 " }{TEXT 259 1 "C" }{TEXT -1 55 " is the concentration. The rate of r
eaction is given by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "Reac
tion := r=k*C: Reaction;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"rG*&%
\"kG\"\"\"%\"CGF'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "which we sub
stitute into the ODE to give" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 32 "ODE2:=subs(Reaction, ODE): ODE2;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/,&*&%\"DG\"\"\"-%%DiffG6%%\"CG%\"zGF,F'F'*&%\"kGF'F+F'F'\"\"!" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 44 "The boundary conditions for thi
s example are" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "BC0 := C(0
)=C[0]: BC0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"CG6#\"\"!&F%F&" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "BC1:= Diff(C(L),z)=0: BC1;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$-%\"CG6#%\"LG%\"zG\"\"!
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 195 "We are going to use a finite
 difference method of solving the ODE. The governing equation is assum
ed to hold for the concentration at certain selected points (nodes), h
ere indicated by the index " }{TEXT 256 1 "i" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 38 "i:='i': ODE3:=subs(C=C[i],ODE2): ODE3;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/,&*&%\"DG\"\"\"-%%DiffG6%&%\"CG6#%\"iG%\"zG
F/F'F'*&%\"kGF'F+F'F'\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "We \+
set the number of nodes" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "n
:=11;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nG\"#6" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 88 "We use a central difference approximation to th
e second derivative in the above equation" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 60 "fddiff:=Diff(C[i],z,z) = (C[i+1]-2*C[i]+C[i-1])/h^2
: fddiff;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6%&%\"CG6#%\"iG%
\"zGF+*&,(&F(6#,&F*\"\"\"F1F1F1F'!\"#&F(6#,&F*F1!\"\"F1F1F1%\"hGF2" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "We can now generate finite diffe
rence approximations for each interior node (the boundaries are dealt \+
with separately below)." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "
for i from 2 to n-1 do\n   eqn[i] :=subs(fddiff,ODE3);\n   print(eqn[i
]); \nod:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*(%\"DG\"\"\",(&%\"CG6
#\"\"$F'&F*6#\"\"#!\"#&F*6#F'F'F'%\"hGF0F'*&%\"kGF'F-F'F'\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*(%\"DG\"\"\",(&%\"CG6#\"\"%F'&F*6#
\"\"$!\"#&F*6#\"\"#F'F'%\"hGF0F'*&%\"kGF'F-F'F'\"\"!" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/,&*(%\"DG\"\"\",(&%\"CG6#\"\"&F'&F*6#\"\"%!\"#&F*6#
\"\"$F'F'%\"hGF0F'*&%\"kGF'F-F'F'\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/,&*(%\"DG\"\"\",(&%\"CG6#\"\"'F'&F*6#\"\"&!\"#&F*6#\"\"%F'F'%\"
hGF0F'*&%\"kGF'F-F'F'\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*(%\"
DG\"\"\",(&%\"CG6#\"\"(F'&F*6#\"\"'!\"#&F*6#\"\"&F'F'%\"hGF0F'*&%\"kGF
'F-F'F'\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*(%\"DG\"\"\",(&%\"
CG6#\"\")F'&F*6#\"\"(!\"#&F*6#\"\"'F'F'%\"hGF0F'*&%\"kGF'F-F'F'\"\"!" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*(%\"DG\"\"\",(&%\"CG6#\"\"*F'&F*
6#\"\")!\"#&F*6#\"\"(F'F'%\"hGF0F'*&%\"kGF'F-F'F'\"\"!" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#/,&*(%\"DG\"\"\",(&%\"CG6#\"#5F'&F*6#\"\"*!\"#&F*6
#\"\")F'F'%\"hGF0F'*&%\"kGF'F-F'F'\"\"!" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/,&*(%\"DG\"\"\",(&%\"CG6#\"#6F'&F*6#\"#5!\"#&F*6#\"\"*F'F'%\"hG
F0F'*&%\"kGF'F-F'F'\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "The b
oundary condition at " }{XPPEDIT 18 0 "z=0" "/%\"zG\"\"!" }{TEXT -1 4 
" is " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "BC0a:=subs(C(0)=C[
1],BC0): BC0a;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/&%\"CG6#\"\"\"&F%6#
\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "C
[0]" "&%\"CG6#\"\"!" }{TEXT -1 152 " is the initial concentration. The
 derivative boundary condition at the other boundary must be replaced \+
using a 2 term backward difference approximation" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 64 "fddiffb2:=Diff(C[i],z)=1/2*(3*C[i]-4*C[i-1]+C
[i-2])/h: fddiffb2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%%DiffG6$&%\"
CG6#\"#6%\"zG,$*&,(F'\"\"$&F(6#\"#5!\"%&F(6#\"\"*\"\"\"F7%\"hG!\"\"#F7
\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "The boundary condition n
ow becomes" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "BC1a:=subs(C(
L)=C[i],fddiffb2,i=n,BC1): BC1a;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,
$*&,(&%\"CG6#\"#6\"\"$&F(6#\"#5!\"%&F(6#\"\"*\"\"\"F3%\"hG!\"\"#F3\"\"
#\"\"!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "We place all the equati
ons into a list. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "eqnlis
t :=[BC0a,seq(eqn[i],i=2..n-1),BC1a];" }}{PARA 12 "" 1 "" {XPPMATH 20 
"6#>%(eqnlistG7-/&%\"CG6#\"\"\"&F(6#\"\"!/,&*(%\"DGF*,(&F(6#\"\"$F*&F(
6#\"\"#!\"#F'F*F*%\"hGF9F**&%\"kGF*F6F*F*F-/,&*(F1F*,(&F(6#\"\"%F*F3F9
F6F*F*F:F9F**&F<F*F3F*F*F-/,&*(F1F*,(&F(6#\"\"&F*FAF9F3F*F*F:F9F**&F<F
*FAF*F*F-/,&*(F1F*,(&F(6#\"\"'F*FIF9FAF*F*F:F9F**&F<F*FIF*F*F-/,&*(F1F
*,(&F(6#\"\"(F*FQF9FIF*F*F:F9F**&F<F*FQF*F*F-/,&*(F1F*,(&F(6#\"\")F*FY
F9FQF*F*F:F9F**&F<F*FYF*F*F-/,&*(F1F*,(&F(6#\"\"*F*F[oF9FYF*F*F:F9F**&
F<F*F[oF*F*F-/,&*(F1F*,(&F(6#\"#5F*FcoF9F[oF*F*F:F9F**&F<F*FcoF*F*F-/,
&*(F1F*,(&F(6#\"#6F*F[pF9FcoF*F*F:F9F**&F<F*F[pF*F*F-/,$*&,(FcpF5F[p!
\"%FcoF*F*F:!\"\"#F*F8F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 165 "In o
rder to obtain a numerical solution we must assign values to the diffu
sion coefficient, reaction rate coefficient and so on. The values used
 here are as follows." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Pa
rams := \{D=1.2e-9,k=1e-3,h=1e-3/(n-1),C[0]=0.2\};" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%'ParamsG<&/%\"hG$\"+++++5!#8/&%\"CG6#\"\"!$\"\"#!\"
\"/%\"kG$\"\"\"!\"$/%\"DG$\"#7!#5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
15 "The step size, " }{XPPEDIT 18 0 "h" "I\"hG6\"" }{TEXT -1 99 ", is \+
determined by the fact that the distance over which diffusion and reac
tion take place is just " }{XPPEDIT 18 0 "1e-3" "$\"\"\"!\"$" }{TEXT 
-1 3 " m." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 67 "We substitute these variables into the finite diff
erence equations." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "eqnlis
t2:=subs(Params,eqnlist);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%)eqnlis
t2G7-/&%\"CG6#\"\"\"$\"\"#!\"\"/,(&F(6#\"\"$$\"+++++7!#5&F(6#F,$!++++!
R#F5F'F3\"\"!/,(&F(6#\"\"%F3F0F8F6F3F:/,(&F(6#\"\"&F3F=F8F0F3F:/,(&F(6
#\"\"'F3FBF8F=F3F:/,(&F(6#\"\"(F3FGF8FBF3F:/,(&F(6#\"\")F3FLF8FGF3F:/,
(&F(6#\"\"*F3FQF8FLF3F:/,(&F(6#\"#5F3FVF8FQF3F:/,(&F(6#\"#6F3FenF8FVF3
F:/,(Fjn$\"+++++:!\"&Fen$!+++++?FaoFV$\"+++++]!\"'F:" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 46 "The variables appearing in these equations are
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "vars:=indets(eqnlist2,n
ame);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%varsG<-&%\"CG6#\"#5&F'6#\"
\"*&F'6#\"#6&F'6#\"\")&F'6#\"\"(&F'6#\"\"'&F'6#\"\"&&F'6#\"\"%&F'6#\"
\"\"&F'6#\"\"$&F'6#\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "and t
hey number" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "nops(vars);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"#6" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 26 "The number of equations is" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "nops(eqnlist2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"
#6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 198 "So there appear to be no f
urther degrees of freedom. Before proceeding it might help to determin
e the structure of the equations. Here is a procedure for looking at t
he structure of equation sytems." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 385 "Eqnanalysis := proc (Eqns:\{set,list,vector\},x:\{se
t,list,vector\})\nlocal neqns, Imat, i, j;\n# Determine number of equa
tions\nneqns := nops(Eqns);\n# Now to construct the incidence matrix\n
Imat := linalg[matrix](neqns,neqns,0);\nfor i from 1 to neqns do\n  fo
r j from 1 to neqns do\n    if has (Eqns[i], x[j]) then \n       Imat[
i,j] := 1; \n    fi;\n  od; \nod;\nplots[sparsematrixplot](Imat);\nend
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Eqnanalysis(eqnlist2,[
seq(C[i],i=1..n)]);" }}{PARA 13 "" 1 "" {INLPLOT "6%-%'POINTSG6A7$$\"
\"(\"\"!F'7$$\"\"&F)$\"\"'F)7$$\"\"\"F)$\"\"#F)7$F2$\"\"$F)7$F+$\"\"%F
)7$F0F07$F'$\"\")F)7$F<$\"\"*F)7$F8F57$F+F+7$$\"#6F)$\"#5F)7$FFF?7$FFF
D7$F?F?7$FFFF7$F5F57$F8F87$F5F27$F<F'7$F5F87$F?FD7$FDFD7$F-F+7$F-F-7$F
<F<7$F8F+7$F?F<7$F2F27$F?FF7$F-F'7$F'F--%&STYLEG6#%&POINTG-%+AXESLABEL
SG6$%'columnG%$rowG" 2 171 173 173 5 0 1 0 2 9 0 4 2 1.000000 
45.000000 45.000000 10020 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 
0 0 0 0 1 1 0 0 0 0 9000 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 306 "Here we see that the system of equations has an almost tridiag
onal form. We could, had we a mind to do so, take advantage of this st
ructure in devising a suitable computational method for calculating th
e concentrations. In this case there is no need to do so since Maple c
an solve these equations directly." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "solve(\{op(eqnlist2)\},vars);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#<-/&%\"CG6#\"\"%$\"+x^xDE!#5/&F&6#\"\")$\"+pc[\\JF+/&F&
6#\"#5$\"+uvtdKF+/&F&6#\"#6$\"+![68F$F+/&F&6#\"\"*$\"+ae,<KF+/&F&6#\"
\"&$\"+&*)eFz#F+/&F&6#\"\"'$\"+B'pk$HF+/&F&6#\"\"($\"+q(4d0$F+/&F&6#\"
\"\"$\"+++++?F+/&F&6#\"\"#$\"+QstFAF+/&F&6#\"\"$$\"+***4pV#F+" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 119 "Maple can also solve the purely s
ymbolic problem in which numerical values have not been given to the m
odel parameters." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "solve(
\{op(eqnlist)\},vars);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#<-/&%\"CG6#
\"\"%,$**%\"DG\"\"$&F&6#\"\"!\"\"\",6*(F+\"\"'%\"kGF,%\"hGF3\"%5B*(F+
\"\"&F4F(F5\"\")!%mL*(F+F(F4F8F5\"#5\"%<F*(F+F,F4F3F5\"#7!%u7*(F+\"\"#
F4\"\"(F5\"#9\"$X$*(F4F9F5\"#;F+F0!#]*&F4\"\"*F5\"#=F,*(F+F9F4F0F5FB\"
#***(F+FCF4FBF5F(!$!y*$F+FJ!\"#!\"\",0*$F+F3FB*(F4F0F5FBF+F8!#[*(F+F(F
4FBF5F(\"$v\"*(F+F,F4F,F5F3!$C#*(F+FBF4F(F5F9\"$E\"*(F+F0F4F8F5F<!#K*&
F4F3F5F?F,F0FR/&F&6#FB,$**,4*&F4F9F5FGF,*(F+F0F4FCF5FD!#W*(F+FBF4F3F5F
?\"$g#*(F+F,F4F8F5F<!$#z*(F+F(F4F(F5F9\"%?8*(F+F8F4F,F5F3!%w6*(F+F3F4F
BF5F(\"$/&*(F+FCF4F0F5FB!#!)*$F+F9FBF0F+F0F-F0F1FRFR/&F&6#FJ**F+F9F-F0
F1FR,&F+FQ*&F4F0F5FBF,F0/&F&6#F3,$**F+F8F-F0F1FR,,*$F+F(FB*(F4F0F5FBF+
F,!#C*(F+FBF4FBF5F(\"#S*(F+F0F4F,F5F3!#?*&F4F(F5F9F,F0FR/&F&6#F0F-/&F&
6#F,**F+FBF-F0,2*(F+F(F4F,F5F3\"$Y&*(F+F,F4F(F5F9!$]%*(F+FBF4F8F5F<\"$
(=*(F+F0F4F3F5F?!#Q*&F4FCF5FDF,*(F+F3F4F0F5FB\"#j*(F+F8F4FBF5F(!$3$*$F
+FCFQF0F1FR/&F&6#\"#6,$**F-F0F+F9,&F+FBFepF0F0F1FRFR/&F&6#F9,$**F+FCF-
F0F1FR,(*$F+FBFB*(F4F0F5FBF+F0!\")*&F4FBF5F(F,F0FR/&F&6#F<,$*(F+FJF-F0
F1FRFQ/&F&6#FC**F+F3F-F0F1FR,**$F+F,FQ*(F4F0F5FBF+FB\"#:*(F+F0F4FBF5F(
!#9*&F4F,F5F3F,F0/&F&6#F8**F+F(F-F0F1FR,.*$F+F8FQ*(F4F0F5FBF+F(\"#N*(F
+F,F4FBF5F(!#!**(F+FBF4F,F5F3\"#x*(F+F0F4F(F5F9!#E*&F4F8F5F<F,F0" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "We can check that this gives the s
ame result as before on substituting the parameter values" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "subs(Params,\");" }}{PARA 12 "" 1 "
" {XPPMATH 20 "6#<-/&%\"CG6#\"\"\"$\"\"#!\"\"/&F&6#\"\"%$\"+z^xDE!#5/&
F&6#F*$\"+SstFAF2/&F&6#\"\"*$\"+be,<KF2/&F&6#\"\"'$\"+D'pk$HF2/&F&6#\"
\"$$\"+,+\"pV#F2/&F&6#\"#6$\"+$[68F$F2/&F&6#\"\")$\"+qc[\\JF2/&F&6#\"#
5$\"+uvtdKF2/&F&6#\"\"($\"+s(4d0$F2/&F&6#\"\"&$\"+(*)eFz#F2" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 135 "As an alternative - not needed in
 this case but necessary in others - we could use Newton's method to s
olve the equations numerically. " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 36 "read `c:/maple/numerics/newton.mpl`:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 37 "A list of initial guesses is created." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "Initial:=[seq(C[i]=0.2,i=1..n)];" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(InitialG7-/&%\"CG6#\"\"\"$\"\"#!\"
\"/&F(6#F,F+/&F(6#\"\"$F+/&F(6#\"\"%F+/&F(6#\"\"&F+/&F(6#\"\"'F+/&F(6#
\"\"(F+/&F(6#\"\")F+/&F(6#\"\"*F+/&F(6#\"#5F+/&F(6#\"#6F+" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 37 "We invoke Newton's method as follows:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "result:=Newton(eqnlist2,Ini
tial,tolerance=1e-6,output=\{variables\});" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6-/&%\"CG6#\"\"\"$\"\"#!\"\"/&F%6#F)F(/&F%6#\"\"$F(/&F%6#
\"\"%F(/&F%6#\"\"&F(/&F%6#\"\"'F(/&F%6#\"\"(F(/&F%6#\"\")F(/&F%6#\"\"*
F(/&F%6#\"#5F(/&F%6#\"#6F(" }}{PARA 12 "" 1 "" {XPPMATH 20 "6-/&%\"CG6
#\"\"\"$\"\"#!\"\"/&F%6#F)$\"+OstFA!#5/&F%6#\"\"$$\"+%**4pV#F0/&F%6#\"
\"%$\"+q^xDEF0/&F%6#\"\"&$\"+'))eFz#F0/&F%6#\"\"'$\"+8'pk$HF0/&F%6#\"
\"($\"+e(4d0$F0/&F%6#\"\")$\"+dc[\\JF0/&F%6#\"\"*$\"+Te,<KF0/&F%6#\"#5
$\"+hvtdKF0/&F%6#\"#6$\"+o9JrKF0" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%
'resultG7-/&%\"CG6#\"\"\"$\"\"#!\"\"/&F(6#F,$\"+PstFA!#5/&F(6#\"\"$$\"
+(**4pV#F3/&F(6#\"\"%$\"+u^xDEF3/&F(6#\"\"&$\"+!*)eFz#F3/&F(6#\"\"'$\"
+<'pk$HF3/&F(6#\"\"($\"+i(4d0$F3/&F(6#\"\")$\"+gc[\\JF3/&F(6#\"\"*$\"+
We,<KF3/&F(6#\"#5$\"+jvtdKF3/&F(6#\"#6$\"+p9JrKF3" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 104 "An advantage of this solution is that the concent
rations at each node are listed in the original order. " }}{PARA 0 "" 
0 "" {TEXT -1 70 "The concentration profile can be plotted as shown in
 the next command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "plot(
[seq([k,rhs(result[k])],k=1..n)],labels=['n','C']);" }}{PARA 13 "" 1 "
" {INLPLOT "6$-%'CURVESG6$7-7$$\"\"\"\"\"!$\"1+++++++?!#;7$$\"\"#F*$\"
1+++PstFAF-7$$\"\"$F*$\"1+++(**4pV#F-7$$\"\"%F*$\"1+++u^xDEF-7$$\"\"&F
*$\"1+++!*)eFz#F-7$$\"\"'F*$\"1+++<'pk$HF-7$$\"\"(F*$\"1+++i(4d0$F-7$$
\"\")F*$\"1+++gc[\\JF-7$$\"\"*F*$\"1+++We,<KF-7$$\"#5F*$\"1+++jvtdKF-7
$$\"#6F*$\"1+++p9JrKF--%'COLOURG6&%$RGBG$FX!\"\"F*F*-%+AXESLABELSG6$%
\"nG%\"CG" 2 183 182 182 2 0 1 0 2 9 0 4 2 1.000000 45.000000 
45.000000 10030 10061 10056 10074 0 0 0 20030 0 12020 0 0 0 0 0 0 0 1 
1 0 0 0 0 -5912 0 0 0 0 0 0 }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 162 "No
te the profile flattens out on the right hand side of the plot, as is \+
to be expected from a boundary condition that forces to zero the conce
ntration derivative." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 266 "Inspection of the iteration histo
ry shows that the solution was obtained in a single step (round off is
 responsible for the procedure taking the extra step).  This means tha
t the equations are linear (we could have deduced this fact if only we
 had bothered to look)." }}}}{MARK "58 0 0" 266 }{VIEWOPTS 1 1 0 1 1 
1803 }