{VERSION 4 0 "IBM INTEL NT" "4.0" }
{USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 
1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 1 135 15 0 0 0 2 0 0 
0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 63 135 1 0 0 0 0 0 0 0 0 
0 0 1 }{CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 }
{CSTYLE "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{PSTYLE "Normal
" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }0 0 0 
-1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" 
-1 -1 "" 0 1 0 0 32 0 0 0 0 0 0 0 0 0 0 1 }3 3 0 -1 -1 -1 0 0 0 0 0 0 
-1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 
0 0 0 1 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 
{CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 
0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 32 32 50 0 
0 0 0 0 0 0 0 0 0 1 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }}
{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 32 "Solution to a convectio
n problem" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 258 "As a
n example of the the convection problem in lecture 15 consider a rod w
hich is kept at temperature T(0,t)=0 at one end. We choose the initial
  rod  temperature to be T(x,0)=0. At the other end (x=a) there is con
vective contact with a fluid at temperature " }{XPPEDIT 18 0 "T[1]" "6
#&%\"TG6#\"\"\"" }{TEXT -1 33 ".The boundary condition at x=a is" }}}
{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "-kappa*diff(T(x,t),x) = h*(T(a,t
)-T[1]);" "6#/,$*&%&kappaG\"\"\"-%%diffG6$-%\"TG6$%\"xG%\"tGF.F'!\"\"*
&%\"hGF',&-F,6$%\"aGF/F'&F,6#F'F0F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 33 "The steady state solution is then" }}}{EXCHG {PARA 0 "" 0 "" 
{XPPEDIT 18 0 "T[S] = x*h*T[1]/(kappa+h*a);" "6#/&%\"TG6#%\"SG*&*(%\"x
G\"\"\"%\"hGF+&F%6#F+F+F+,&%&kappaGF+*&F,F+%\"aGF+F+!\"\"" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 38 "The initial condition on the transient" }
}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "u(x,t)=T(x,t)-T[S]" "6#/-%\"uG
6$%\"xG%\"tG,&-%\"TG6$F'F(\"\"\"&F+6#%\"SG!\"\"" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 8 "is thus " }{XPPEDIT 18 0 "u(x,0)=-T[S]" "6#/-%\"uG6$%
\"xG\"\"!,$&%\"TG6#%\"SG!\"\"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "
The eigenvalue problem is thus" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "tan(l
ambda*a)=-kappa*lambda/h" "6#/-%$tanG6#*&%'lambdaG\"\"\"%\"aGF),$*(%&k
appaGF)F(F)%\"hG!\"\"F/" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "To be \+
specific let us choose some values for the constants" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 32 "a:=Pi;kappa:=1;h:=1;t1:=1;\nk:=1;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%\"aG%#PiG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&kap
paG\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"hG\"\"\"" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#t1G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%\"kG\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "plot(\{-ka
ppa*lambda/h,tan(lambda*a)\},lambda=0..5,-5..5);" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}{PARA 13 "" 1 "" {GLPLOT2D 439 303 303 {PLOTDATA 
2 "6&-%'CURVESG6$7S7$$\"\"!F)F(7$$\"3GLLL3x&)*3\"!#=$!3GLLL3x&)*3\"F-7
$$\"3umm\"H2P\"Q?F-$!3umm\"H2P\"Q?F-7$$\"3MLL$eRwX5$F-$!3MLL$eRwX5$F-7
$$\"33ML$3x%3yTF-$!33ML$3x%3yTF-7$$\"3emm\"z%4\\Y_F-$!3emm\"z%4\\Y_F-7
$$\"3`LLeR-/PiF-$!3`LLeR-/PiF-7$$\"3]***\\il'pisF-$!3]***\\il'pisF-7$$
\"3>MLe*)>VB$)F-$!3>MLe*)>VB$)F-7$$\"3Y++DJbw!Q*F-$!3Y++DJbw!Q*F-7$$\"
3%ommTIOo/\"!#<$!3%ommTIOo/\"Fen7$$\"3YLL3_>jU6Fen$!3YLL3_>jU6Fen7$$\"
37++]i^Z]7Fen$!37++]i^Z]7Fen7$$\"33++](=h(e8Fen$!33++](=h(e8Fen7$$\"3/
++]P[6j9Fen$!3/++]P[6j9Fen7$$\"3UL$e*[z(yb\"Fen$!3UL$e*[z(yb\"Fen7$$\"
3wmm;a/cq;Fen$!3wmm;a/cq;Fen7$$\"3%ommmJ<gw\"Fen$!3%ommmJ<gw\"Fen7$$\"
3/+]iSj0x=Fen$!3/+]iSj0x=Fen7$$\"3gmmm\"pW`(>Fen$!3gmmm\"pW`(>Fen7$$\"
3K+]i!f#=$3#Fen$!3K+]i!f#=$3#Fen7$$\"3?+](=xpe=#Fen$!3?+](=xpe=#Fen7$$
\"37nm\"H28IH#Fen$!37nm\"H28IH#Fen7$$\"3um;zpSS\"R#Fen$!3um;zpSS\"R#Fe
n7$$\"3GLL3_?`(\\#Fen$!3GLL3_?`(\\#Fen7$$\"3fL$e*)>pxg#Fen$!3fL$e*)>px
g#Fen7$$\"33+]Pf4t.FFen$!33+]Pf4t.FFen7$$\"3uLLe*Gst!GFen$!3uLLe*Gst!G
Fen7$$\"30+++DRW9HFen$!30+++DRW9HFen7$$\"3:++DJE>>IFen$!3:++DJE>>IFen7
$$\"3F+]i!RU07$Fen$!3F+]i!RU07$Fen7$$\"3+++v=S2LKFen$!3+++v=S2LKFen7$$
\"3Jmmm\"p)=MLFen$!3Jmmm\"p)=MLFen7$$\"3B++](=]@W$Fen$!3B++](=]@W$Fen7
$$\"35L$e*[$z*RNFen$!35L$e*[$z*RNFen7$$\"3e++]iC$pk$Fen$!3e++]iC$pk$Fe
n7$$\"3[m;H2qcZPFen$!3[m;H2qcZPFen7$$\"3O+]7.\"fF&QFen$!3O+]7.\"fF&QFe
n7$$\"3Ymm;/OgbRFen$!3Ymm;/OgbRFen7$$\"3w**\\ilAFjSFen$!3w**\\ilAFjSFe
n7$$\"3yLLL$)*pp;%Fen$!3yLLL$)*pp;%Fen7$$\"3)RL$3xe,tUFen$!3)RL$3xe,tU
Fen7$$\"3Cn;HdO=yVFen$!3Cn;HdO=yVFen7$$\"3a+++D>#[Z%Fen$!3a+++D>#[Z%Fe
n7$$\"3SnmT&G!e&e%Fen$!3SnmT&G!e&e%Fen7$$\"3#RLLL)Qk%o%Fen$!3#RLLL)Qk%
o%Fen7$$\"37+]iSjE!z%Fen$!37+]iSjE!z%Fen7$$\"3a+]P40O\"*[Fen$!3a+]P40O
\"*[Fen7$$\"\"&F)$!\"&F)-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-F$6$7[uF'7$F+
$\"3u]rZb(pUc$F-7$F1$\"3i_yY)y'4]uF-7$$\"3!***\\PMnNrDF-$\"3Uc)yXuqe/
\"Fen7$F6$\"3#)\\O1,c,w9Fen7$$\"3_LLe*[`HP$F-$\"3)[Do!z'QHy\"Fen7$$\"3
rLLL$eI8k$F-$\"3wc*z4zn()>#Fen7$$\"3*QL$3xwq4RF-$\"3@AUAYHT/GFen7$F;$
\"3o`+b]mK'y$Fen7$$\"31nT5:j=XWF-$\"3]@'oTI@!zcFen7$$\"3h+]PfyG7ZF-$\"
3t9\"R=\"RL.6!#;7$$\"3+4FWXK1zZF-$\"3tQZ&[99%Q9Fj]l7$$\"3Q</^J'Qe%[F-$
\"3SlE<3\"pJ1#Fj]l7$$\"3IrUaCjAz[F-$\"3w&H9t4DVj#Fj]l7$$\"3@D\"yv,9E\"
\\F-$\"3L\"HCege;k$Fj]l7$$\"39z>h5<+Y\\F-$\"3#4lK#y([U*eFj]l7$$\"3gLek
.%*Qz\\F-$\"3g&H.8ixVa\"!#:7$$\"32)ozm4xF,&F-$!3Mi![.ES7\\#Fi_l7$$\"3)
>a8(*ykh/&F-$!3=ke*='ef%*oFj]l7$$\"3!fRZF[_&z]F-$!3)*)yEAOB/+%Fj]l7$$
\"3#)\\7yv,%H6&F-$!3/p-i\\4@<GFj]l7$$\"3we*[=c:(z^F-$!3_GleIZIp<Fj]l7$
F@$!3q2=7RJy)G\"Fj]l7$$\"3'QLL3FGT\\&F-$!3?mrVw*=+R'Fen7$$\"32++v$fl<u
&F-$!3V*oW^6'G8UFen7$$\"3Cmmm;HS*)fF-$!3E,u=C\"3H6$Fen7$FE$!3Kb\"*\\b6
HUCFen7$$\"31nm\"zWo)\\nF-$!3X.3W!3.?j\"Fen7$FJ$!3QPSUUlVh6Fen7$FO$!3^
$G1k4_]\"eF-7$FT$!3u>6!\\#3Iq>F-7$FY$\"3sT'\\L[<@[\"F-7$Fin$\"3IUP$Q)Q
.2[F-7$F^o$\"3I(GK'****)H+\"Fen7$$\"35+++v\"=YI\"Fen$\"3c;l_HUB>9Fen7$
Fco$\"3*)e4L=k\"Q5#Fen7$$\"3!********f\\[Q\"Fen$\"3X&)>P)GYEk#Fen7$$\"
3&*****\\7!Q4T\"Fen$\"3)f<+ORq-[$Fen7$$\"3*******\\UEqV\"Fen$\"3U\\Q#R
`R&))\\Fen7$Fho$\"3g\"Q89`$4\"f)Fen7$$\"3'=HKkAg\\Z\"Fen$\"3!H%QzpQfo7
Fj]l7$$\"3W$ek`h0o[\"Fen$\"3_]!e'QH36CFj]l7$$\"3=cwfipw*[\"Fen$\"3-Iw3
R$Q&4JFj]l7$$\"37H2$)4$GF\\\"Fen$\"3)Q!QfTOiwVFj]l7$$\"33-Q1d'*o&\\\"F
en$\"3!3e^j@TVQ(Fj]l7$$\"3/voH/5l)\\\"Fen$\"3;sy**[AffBFi_l7$$\"3*z%*H
:N7;]\"Fen$!3+*yplozT(>Fi_l7$$\"3%4-j()ptX]\"Fen$!3)y95Ek$4fpFj]l7$$\"
3*Q4'*f/Nv]\"Fen$!3M6z-\")HgBUFj]l7$$\"3%o;HKR'\\5:Fen$!3?Gf.e^YJIFj]l
7$$\"3v7`p(3>k^\"Fen$!3%Rk\")*fy$p$>Fj]l7$$\"3Ve9;#yTB_\"Fen$!3)p4_\"p
()QA9Fj]l7$$\"37/wiwWEG:Fen$!3'[h>O9AK7\"Fj]l7$$\"3-]P4rr=M:Fen$!3kicI
V4(\\F*Fen7$$\"3%=/E+cKga\"Fen$!3#ePRg18m'oFen7$F]p$!3qw$Gw%4$*QaFen7$
$\"3w;/Ev&[ge\"Fen$!3ZSg1+kj3OFen7$$\"34+Dc,#>Uh\"Fen$!3)QTUV&Q=mEFen7
$$\"3V$eky#)*QU;Fen$!3'*pX\"4yVV3#Fen7$Fbp$!371*z\"3H8%o\"Fen7$$\"3\"p
m;a)))G=<Fen$!3lcD2)Q4@A\"Fen7$Fgp$!3B\"H)[&=Z5/*F-7$F\\q$!3w\"*Gs#z]m
1%F-7$Faq$!3![F&Q76Ahx!#>7$Ffq$\"3onVaOoTuEF-7$F[r$\"3IMT`6w&zg'F-7$F`
r$\"3FREgZDq98Fen7$$\"3$p;a8d3AM#Fen$\"3Yudr\\<F\\=Fen7$Fer$\"3PxP;!RE
l\"GFen7$$\"3;$ek`1OzT#Fen$\"3Y6csPK[#z$Fen7$$\"3-+v$41oWW#Fen$\"3!)>%
)ejLutcFen7$$\"3WeRseStdCFen$\"3%)>\"4mM2o[(Fen7$$\"3)oT5l0+5Z#Fen$\"3
;]$zs)Qe%4\"Fj]l7$$\"35YOSbIjxCFen$\"3^75INLy?9Fj]l7$$\"3IvoHagE%[#Fen
$\"3aO5/\"RH9-#Fj]l7$$\"3#**[VPb#e([#Fen$\"39Y]sB\"3@c#Fj]l7$$\"3_/,>`
!**3\\#Fen$\"3a#*o*Q$Qf'\\$Fj]l7$$\"37>nj_b@%\\#Fen$\"3geAfy:D-bFj]l7$
Fjr$\"3]$o-wp\\(*G\"Fi_l7$$\"3GxJ#H'p(4]#Fen$!3/*=$p]q9eKFi_l7$$\"3$3-
jP(=U/DFen$!3%f:G;Tm!)>(Fj]l7$$\"3RkGg%ymy]#Fen$!3irnC5wUXSFj]l7$$\"3%
zqUap68^#Fen$!3MD%*QDX!G\"GFj]l7$$\"3[&RAr^,#=DFen$!3We1jmJ!pu\"Fj]l7$
$\"3.$3-)Q84DDFen$!3s90p$)f(fE\"Fj]l7$$\"37e9;#)4()QDFen$!3)fLYz(e8[\"
)Fen7$$\"3AL3_D1l_DFen$!3=P<\"fbj/*fFen7$$\"3S$eRA\"*4-e#Fen$!3Q.7%oO2
T)QFen7$F_s$!3efRE2#)*)RGFen7$$\"3%omm\"z+vbEFen$!3-T&[M%p&z(=Fen7$Fds
$!3^Rr!*=I*HM\"Fen7$Fis$!3\\dge>cN<pF-7$F^t$!3G(y&[lkZaFF-7$Fct$\"3sET
La/'o.'Fg\\m7$Fht$\"3OfBvWU+zRF-7$F]u$\"3Ee\"=d'**H*)*)F-7$$\"3;L$3_NJ
OG$Fen$\"3\\C!=9$REP7Fen7$Fbu$\"3)z!)Qe%)\\Gu\"Fen7$$\"3e**\\il!z6O$Fe
n$\"3kO0a`FqX@Fen7$$\"3GLLeR%p\")Q$Fen$\"3,eOvr[FGFFen7$$\"3)pmTN\")f^
T$Fen$\"3ikemX#3Em$Fen7$Fgu$\"3!=iQ))[N;W&Fen7$$\"3u\"H#oK)yVX$Fen$\"3
!oU[*G7SHpFen7$$\"3C$ekyZ2mY$Fen$\"3)o3j![gP(\\*Fen7$$\"3?HdX+=ssMFen$
\"3[XubWT/k6Fj]l7$$\"3=vo/Bh$)yMFen$\"3=hj8Jv!=]\"Fj]l7$$\"3;[CM%G$*=[
$Fen$\"3%y.DiNpgv\"Fj]l7$$\"39@!QcW]\\[$Fen$\"3fed,jA]8@Fj]l7$$\"3o$fL
pg2!)[$Fen$\"3#**))[O**3Il#Fj]l7$$\"3mm\"H#oZ1\"\\$Fen$\"3O-/#*ovZhNFj
]l7$$\"3mRZ_H>7%\\$Fen$\"3%)ySBy[f9aFj]l7$$\"3?7.#34zr\\$Fen$\"3sg?)=7
l$G6Fi_l7$$\"3=&)e6_iB+NFen$!3g0\"3$>6LZ8!#97$$\"3<e9T8MH.NFen$!3y)Q\\
#\\+qk'*Fj]l7$$\"3:Jqqu0N1NFen$!3^%HyjUN;,&Fj]l7$$\"39/E+OxS4NFen$!3Mm
d$)4c]#Q$Fj]l7$$\"37x\")H(*[Y7NFen$!3SKA*p$[M_DFj]l7$$\"36]Pfe?_:NFen$
!3ur)*yI!o!\\?Fj]l7$$\"3hTgx.2vFNFen$!3x5-<Ug7W6Fj]l7$F\\v$!3z*)[#))HZ
*>zFen7$$\"3)*\\PMFwrmNFen$!3:E!>*p#H4q%Fen7$$\"3%o;Hd!fX$f$Fen$!3*>;u
P>cvI$Fen7$$\"3r$e9T=%>?OFen$!3v]@\"3%p@@DFen7$Fav$!3Llk!\\Nq-,#Fen7$$
\"3ILe*[t\\sp$Fen$!3h\\G>5?p,9Fen7$Ffv$!3$)y:m\"*\\S:5Fen7$F[w$!3q*G)*
Gi[l)\\F-7$F`w$!3.Z['H2pQS\"F-7$Few$\"3L!>BAFdV,#F-7$Fjw$\"3O&\\2$))\\
@'y&F-7$F_x$\"3WA*f%yz<c6Fen7$$\"3h+v=n(*fDVFen$\"38K&)H#yu(Q;Fen7$Fdx
$\"3KI]^;@?%[#Fen7$$\"3N](=UAVBS%Fen$\"3[J]!QAil:$Fen7$$\"3MMe9\"z-lU%
Fen$\"3/:WXk,n`UFen7$$\"3W<H2eBm]WFen$\"3YsRQfZ\"**R'Fen7$Fix$\"35*z+3
m'fh7Fj]l7$$\"3GUg_AVu\"[%Fen$\"3PRSbz\"4<u\"Fj]l7$$\"3+%3_+sm')[%Fen$
\"3L=O<aHW2GFj]l7$$\"3'[5:)=z7#\\%Fen$\"3%[Z*>6YqUSFj]l7$$\"3uD\"yv6*e
&\\%Fen$\"39/)*3b`+;sFj]l7$$\"3gY6M;.0*\\%Fen$\"3+jT:Olt^LFi_l7$$\"3Zn
T5::^-XFen$!3!*e=UQbPn7Fi_l7$$\"3M)=nQrsf]%Fen$!3q-2?_dxG`Fj]l7$$\"3@4
-j7RV4XFen$!3guE\\qX6tLFj]l7$$\"32IKR6^*G^%Fen$!3?L%QlU.rY#Fj]l7$$\"3%
4Dc,Jcj^%Fen$!3N^nwW_QW>Fj]l7$$\"3o#H#o2(yK_%Fen$!31i4s.y%\\O\"Fj]l7$$
\"3TM$3_5,-`%Fen$!3%QG5B7.30\"Fj]l7$$\"3O,DJ&p!*yb%Fen$!3*)RKR%f1xV&Fe
n7$F^y$!30*Hw`sv$HOFen7$$\"3fLe*[=Y.h%Fen$!3/+za)4d\"oFFen7$$\"3m+]P%3
7^j%Fen$!3We<p!ywE@#Fen7$$\"3unT&Q)z()fYFen$!3#=L8j/Y1#=Fen7$Fcy$!3g%z
yDWsg_\"Fen7$$\"3-n\"z>6but%Fen$!35!H\\V++@3\"Fen7$Fhy$!37\"\\$p8?SVxF
-7$F]z$!3m\"4^av8?b$F-7$Fbz$!3Mint&*RBBh!#L-Fgz6&FizF(FjzF(-%+AXESLABE
LSG6$Q'lambda6\"Q!6\"-%%VIEWG6$;F(Fbz;FdzFbz" 1 2 0 1 10 0 2 9 1 4 2 
1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 33 "We see that there are eigenvalues" }}{PARA 0 "
" 0 "" {XPPEDIT 18 0 "-pi/(2*a)" "6#,$*&%#piG\"\"\"*&\"\"#F&%\"aGF&!\"
\"F*" }{TEXT -1 1 "<" }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 
4 "(1)<" }{XPPEDIT 18 0 "pi/(2*a)" "6#*&%#piG\"\"\"*&\"\"#F%%\"aGF%!\"
\"" }{TEXT -1 2 "; " }{XPPEDIT 18 0 "pi/(2*a)" "6#*&%#piG\"\"\"*&\"\"#
F%%\"aGF%!\"\"" }{TEXT -1 1 "<" }{XPPEDIT 18 0 "lambda(2)" "6#-%'lambd
aG6#\"\"#" }{TEXT -1 1 "<" }{XPPEDIT 18 0 "3*pi/(2*a)" "6#*(\"\"$\"\"
\"%#piGF%*&\"\"#F%%\"aGF%!\"\"" }{TEXT -1 4 ",..\n" }{XPPEDIT 18 0 "(2
*n-3)/(2*a)" "6#*&,&*&\"\"#\"\"\"%\"nGF'F'\"\"$!\"\"F'*&F&F'%\"aGF'F*
" }{TEXT -1 1 "<" }{XPPEDIT 18 0 "lambda(n)" "6#-%'lambdaG6#%\"nG" }
{TEXT -1 1 "<" }{XPPEDIT 18 0 "(2*n-1)/(2*a)" "6#*&,&*&\"\"#\"\"\"%\"n
GF'F'F'!\"\"F'*&F&F'%\"aGF'F)" }{TEXT -1 3 "..." }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 74 "i.e there is one eigenvalue in each interval when th
e tangent goes from  -" }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }
{TEXT -1 6 " to + " }{XPPEDIT 18 0 "infinity" "6#%)infinityG" }{TEXT 
-1 2 " ." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Let us find the first
 10 eigenvalues" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "ne:=10;" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#neG\"#5" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 40 "f:=lambda->tan(lambda*a)+kappa*lambda/h;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGR6#%'lambdaG6\"6$%)operatorG%&ar
rowGF(,&-%$tanG6#*&9$\"\"\"%\"aGF2F2*&*&%&kappaGF2F1F2F2%\"hG!\"\"F2F(
F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "s:=seq(fsolve(f(lam
bda)=0,lambda=(2*n-3)*Pi/(2*a)..(2*n-1)*Pi/(2*a)),n=1..ne);" }}{PARA 
12 "" 1 "" {XPPMATH 20 "6#>%\"sG6,$\"\"!F'$\"+UHPwy!#5$\"+Dcgr;!\"*$\"
+&e8ih#F-$\"+YFb'e$F-$\"+Y<foXF-$\"+PvncbF-$\"+ItB[lF-$\"+R/'>a(F-$\"+
ti6P&)F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "We must also find the
 normalizing factors" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "b:=
seq(int((sin(s[n]*x))^2,x=0..a),n=1..ne);" }}{PARA 12 "" 1 "" 
{XPPMATH 20 "6#>%\"bG6,$\"\"!F'$\"+\\tOz=!\"*$\"+UTd-<F*$\"+bY`M;F*$\"
+=E'og\"F*$\"+tll$f\"F*$\"+#y\"['e\"F*$\"+#>\">#e\"F*$\"+:ZVz:F*$\"+VQ
cx:F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "The steady state solutio
n is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ts:=x->(x*h*t1)/(ka
ppa+h*a);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#tsGR6#%\"xG6\"6$%)oper
atorG%&arrowGF(*&*(9$\"\"\"%\"hGF/%#t1GF/F/,&%&kappaGF/*&F0F/%\"aGF/F/
!\"\"F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "The five first coe
fficients are (we must omit the zero eigenvector for n=0)" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "alpha:=seq(int(-ts(x)*sin(s[n]*x)/b
[n],x=0..a),n=2..ne);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%&alphaG6+$!
+\"enqI&!#5$\"+\\G$Q!=F($!+!Hg#\\$)!#6$\"+RfDgYF-$!+tp#o$HF-$\"+DU:4?F
-$!+)[*3d9F-$\"+R;V.6F-$!+HROQ')!#7" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 78 "temp:=(x,t)->ts(x)+sum(alpha[n]*sin(s[n+1]*x)*exp(-(s
[n+1])^2*t*k),n=1..ne-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%tempGR
6$%\"xG%\"tG6\"6$%)operatorG%&arrowGF),&-%#tsG6#9$\"\"\"-%$sumG6$*(&%&
alphaG6#%\"nGF2-%$sinG6#*&&%\"sG6#,&F:F2F2F2F2F1F2F2-%$expG6#,$*()F?\"
\"#F29%F2%\"kGF2!\"\"F2/F:;F2,&%#neGF2F2FLF2F)F)F)" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 83 "plot([temp(x,0.25),temp(x,1),temp(x,4),ts(x
)],x=0..a,color=[black,blue,green,red]);" }}{PARA 13 "" 1 "" 
{GLPLOT2D 281 281 281 {PLOTDATA 2 "6(-%'CURVESG6$7U7$$\"\"!F)F(7$$\"3%
)eD2LzxZo!#>$\"3o3M^aH<%4\"!#B7$$\"3)\\$px*G*f!G\"!#=$\"39A0YDpu4AF07$
$\"3+5@exGm]>F4$\"3'\\Oy%o]%=&QF07$$\"3[99!=3o^i#F4$\"3_^WS<!G\\>'F07$
$\"35!\\D0[nkH$F4$\"3gY80I9K&f*F07$$\"37\"=Za&z%)=RF4$\"3v\"z#3'3]bT\"
!#A7$$\"3edXa()oGjXF4$\"3IDpqR)oK4#FK7$$\"3W%3**Hbm(H_F4$\"3\"y8-Ei)p1
JFK7$$\"37PRr4)3T*eF4$\"390iD#e+Xc%FK7$$\"3y\"[)yykYxlF4$\"31KMo!zk<s'
FK7$$\"3[s'ocGo$zrF4$\"3khf39)))[Q*FK7$$\"3UQ0;gr'p&yF4$\"3OSK$\\d\"pb
8!#@7$$\"3vFMt?$[t`)F4$\"3$*3LDz>$\\%>Fdo7$$\"3u\"p30k@I>*F4$\"3!43G!>
rBKFFdo7$$\"3#*R7\\HeV)y*F4$\"3uWrJ!Q[`p$Fdo7$$\"3#G[))*)3W'\\5!#<$\"3
cnIVrS1[_Fdo7$$\"30'[@XS@'46Fgp$\"3)\\P$fw$*y9qFdo7$$\"3G9w$3G*Qz6Fgp$
\"3!4y\"\\r&HBv*Fdo7$$\"3>%3*3tc9T7Fgp$\"3Y6%>^#p8'H\"!#?7$$\"3Ey0DBA!
*38Fgp$\"3_Q&HdI'Gd<Fiq7$$\"3qjE.#[AMP\"Fgp$\"3VnU5wejIBFiq7$$\"3*R:_M
gU2W\"Fgp$\"30T'4Ywk]5$Fiq7$$\"3\"Qu#)**[jD]\"Fgp$\"3aj#>n4_J,%Fiq7$$
\"3=hw\"ymX#p:Fgp$\"3)=f?&*4-OD&Fiq7$$\"3mwM!*4(4&Q;Fgp$\"31e*o4<&z$*o
Fiq7$$\"3CDL:iU!))p\"Fgp$\"3qY_$oH.bn)Fiq7$$\"3o(R1/.CRw\"Fgp$\"3x*)GU
k!*R/6F-7$$\"32Id)H7*>J=Fgp$\"3;cG%Q=XmS\"F-7$$\"3Gfb7wY,(*=Fgp$\"3%>u
6x@$Hp<F-7$$\"3cM5'zg%pg>Fgp$\"3Gzu#\\8eR>#F-7$$\"3**oJ8:.SJ?Fgp$\"3'e
,S?2MVw#F-7$$\"3)**p!zPD$\\4#Fgp$\"3suyI;g&)yLF-7$$\"3k<!fhumF;#Fgp$\"
3[>L*=*=pcTF-7$$\"3wak3@YBCAFgp$\"37cLex*pN)\\F-7$$\"39/b<W_V\"H#Fgp$
\"33j%))\\Pld.'F-7$$\"3#*z_V$zlYN#Fgp$\"3W2j)\\&fq\"=(F-7$$\"3cmjYO*f2
U#Fgp$\"3+AlP!)eXc&)F-7$$\"3)eq)=U!z`[#Fgp$\"3k&*[_zL)*35F47$$\"3#Q'HH
d#HIb#Fgp$\"3#)>)\\6nZ5>\"F47$$\"3z^r&[X%==EFgp$\"3ID^9K^^)Q\"F47$$\"3
'=BS\\0:[o#Fgp$\"3)zr0(4(\\Sh\"F47$$\"3[gN,?R*3v#Fgp$\"3GsQYBI?i=F47$$
\"3@/0LMNh6GFgp$\"3Npv;>]I7@F47$$\"3w#oUX107)GFgp$\"3%39+(pWMDCF47$$\"
3W46xe&[M%HFgp$\"3aaq&>)*y#HFF47$$\"3H.6)e58)4IFgp$\"3M\\6!pQ!*z2$F47$
$\"34QfjvFdTIFgp$\"3ajPnP%4PD$F47$$\"3Kt2RXCLtIFgp$\"3wn9F,^-NMF47$$\"
3M'Q?zaiu5$Fgp$\"3\"\\$3,Q)zfj$F47$$\"3!)***\\/l#fTJFgp$\"3Osi_Sl4VQF4
-%'COLOURG6&%$RGBGF)F)F)-F$6$7SF'7$F+$\"3]BDs!***[$R#Fiq7$F2$\"3#Q;%f=
]V)\\%Fiq7$F8$\"3;PQgqwH:pFiq7$F=$\"3iBj&eJ&oF%*Fiq7$FB$\"36b%34o'[.7F
-7$FG$\"3-\">xl*Q8d9F-7$FM$\"3H1di_2IM<F-7$FR$\"3^MSZn`7R?F-7$FW$\"3-#
Hm/,<TO#F-7$Ffn$\"3&RaOD%RRBFF-7$F[o$\"3ka=G$flK1$F-7$F`o$\"3+L!>yC4\\
Z$F-7$Ffo$\"318ex9E<ARF-7$F[p$\"3CA=U)y4$)Q%F-7$F`p$\"3%>nV)=B*Q%[F-7$
Fep$\"3LdH2$o(oGaF-7$F[q$\"3Qz[_)HKK'fF-7$F`q$\"3**\\]&RP4Lj'F-7$Feq$
\"3mO5$pu2CF(F-7$F[r$\"3HcUccLAE!)F-7$F`r$\"3A$[t!Q+*zz)F-7$Fer$\"3!3E
TkX&Gi'*F-7$Fjr$\"35W1^l&e60\"F47$F_s$\"3)o2'z6)***[6F47$Fds$\"3;D9.*)
=wd7F47$Fis$\"3w-oEd0ee8F47$F^t$\"3]slL\"=3TZ\"F47$Fct$\"3H#Q-eJD4g\"F
47$Fht$\"3[z&Hx*HaK<F47$F]u$\"3%\\[47%z<n=F47$Fbu$\"3c:Iut8DD?F47$Fgu$
\"3aT8K4f8v@F47$F\\v$\"3d>4#p:)\\VBF47$Fav$\"3i.O27Md.DF47$Ffv$\"3MR*y
9$3#oo#F47$F[w$\"3yUlPP>9nGF47$F`w$\"3]$\\e!*\\FQ1$F47$Few$\"3'=;R&3c?
kKF47$Fjw$\"3wB&>W(3\\#[$F47$F_x$\"3*4xp)4\"o3q$F47$Fdx$\"3=D\"H;3KB$R
F47$Fix$\"3o%3&p(\\b)pTF47$F^y$\"3!3'p)zvb\\R%F47$Fcy$\"3M$4/H7v2m%F47
$Fhy$\"3i![(zH%y`!\\F47$F]z$\"3i-L2S:-t^F47$Fgz$\"3v986dS^NaF47$Fa[l$
\"3$\\_?M\"G8CdF4-Ff[l6&Fh[lF(F($\"*++++\"!\")-F$6$7SF'7$F+$\"3I-_q&f6
UT\"F-7$F2$\"3+Q$\\i%3DXEF-7$F8$\"3K(\\X&o1)3.%F-7$F=$\"3u`z]5egFaF-7$
FB$\"3)e:]llU-#oF-7$FG$\"3*>P`j&[A9\")F-7$FM$\"3q$fXj/'[d%*F-7$FR$\"3i
x.$yA#4&3\"F47$FW$\"37%R1v!eZC7F47$Ffn$\"3;i?&fs;%o8F47$F[o$\"3S5d(\\&
\\s&\\\"F47$F`o$\"33J-7,\\oR;F47$Ffo$\"37#[0/Kt\\y\"F47$F[p$\"33o'*f`c
tD>F47$F`p$\"3K8\"4:)*QU0#F47$Fep$\"3-\"eXG@Hz?#F47$F[q$\"3HGBvHo\"*QB
F47$F`q$\"3%e#\\znOC#\\#F47$Feq$\"3AE<!\\dd)GEF47$F[r$\"398w,5ouzFF47$
F`r$\"3C4H@qdVCHF47$Fer$\"3q'fXo)>[wIF47$Fjr$\"3k(Rx.%H5<KF47$F_s$\"3)
>%>\\LR()pLF47$Fds$\"3Gl>!H2%yHNF47$Fis$\"3#>#*>%>'=+n$F47$F^t$\"3jw\\
kn'pD#QF47$Fct$\"3iRN8IbP\")RF47$Fht$\"3cYb1N:$z8%F47$F]u$\"3EO**\\4!R
0H%F47$Fbu$\"3o]o0t3HhWF47$Fgu$\"3xUr$zK#*eh%F47$F\\v$\"3QDOD')[=#y%F4
7$Fav$\"3Wlw\"y@bR$\\F47$Ffv$\"3*)4$f;i]55&F47$F[w$\"3KI$[a))y$f_F47$F
`w$\"3'QH.wx1gU&F47$Few$\"37R<SZQ,!f&F47$Fjw$\"3)4i3+TaGw&F47$F_x$\"35
&y%e$Q)RIfF47$Fdx$\"3!\\<SR#=!G5'F47$Fix$\"3y7(=)p2\"[F'F47$F^y$\"3/Xk
>yruLkF47$Fcy$\"3A>WN_I!ph'F47$Fhy$\"3Gp(pKK'e\"y'F47$F]z$\"3q+v=_Q.ep
F47$Fgz$\"3PHCE)p3x7(F47$Fa[l$\"3A%**4avj3J(F4-Ff[l6&Fh[lF(F^elF(-F$6$
7SF'7$F+$\"3!fQWKd;Ml\"F-7$F2$\"3))\\R&G\\X?4$F-7$F8$\"3^LmXuT$*4ZF-7$
F=$\"38q3XssaQjF-7$FB$\"3oP\"ym&)>%fzF-7$FG$\"35a**QGi<i%*F-7$FM$\"3FU
^#3M>=5\"F47$FR$\"3Ytp:gGui7F47$FW$\"3%4T@dH]JU\"F47$Ffn$\"3IS&z(f!\\
\")e\"F47$F[o$\"3,SE(41![L<F47$F`o$\"3Qc?wY$)3(*=F47$Ffo$\"3e%3#*QUo81
#F47$F[p$\"3$)>\\c=Fo>AF47$F`p$\"3oK$3\\EZMO#F47$Fep$\"3Vz_r:sRMDF47$F
[q$\"37F2fZU@zEF47$F`q$\"3G/(zG)3nZGF47$Feq$\"3[>)))*[Ny'*HF47$F[r$\"3
k\\nzwPQgJF47$F`r$\"3pYOv\"))phJ$F47$Fer$\"3+u0CRjryMF47$Fjr$\"3LqfsGZ
)zi$F47$F_s$\"3Y\\\\C_3**)y$F47$Fds$\"3_9Xjk4BcRF47$Fis$\"3z4@5uR\"=5%
F47$F^t$\"3o')Rq7w/fUF47$Fct$\"3&HL-nM&[@WF47$Fht$\"3Wrb$)y*)R!e%F47$F
]u$\"3M.76*3cTt%F47$Fbu$\"3=IE?Xr([!\\F47$Fgu$\"3M#\\Esgx#e]F47$F\\v$
\"30&)*4JMl?A&F47$Fav$\"3+#\\kaP\"[q`F47$Ffv$\"3!*[#R1IRF`&F47$F[w$\"3
U'*G<j8T&o&F47$F`w$\"30*yv*ew*\\%eF47$Few$\"3AhA%GVA5+'F47$Fjw$\"3'[*p
X6fOkhF47$F_x$\"3P@f>&4&o@jF47$Fdx$\"3kU@o#omD['F47$Fix$\"38>mR*3;@k'F
47$F^y$\"33bo+Cas)y'F47$Fcy$\"393^KmivcpF47$Fhy$\"36oZ![0Xq5(F47$F]z$
\"3#eC(Q\\UGnsF47$Fgz$\"37-Km/Ol?uF47$Fa[l$\"3aNmOA*pae(F4-Ff[l6&Fh[lF
^elF(F(-%+AXESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F($\"+aEfTJ!\"*%(DEFAULT
G" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "
Curve 2" "Curve 3" "Curve 4" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "We see that the solution
 gradually approaches the steady state solution!" }}}}{MARK "28 0 0" 
72 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }