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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 256 48 "Cooking a cylindrical r
oast: a numerical example" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "restart;with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the \+
name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 72 "Let us first select values for the height b and radius a \+
of the cylinder" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "a:=1:b:=
2:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 128 "we also choose for the the
rmal diffusion constant k=1.\n\nWe have a formal solution for the temp
erature of the object on the form\n" }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "
u(z,r,t) = sum(sum(J[0](alpha[0,i]*r/a)*B[j,i]*sin((2*j-1)*pi*z/b)*exp
(-t*((alpha[0,i]/a)^2+((2*j-1)*pi/b)^2)),i = 1 .. infinity),j = 1 .. i
nfinity);" "6#/-%\"uG6%%\"zG%\"rG%\"tG-%$sumG6$-F+6$**-&%\"JG6#\"\"!6#
*(&%&alphaG6$F4%\"iG\"\"\"F(F;%\"aG!\"\"F;&%\"BG6$%\"jGF:F;-%$sinG6#**
,&*&\"\"#F;FAF;F;F;F=F;%#piGF;F'F;%\"bGF=F;-%$expG6#,$*&F)F;,&*$*&&F86
$F4F:F;F<F=FHF;*$*(,&*&FHF;FAF;F;F;F=F;FIF;FJF=FHF;F;F=F;/F:;F;%)infin
ityG/FA;F;Fen" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 27 "The expansion coefficients " }{XPPEDIT 18 0 "B[j,i];" "6#
&%\"BG6$%\"jG%\"iG" }{TEXT -1 13 " are given by" }}{PARA 0 "" 0 "" 
{XPPEDIT 18 0 "B[j,i] = 8*Delta/(2*j-1)/(Pi*alpha[0,i]*J[1](alpha[0,i]
));" "6#/&%\"BG6$%\"jG%\"iG**\"\")\"\"\"%&DeltaGF+,&*&\"\"#F+F'F+F+F+!
\"\"F0*(%#PiGF+&%&alphaG6$\"\"!F(F+-&%\"JG6#F+6#&F46$F6F(F+F0" }}
{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "J[0]" "6#&%\"JG6#
\"\"!" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "J[1]" "6#&%\"JG6#\"\"\"" }
{TEXT -1 76 " are Bessel functions of the first kind of order 0 and 1,
 respectively, and " }{XPPEDIT 18 0 "alpha[0,i]" "6#&%&alphaG6$\"\"!%
\"iG" }}{PARA 0 "" 0 "" {TEXT -1 20 "is the i'th zero of " }{XPPEDIT 
18 0 "J[0]" "6#&%\"JG6#\"\"!" }{TEXT -1 36 ". Without loss of generali
ty we put " }{XPPEDIT 18 0 "Delta = 1;" "6#/%&DeltaG\"\"\"" }{TEXT -1 
58 ".\nFor moderate times the terms with the smallest values of" }}
{PARA 0 "" 0 "" {XPPEDIT 18 0 "x=(alpha[0,i]/a)^2+((2*j-1)*pi/b)^2)" "
6#/%\"xG,&*$*&&%&alphaG6$\"\"!%\"iG\"\"\"%\"aG!\"\"\"\"#F-*$*(,&*&F0F-
%\"jGF-F-F-F/F-%#piGF-%\"bGF/F0F-" }}{PARA 0 "" 0 "" {TEXT -1 71 "will
 dominate. We first need to find the lowest values of the two terms" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "z1:=evalf((1*Pi/b)^2);\nz2
:=evalf((3*Pi/b)^2);\nz3:=evalf((5*Pi/b)^2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#z1G$\"+,6SnC!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%#z2G$\"+\"*4m?A!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#z3G$\"+_F]
oh!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 61 "To find the first zeroes of J0 let us first plot the func
tion" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot(BesselJ(0,x),x
=0..12);" }}{PARA 13 "" 1 "" {GLPLOT2D 303 303 303 {PLOTDATA 2 "6%-%'C
URVESG6$7do7$$\"\"!F)$\"\"\"F)7$$\"+]i9Rl!#6$\"+[FJ*)**!#57$$\"+]#HyI
\"F2$\"+c_Gd**F27$$\"+vQuh>F2$\"+\">?S!**F27$$\"++&ech#F2$\"+f#)oH)*F2
7$$\"+QPf`PF2$\"+0I&3l*F27$$\"+v*G:*[F2$\"+\\(42T*F27$$\"+]L)4X(F2$\"+
qN\\f')F27$$\"+X.u-5!\"*$\"+kd!*RwF27$$\"+Gy:f7FT$\"+!y*>7kF27$$\"+VP-
y8FT$\"+:+@(y&F27$$\"+e'*)o\\\"FT$\"+lOiN^F27$$\"+G%o*>;FT$\"+V+.SWF27
$$\"+)>ZIu\"FT$\"+:JlIPF27$$\"+QaLq=FT$\"+\"Qr1*HF27$$\"+yOi(*>FT$\"+T
Wh_AF27$$\"+.P]C@FT$\"+y*Qr_\"F27$$\"+GPQ^AFT$\"+2*>\"*>)F/7$$\"+Ha*=Q
#FT$\"+J(=g>\"F/7$$\"+IrS7DFT$!+;e@`aF/7$$\"+&o;Bu#FT$!+1:f3;F27$$\"+!
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\"+5cZ6NFT$!+dhw;QF27$$\"+W8>DOFT$!+kCTSRF27$$\"+yq!*QPFT$!+W=75SF27$$
\"+\"e;l!QFT$!+YMJESF27$$\"+%3ET(QFT$!+,S)R-%F27$$\"+(eN<%RFT$!+]mY.SF
27$$\"+!4X$4SFT$!+(Hg^'RF27$$\"+DL*Q7%FT$!+/\"))4'QF27$$\"+g:WQUFT$!+w
Wv4PF27$$\"+=_$\\]%FT$!+F!*)R>$F27$$\"+gs#3u%FT$!+0DlxDF27$$\"+=#Q'**
\\FT$!+iF:x<F27$$\"+_u3Y_FT$!+dg1V%*F/7$$\"+v8B.bFT$!+%oW3u&!#77$$\"+o
(p$RdFT$\"+M[\")esF/7$$\"+Dp2%*fFT$\"+[*H+\\\"F27$$\"+ygkeiFT$\"+%>\"*
)\\@F27$$\"+-V&*)['FT$\"+Y.y$e#F27$$\"+**QK8mFT$\"+xa#ov#F27$$\"+&\\$p
PnFT$\"+w[X%)GF27$$\"+e%zh'oFT$\"+')>[nHF27$$\"+?am%*pFT$\"+%4+0+$F27$
$\"+oQO?rFT$\"+g)zZ)HF27$$\"+:B1YsFT$\"+eykAHF27$$\"+F?ontFT$\"+Xa+?GF
27$$\"+Q<I*[(FT$\"+wLsxEF27$$\"+XwPfxFT$\"+5]GMAF27$$\"+gG0-!)FT$\"+?$
)o6<F27$$\"+]/;h#)FT$\"+@Swi5F27$$\"+Q/&f\\)FT$\"+w#>XI%F/7$$\"+5zj_()
FT$!+_bNlEF/7$$\"+=3;%**)FT$!+%R>**)))F/7$$\"+[=iY#*FT$!+Zw1n9F27$$\"+
]'[M\\*FT$!+%Gx'G>F27$$\"+QM&=v*FT$!+@?FvAF27$$\"+*p!Hw)*FT$!+,r]'Q#F2
7$$\"+'zs++\"!\")$!+YMnfCF27$$\"+.$)z75Fd^l$!+*GiW\\#F27$$\"+5Q_D5Fd^l
$!+'fF()[#F27$$\"+WR9Q5Fd^l$!+JFgVCF27$$\"+ySw]5Fd^l$!+w.RgBF27$$\"+q1
Oi5Fd^l$!+Q([=D#F27$$\"+is&R2\"Fd^l$!+s-S9@F27$$\"+o#R05\"Fd^l$!+LqM-<
F27$$\"+KXJC6Fd^l$!+Ek#4C\"F27$$\"+ARm\\6Fd^l$!+.27UoF/7$$\"+Al#R<\"Fd
^l$!+of><7F/7$$\"#7F)$\"+!3J*oZF/-%'COLOURG6&%$RGBG$\"#5!\"\"F(F(-%+AX
ESLABELSG6$Q\"x6\"Q!6\"-%%VIEWG6$;F(Fjal%(DEFAULTG" 1 2 0 1 10 0 2 9 
1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 100 "\na1:=fsolve(BesselJ(0,x),x=0..4);\na2:=fsol
ve(BesselJ(0,x),x=4..7);\na3:=fsolve(BesselJ(0,x),x=7..10);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#a1G$\"+eb#[S#!\"*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#a2G$\"+5\"y+_&!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6#>%#a3G$\"+8zs`')!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "r
1:=a1^2/a^2;r2:=a2^2/a^2;r3:=a3^2/a^2;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%#r1G$\"+kf=$y&!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r2G$
\"+Mi7ZI!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#r3G$\"+z1q)[(!\")" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "The four lowest damping coeffic
ients are then " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 47 "d11:=z1+r1;\nd21:=z2+r1;\nd12:=z1+r2;\nd22:=z2
+r2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$d11G$\"+lqe]#)!\"*" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$d21G$\"+(ez*)z#!\")" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%$d12G$\"+Wj'QH$!\")" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$d22G$\"+Dsyn_!\")" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 19 "The coefficients B/" }{XPPEDIT 18 0 "Delta" "6#%&DeltaG" }
{TEXT -1 58 " corresponding to the four lowest damping coefficients ar
e" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "b11:=8/(a1*BesselJ(1,
a1)*Pi);\nb21:=8/(3*a1*BesselJ(1,a1)*Pi);\nb12:=8/(a2*BesselJ(1,a2)*Pi
);\nb22:=8/(3*a2*BesselJ(1,a2)*Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6
#>%$b11G,$*&\"\"\"F'%#PiG!\"\"$\"+*y)*yS'!\"*" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%$b21G,$*&\"\"\"F'%#PiG!\"\"$\"+ji'f8#!\"*" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%$b12G,$*&\"\"\"F'%#PiG!\"\"$!+Mq>fU!\"*" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$b22G,$*&\"\"\"F'%#PiG!\"\"$!+XBt>9
!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 63 "Let us construct the first four spatial terms in the expansion:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "r11:=(r,z)->b11*BesselJ
(0,a1*r/a)*sin(Pi*z/b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$r11GR6$%
\"rG%\"zG6\"6$%)operatorG%&arrowGF)*(%$b11G\"\"\"-%(BesselJG6$\"\"!*&*
&%#a1GF/9$F/F/%\"aG!\"\"F/-%$sinG6#*&*&%#PiGF/9%F/F/%\"bGF9F/F)F)F)" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "r21:=(r,z)->b21*BesselJ(0,
a1*r/a)*sin(3*Pi*z/b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$r21GR6$%
\"rG%\"zG6\"6$%)operatorG%&arrowGF)*(%$b21G\"\"\"-%(BesselJG6$\"\"!*&*
&%#a1GF/9$F/F/%\"aG!\"\"F/-%$sinG6#,$*&*&%#PiGF/9%F/F/%\"bGF9\"\"$F/F)
F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "r12:=(r,z)->b12*Bes
selJ(0,a2*r/a)*sin(Pi*z/b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$r12G
R6$%\"rG%\"zG6\"6$%)operatorG%&arrowGF)*(%$b12G\"\"\"-%(BesselJG6$\"\"
!*&*&%#a2GF/9$F/F/%\"aG!\"\"F/-%$sinG6#*&*&%#PiGF/9%F/F/%\"bGF9F/F)F)F
)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "r22:=(r,z)->b22*Bessel
J(0,a2*r/a)*sin(3*Pi*z/b);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$r22GR
6$%\"rG%\"zG6\"6$%)operatorG%&arrowGF)*(%$b22G\"\"\"-%(BesselJG6$\"\"!
*&*&%#a2GF/9$F/F/%\"aG!\"\"F/-%$sinG6#,$*&*&%#PiGF/9%F/F/%\"bGF9\"\"$F
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{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "The series at t=0 is slow
ly converging" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "plot3d(r11
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