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{SECT 0 {EXCHG {PARA 262 "" 0 "" {TEXT 260 25 "Analysis of Heat Equati
on" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 257 "" 0 "" {TEXT -1 60 "3.1
0  Example Diffusion Problems in Cylindrical Coordinates " }}{PARA 
259 "" 0 "" {TEXT -1 1 " " }}{PARA 259 "" 0 "" {TEXT -1 384 "We now co
nsider several examples of partial differential equations for heat or \+
diffusion phenomena under various homogeneous boundary conditions over
 finite intervals in the cylindrical coordinate system. We note that a
ll the spatial ordinary differential equations in the cylindrical coor
dinate system are of the Bessel type and the solutions are Bessel func
tions of the first kind. " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 256 
14 "EXAMPLE 3.10.1" }{TEXT -1 39 ": We seek the temperature distributi
on " }{XPPEDIT 18 0 "u[](r,t)" "6#-&%\"uG6\"6$%\"rG%\"tG" }{TEXT -1 
56 " in a thin circularly symmetric plate over the interval " }
{XPPEDIT 18 0 "I" "6#%\"IG" }{TEXT -1 5 " = \{ " }{XPPEDIT 18 0 "r" "6
#%\"rG" }{TEXT -1 3 " | " }{XPPEDIT 18 0 "0" "6#\"\"!" }{TEXT -1 3 " <
 " }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "1" "
6#\"\"\"" }{TEXT -1 147 " \} whose lateral surface is insulated. The p
eriphery (edge) of the plate is at the fixed temperature of zero. The \+
initial temperature distribution " }{XPPEDIT 18 0 "f(r)" "6#-%\"fG6#%
\"rG" }{TEXT -1 39 " is given below and the diffusivity is " }
{XPPEDIT 18 0 "k " "6#%\"kG" }{TEXT -1 4 "  = " }{XPPEDIT 18 0 "1" "6#
\"\"\"" }{TEXT -1 1 "/" }{XPPEDIT 18 0 "20" "6#\"#?" }{TEXT -1 1 "." }
}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 257 8 "SOLU
TION" }{TEXT -1 39 ": The homogeneous diffusion equation is" }}{PARA 
259 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "diff(u[](
r,t),t)=k*(diff(u[](r,t),r)+r*diff(u[](r,t),r,r))/r" "6#/-%%diffG6$-&%
\"uG6\"6$%\"rG%\"tGF-*(%\"kG\"\"\",&-F%6$-&F)F*6$F,F-F,F0*&F,F0-F%6%-&
F)F*6$F,F-F,F,F0F0F0F,!\"\"" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}
{PARA 259 "" 0 "" {TEXT -1 38 "The boundary conditions are type 1 at \+
" }{XPPEDIT 18 0 "r = 1" "6#/%\"rG\"\"\"" }{TEXT -1 37 " and we requir
e a finite solution at " }{XPPEDIT 18 0 "r = 0" "6#/%\"rG\"\"!" }
{TEXT -1 1 "." }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{XPPEDIT 18 0 "abs(u[](0,t))<infinity and u[](1,t)=0" "6#32-%$absG6#-&
%\"uG6\"6$\"\"!%\"tG%)infinityG/-&F*F+6$\"\"\"F.F-" }}{PARA 259 "" 0 "
" {TEXT -1 24 "The initial condition is" }}{PARA 256 "" 0 "" {XPPEDIT 
18 0 "u(r,0)=1-r^2" "6#/-%\"uG6$%\"rG\"\"!,&\"\"\"F**$F'\"\"#!\"\"" }}
{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 87 "Ordin
ary differential equations obtained from the method of separation of v
ariables are" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{XPPEDIT 18 0 "diff(T[](t),t)+k*lambda^2*T[](t)=0" "6#/,&-%%diffG6$-&%
\"TG6\"6#%\"tGF-\"\"\"*(%\"kGF.*$%'lambdaG\"\"#F.-&F*F+6#F-F.F.\"\"!" 
}}{PARA 258 "" 0 "" {TEXT -1 3 "and" }}{PARA 260 "" 0 "" {XPPEDIT 18 
0 "diff(R[](r),r,r)+diff(R[](r),r)/r+lambda^2*R[](r)=0" "6#/,(-%%diffG
6%-&%\"RG6\"6#%\"rGF-F-\"\"\"*&-F&6$-&F*F+6#F-F-F.F-!\"\"F.*&%'lambdaG
\"\"#-&F*F+6#F-F.F.\"\"!" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 
259 "" 0 "" {TEXT -1 47 "Boundary conditions on the spatial equation a
re" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 
0 "abs(R[](0))<infinity and R[](1)=0" "6#32-%$absG6#-&%\"RG6\"6#\"\"!%
)infinityG/-&F*F+6#\"\"\"F-" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}
{PARA 259 "" 0 "" {TEXT -1 32 "Assignment of system parameters " }}
{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "> " 0 "" {MPLTEXT 1 0 42 
"restart:with(plots):a:=0:b:=1:k:=0.000001;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"kG$\"\"\"!\"'
" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 
-1 131 "Allowed eigenvalues and orthonormal eigenfunctions are from Ex
ample 2.6.2. The eigenvalues are the roots of the eigenvalue equation
" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "> " 0 "" {MPLTEXT 1 
0 25 "BesselJ(0,lambda[n]*b)=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%
(BesselJG6$\"\"!&%'lambdaG6#%\"nGF'" }}}{EXCHG {PARA 259 "" 0 "" 
{TEXT -1 4 "for " }{XPPEDIT 18 0 "n=1" "6#/%\"nG\"\"\"" }{TEXT -1 13 "
, 2, 3, ... ." }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" 
{TEXT -1 27 "Orthonormal eigenfunctions " }}{PARA 259 "" 0 "" {TEXT 
-1 0 "" }}{PARA 259 "> " 0 "" {MPLTEXT 1 0 87 "R[n](r):=simplify(Besse
lJ(0,lambda[n]*r)/sqrt(int(BesselJ(0,lambda[n]*r)^2*r,r=a..b)));" }}
{PARA 260 "" 0 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%
\"RG6#%\"nG6#%\"rG,$*(\"\"#\"\"\"-%(BesselJG6$\"\"!*&&%'lambdaGF'F.F*F
.F.,&*&F-F.)-F06$F2F4F-F.F.*&F-F.)-F06$F.F4F-F.F.#!\"\"F-F." }}}
{EXCHG {PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 
74 "Substitution of the eigenvalue equation simplifies the prceding eq
uation  " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "> " 0 "" 
{MPLTEXT 1 0 82 "R[n](r):=radsimp(subs(BesselJ(0,lambda[n])=0,R[n](r))
);R[m](r):=subs(n=m,R[n](r)):" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%\"RG6#%\"nG6#%\"rG*(-%(BesselJG6$
\"\"!*&&%'lambdaGF'\"\"\"F*F3F3\"\"##F3F4-F-6$F3F1!\"\"" }}}{EXCHG 
{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 64 "State
ment of orthonormality with respect to the weight function " }
{XPPEDIT 18 0 "w(r) = r" "6#/-%\"wG6#%\"rGF'" }{TEXT -1 1 " " }}{PARA 
259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "> " 0 "" {MPLTEXT 1 0 53 "w(r):
=r:Int(R[n](r)*R[m](r)*w(r),r=a...b)=delta(n,m);" }}{PARA 260 "" 0 "" 
{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$,$*.\"\"#\"
\"\"-%(BesselJG6$\"\"!*&&%'lambdaG6#%\"nGF*%\"rGF*F*-F,6$F*F0!\"\"-F,6
$F.*&&F16#%\"mGF*F4F*F*-F,6$F*F;F7F4F*F*/F4;F.F*-%&deltaG6$F3F=" }}}
{EXCHG {PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 
24 "Time-dependent solution " }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}
{PARA 259 "> " 0 "" {MPLTEXT 1 0 63 "T[n](t):=C(n)*exp(-k*lambda[n]^2*
t);u[n](r,t):=T[n](t)*R[n](r):" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>-&%\"TG6#%\"nG6#%\"tG*&-%\"CGF'\"\"\"
-%$expG6#,$*($F.!\"'F.)&%'lambdaGF'\"\"#F.F*F.!\"\"F." }}}{EXCHG 
{PARA 259 "" 0 "" {TEXT -1 24 "Eigenfunction expansion " }}{PARA 259 "
" 0 "" {TEXT -1 0 "" }}{PARA 259 "> " 0 "" {MPLTEXT 1 0 37 "u(r,t):=Su
m(u[n](r,t),n=1..infinity);" }}{PARA 260 "" 0 "" {TEXT -1 0 "" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"uG6$%\"rG%\"tG-%$SumG6$*,-%\"CG6#
%\"nG\"\"\"-%$expG6#,$*($F1!\"'F1)&%'lambdaGF/\"\"#F1F(F1!\"\"F1-%(Bes
selJG6$\"\"!*&F:F1F'F1F1F<#F1F<-F?6$F1F:F=/F0;F1%)infinityG" }}}
{EXCHG {PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT -1 
69 "Evaluation of Fourier coefficients for the specific initial condit
ion" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "> " 0 "" {MPLTEXT 
1 0 22 "f1(r):=sin(10*Pi*r)/r;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f
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1G6#%\"rG*&-%$sinG6#,$*(\"#5\"\"\"%#PiGF/F'F/F/F/F'!\"\"" }}{PARA 11 "
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{MPLTEXT 1 0 18 "plot(f(r),r=0..1);" }}{PARA 13 "" 1 "" {GLPLOT2D 317 
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{XPPEDIT 18 0 "u[](r,t)" "6#-&%\"uG6\"6$%\"rG%\"tG" }{TEXT -1 70 ". Th
e animation sequence shown in Figure 3.9 shows snapshots at times " }
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{PARA 259 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 18 "Anima
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