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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "Unit 5: Boundary Value Pro
blems for Partial Differential Equations" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 64 "Chapter 29: Separation of Variable
s in Non-Cartesian Coordinates" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 47 "Section 29.2:  Laplace's equation in a cy
linder" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" 
{TEXT -1 9 "Copyright" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 48 "Copyright * 2001 by Addison Wesley Longman, Inc." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 302 "All righ
ts reserved.  No part of this publication may be reproduced, stored in
 a retrieval system, or transmitted, in any form or by any means, elec
tronic, mechanical, photocopying, recording, or otherwise, without the
 prior written permission of the publisher. Printed in the United Stat
es of America." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 
0 "" {TEXT -1 15 "Initializations" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 73 "with(linalg):\nwith(plots):\nwith(plottools)
:\nwith(student):\nwith(DEtools):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "with(FPS):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 33 "Problem Statement and Fo
rmulation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 78 "Find the steady-state, bounded temperatures inside a soli
d cylinder of radius " }{XPPEDIT 18 0 "sigma" "6#%&sigmaG" }{TEXT -1 
12 " and height " }{XPPEDIT 18 0 "h" "6#%\"hG" }{TEXT -1 114 ", if the
 base is insulated, and the curved lateral surface and the top are mai
ntained at temperatures of zero and " }{XPPEDIT 18 0 "f(r)" "6#-%\"fG6
#%\"rG" }{TEXT -1 15 ", respectively." }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 46 "Because none of the data depends on t
he angle " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 51 ", the tem
perature will be a function of the height " }{XPPEDIT 18 0 "z" "6#%\"z
G" }{TEXT -1 12 " and radius " }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 
59 ".  Thus, the temperature function, which can be written as " }
{XPPEDIT 18 0 "u(r,z)" "6#-%\"uG6$%\"rG%\"zG" }{TEXT -1 256 " in cylin
drical coordinates, satisfies Laplace's equation inside the cylinder. \+
 The insulation condition on the bottom of the cylinder, a homogeneous
 Neumann condition, is expressed by the vanishing of the temperature g
radient, a partial derivative in the " }{XPPEDIT 18 0 "z" "6#%\"zG" }
{TEXT -1 75 "-direction, on that face.  The complete boundary value pr
oblem determining " }{XPPEDIT 18 0 "u(r,z)" "6#-%\"uG6$%\"rG%\"zG" }
{TEXT -1 56 " therefore consists of the partial differential equation
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 276 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "grad(u(r,z))=u[rr](r,z)+1/r" "6#/-%%gradG6#-%\"uG6$%\"r
G%\"zG,&-&F(6#%#rrG6$F*F+\"\"\"*&F2F2F*!\"\"F2" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "u[r](r,z)+u[zz](r,z)=0" "6#/,&-&%\"uG6#%\"rG6$F)%\"zG\"
\"\"-&F'6#%#zzG6$F)F+F,\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 67 "q := expand(laplacian(u(r,z),[r,theta,z], coor
ds=cylindrical)) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "and the boundary conditions" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {XPPEDIT 18 0 "u(sig
ma,z)=0" "6#/-%\"uG6$%&sigmaG%\"zG\"\"!" }}{PARA 259 "" 0 "" {XPPEDIT 
18 0 "u[z](r,0)=0" "6#/-&%\"uG6#%\"zG6$%\"rG\"\"!F+" }}{PARA 260 "" 0 
"" {XPPEDIT 18 0 "u(r,h)=f(r)" "6#/-%\"uG6$%\"rG%\"hG-%\"fG6#F'" }}
{PARA 261 "" 0 "" {XPPEDIT 18 0 "u(r,z)" "6#-%\"uG6$%\"rG%\"zG" }
{TEXT -1 8 " bounded" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 
"Statement and Analysis of the Solution" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "The solution is given by" }}
{PARA 262 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(r,z)=Sum(A[k]*J[0](
lambda[k]*r)*cosh(lambda[k]*z),k=1..infinity)" "6#/-%\"uG6$%\"rG%\"zG-
%$SumG6$*(&%\"AG6#%\"kG\"\"\"-&%\"JG6#\"\"!6#*&&%'lambdaG6#F0F1F'F1F1-
%%coshG6#*&&F:6#F0F1F(F1F1/F0;F1%)infinityG" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 263 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "A[k]=Int(r*
f(r)*J[0](lambda[k]*r),r=0..sigma)/cosh(lambda[k]*h)/Int(r*[J[0](lambd
a[k]*r)]^2,r=0..sigma)" "6#/&%\"AG6#%\"kG*(-%$IntG6$*(%\"rG\"\"\"-%\"f
G6#F-F.-&%\"JG6#\"\"!6#*&&%'lambdaG6#F'F.F-F.F./F-;F6%&sigmaGF.-%%cosh
G6#*&&F:6#F'F.%\"hGF.!\"\"-F*6$*&F-F.*$7#-&F46#F66#*&&F:6#F'F.F-F.\"\"
#F./F-;F6F>FF" }{TEXT -1 5 "  =  " }{XPPEDIT 18 0 "2*Int(r*f(r)*J[0](l
ambda[k]*r),r=0..sigma)/sigma^2/cosh(lambda[k]*h)/[J[1](lambda[k]*sigm
a)]^2" "6#*,\"\"#\"\"\"-%$IntG6$*(%\"rGF%-%\"fG6#F*F%-&%\"JG6#\"\"!6#*
&&%'lambdaG6#%\"kGF%F*F%F%/F*;F2%&sigmaGF%*$F;F$!\"\"-%%coshG6#*&&F66#
F8F%%\"hGF%F=*$7#-&F06#F%6#*&&F66#F8F%F;F%F$F=" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "J[
0](x)" "6#-&%\"JG6#\"\"!6#%\"xG" }{TEXT -1 43 " is the Bessel function
 of order zero, and " }{XPPEDIT 18 0 "lambda[k]" "6#&%'lambdaG6#%\"kG
" }{TEXT -1 8 " is the " }{TEXT 259 1 "k" }{TEXT -1 11 "th zero of " }
{XPPEDIT 18 0 "J[0](sigma*x)" "6#-&%\"JG6#\"\"!6#*&%&sigmaG\"\"\"%\"xG
F+" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 14 "The functions " }{XPPEDIT 18 0 "J[nu](x)" "6#-&%\"JG6#%#n
uG6#%\"xG" }{TEXT -1 26 " satisfy Bessel's equation" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 277 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2
*`y''`(x)+x*`y'`(x)+(x^2-nu^2)*y(x)=0" "6#/,(*&%\"xG\"\"#-%$y''G6#F&\"
\"\"F+*&F&F+-%#y'G6#F&F+F+*&,&*$F&F'F+*$%#nuGF'!\"\"F+-%\"yG6#F&F+F+\"
\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "B
essel_eqn := x^2*diff(y(x),x,x) + x*diff(y(x),x) + (x^2-nu^2)*y(x) = 0
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 41 "and are called Bessel functions of order " }
{XPPEDIT 18 0 "nu" "6#%#nuG" }{TEXT -1 88 ".  As seen in Section 16.2,
 two linearly independent solutions of Bessel's equation are " }
{XPPEDIT 18 0 "J[nu](x)" "6#-&%\"JG6#%#nuG6#%\"xG" }{TEXT -1 5 " and \+
" }{XPPEDIT 18 0 "Y[nu](x)" "6#-&%\"YG6#%#nuG6#%\"xG" }{TEXT -1 35 ", \+
as we see from the Maple solution" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "dsolve(Bessel_eqn,y(x), outp
ut=basis);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 29 "Only one of these solutions, " }{XPPEDIT 
18 0 "J[nu](x)" "6#-&%\"JG6#%#nuG6#%\"xG" }{TEXT -1 69 ", is bounded a
t the origin.  The Bessel function of the second kind, " }{XPPEDIT 18 
0 "Y[nu](x)" "6#-&%\"YG6#%#nuG6#%\"xG" }{TEXT -1 35 ", has a logarithm
ic singularity at " }{XPPEDIT 18 0 "x=0" "6#/%\"xG\"\"!" }{TEXT -1 3 "
.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 22 "Se
ries expansions for " }{XPPEDIT 18 0 "J[0](x)" "6#-&%\"JG6#\"\"!6#%\"x
G" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "J[1](x)" "6#-&%\"JG6#\"\"\"6#%
\"xG" }{TEXT -1 82 ", the Bessel functions of order zero and one, resp
ectively, can be found with the " }{TEXT 256 17 "FormalPowerSeries" }
{TEXT -1 18 " command from the " }{TEXT 257 3 "FPS" }{TEXT -1 35 " pac
kage.  The series expansion of " }{XPPEDIT 18 0 "J[0](x)" "6#-&%\"JG6#
\"\"!6#%\"xG" }{TEXT -1 3 " is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "FormalPowerSeries(BesselJ(0,
x),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 30 "while the series expansion of " }{XPPEDIT 18 0 
"J[1](x)" "6#-&%\"JG6#\"\"\"6#%\"xG" }{TEXT -1 3 " is" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "FormalPo
werSeries(BesselJ(1,x),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "In Figure 29.4, " }
{XPPEDIT 18 0 "J[0](x)" "6#-&%\"JG6#\"\"!6#%\"xG" }{TEXT -1 25 " is th
e black curve, and " }{XPPEDIT 18 0 "J[1](x)" "6#-&%\"JG6#\"\"\"6#%\"x
G" }{TEXT -1 18 " is the red curve." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "plot([BesselJ(0,x),Bessel
J(1,x)],x=0..10, color=[black,red], scaling=constrained, xtickmarks=5,
 ytickmarks=2, labels=[x,``], labelfont=[TIMES,ITALIC,12]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 48 "It should be recalled from Section 16.2 that if " }
{XPPEDIT 18 0 "lambda[k],k=1,2,`...`" "6&&%'lambdaG6#%\"kG/F&\"\"\"\"
\"#%$...G" }{TEXT -1 19 ", are the roots of " }{XPPEDIT 18 0 "J[0](sig
ma*x)" "6#-&%\"JG6#\"\"!6#*&%&sigmaG\"\"\"%\"xGF+" }{TEXT -1 21 ", the
n the functions " }{XPPEDIT 18 0 "J[0](lambda[k]*x)" "6#-&%\"JG6#\"\"!
6#*&&%'lambdaG6#%\"kG\"\"\"%\"xGF." }{TEXT -1 22 " are, on the interva
l " }{XPPEDIT 18 0 "0<=x" "6#1\"\"!%\"xG" }{XPPEDIT 18 0 "``<=sigma" "
6#1%!G%&sigmaG" }{TEXT -1 49 ", orthogonal with respect to the weight \+
function " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 13 ".  Thus, for " }
{XPPEDIT 18 0 "lambda[k]<>lambda[j]" "6#0&%'lambdaG6#%\"kG&F%6#%\"jG" 
}{TEXT -1 9 ", we have" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 264 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*J[0](lambda[k]*x)*J[0](lam
bda[j]*x),x=0..sigma)=0" "6#/-%$IntG6$*(%\"xG\"\"\"-&%\"JG6#\"\"!6#*&&
%'lambdaG6#%\"kGF)F(F)F)-&F,6#F.6#*&&F26#%\"jGF)F(F)F)/F(;F.%&sigmaGF.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 12 "Example 29.
2" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
38 "For the cylinder described above, let " }{XPPEDIT 18 0 "sigma=2" "
6#/%&sigmaG\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "h=3" "6#/%\"hG\"\"$
" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "f(r)=1" "6#/-%\"fG6#%\"rG\"\"\"
" }{TEXT -1 34 ". Then, the first ten eigenvalues " }{XPPEDIT 18 0 "la
mbda[k]" "6#&%'lambdaG6#%\"kG" }{TEXT -1 28 " are the first ten zeros \+
of " }{XPPEDIT 18 0 "J[0](2*x)" "6#-&%\"JG6#\"\"!6#*&\"\"#\"\"\"%\"xGF
+" }{TEXT -1 12 ", listed as " }{XPPEDIT 18 0 "mu[k]" "6#&%#muG6#%\"kG
" }{TEXT -1 32 " in Table 16.1 in Section 16.2. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 127 "For the convenience of t
he reader executing these calculations in Maple, we reconstruct these \+
first ten eigenvalues as follows." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 25 "As seen \+
in Section 16.2, " }{XPPEDIT 18 0 "J[0](2*x)" "6#-&%\"JG6#\"\"!6#*&\"
\"#\"\"\"%\"xGF+" }{TEXT -1 18 " is asymptotic to " }{XPPEDIT 18 0 "co
s(2*x-Pi/4)/sqrt(Pi*x)" "6#*&-%$cosG6#,&*&\"\"#\"\"\"%\"xGF*F**&%#PiGF
*\"\"%!\"\"F/F*-%%sqrtG6#*&F-F*F+F*F/" }{TEXT -1 21 ", so the eigenval
ues " }{XPPEDIT 18 0 "lambda[k]" "6#&%'lambdaG6#%\"kG" }{TEXT -1 25 ",
 the first ten roots of " }{XPPEDIT 18 0 "J[0](2*x)=0" "6#/-&%\"JG6#\"
\"!6#*&\"\"#\"\"\"%\"xGF,F(" }{TEXT -1 17 ", are computed by" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "f
or k from 1 to 10 do\nm[k] := fsolve(BesselJ(0,2*x)=0, x, (4*k-1)*Pi/8
-.1 .. (4*k-1)*Pi/8+.1);\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 17 "The coefficients " }
{XPPEDIT 18 0 "A[k]" "6#&%\"AG6#%\"kG" }{TEXT -1 10 ", given by" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 265 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "A[k]=Int(r*J[0](lambda[k]*r),r=0..2)/2/cosh(3*lambda[k]
)/[J[1](2*lambda[k])]^2" "6#/&%\"AG6#%\"kG**-%$IntG6$*&%\"rG\"\"\"-&%
\"JG6#\"\"!6#*&&%'lambdaG6#F'F.F-F.F./F-;F3\"\"#F.F;!\"\"-%%coshG6#*&
\"\"$F.&F76#F'F.F<*$7#-&F16#F.6#*&F;F.&F76#F'F.F;F<" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 66 "can be evaluated with n
umeric integration, yielding, for example, " }{XPPEDIT 18 0 "A[1]" "6#
&%\"AG6#\"\"\"" }{TEXT -1 3 " as" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "evalf(Int(r*BesselJ(0,m[1]*r
),r=0..2))/2/ cosh(3*m[1])/BesselJ(1,2*m[1])^2;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Altern
atively, because " }{XPPEDIT 18 0 "f(r)=1" "6#/-%\"fG6#%\"rG\"\"\"" }
{TEXT -1 72 ", the integral in the numerator can be evaluated exactly \+
via the formula" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 266 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "Int(x*J[0](x),x=0..alpha)=alpha*J[1](al
pha)" "6#/-%$IntG6$*&%\"xG\"\"\"-&%\"JG6#\"\"!6#F(F)/F(;F.%&alphaG*&F2
F)-&F,6#F)6#F2F)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 27 "if the change of variables " }{XPPEDIT 18 0 "lambda[k]*r=
x" "6#/*&&%'lambdaG6#%\"kG\"\"\"%\"rGF)%\"xG" }{TEXT -1 45 " is made i
n the integral in the numerator of " }{XPPEDIT 18 0 "A[k]" "6#&%\"AG6#
%\"kG" }{TEXT -1 18 ", as summarized by" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 278 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "\{Int(r*J[0](lam
bda[k]*r),r=0..2)\}*`|`[x=lambda[k]*r]=1/lambda[k]^2" "6#/*&<#-%$IntG6
$*&%\"rG\"\"\"-&%\"JG6#\"\"!6#*&&%'lambdaG6#%\"kGF+F*F+F+/F*;F0\"\"#F+
&%\"|grG6#/%\"xG*&&F46#F6F+F*F+F+*&F+F+*$&F46#F6F9!\"\"" }{TEXT -1 1 "
 " }{XPPEDIT 18 0 "Int(x*J[0](x),x=0..2*lambda[k])" "6#-%$IntG6$*&%\"x
G\"\"\"-&%\"JG6#\"\"!6#F'F(/F';F-*&\"\"#F(&%'lambdaG6#%\"kGF(" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "``=2/lambda[k]" "6#/%!G*&\"\"#\"\"\"&%'lambda
G6#%\"kG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "J[1](2*lambda[k]" "6#-&
%\"JG6#\"\"\"6#*&\"\"#F'&%'lambdaG6#%\"kGF'" }{TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "In fact, the in
tegral" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 57 "unassign('k'):\nX := Int(r*BesselJ(0,lambda[k]*r),r=0
..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 39 "under that change of variables, becomes" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 34 "Y := changevar(x=lambda[k]*r,X,x);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 49 "Maple knows t
he value of this integral, and gives" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(Y);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 62 "In fact, Maple can also evaluate the original integral, giving
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "value(X);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Consequently, the expression
 for " }{XPPEDIT 18 0 "A[k]" "6#&%\"AG6#%\"kG" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 89 "Ak := Int(r*BesselJ(0,lambda[k]*r),r=0..2)/2/ cosh(3*lambda[k])/
BesselJ(1,2*lambda[k])^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 
"" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "simplifies to" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "AAk :=
 value(Ak);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 11 "that is, to" }}{PARA 279 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "A[k]=sech(3*lambda[k])/lambda[k]/J[1](2*lambd
a[k])" "6#/&%\"AG6#%\"kG*(-%%sechG6#*&\"\"$\"\"\"&%'lambdaG6#F'F.F.&F0
6#F'!\"\"-&%\"JG6#F.6#*&\"\"#F.&F06#F'F.F4" }{TEXT -1 1 " " }}{PARA 0 
"" 0 "" {TEXT -1 37 "Hence, the first ten coefficients are" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "for
 k from 1 to 10 do\nA[k] := evalf(subs(lambda[k]=m[k],AAk));\nod;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 96 "The coefficients rapidly decrease in magnitude, so the co
nvergence is rapid, and the partial sum" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 280 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u[10](r,z)=sum(A
[k]*J[0](lambda[k]*r)*cosh(lambda[k]*z),k=1..10)" "6#/-&%\"uG6#\"#56$%
\"rG%\"zG-%$sumG6$*(&%\"AG6#%\"kG\"\"\"-&%\"JG6#\"\"!6#*&&%'lambdaG6#F
3F4F*F4F4-%%coshG6#*&&F=6#F3F4F+F4F4/F3;F4F(" }{TEXT -1 1 " " }}{PARA 
0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "U10 := add(A[k]*BesselJ(0,m[
k]*r)*cosh(m[k]*z),k=1..10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "is adequate, except on t
he boundary " }{XPPEDIT 18 0 "z=3" "6#/%\"zG\"\"$" }{TEXT -1 68 " wher
e a much slower convergence is evident, as seen in Figure 29.5." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 175 "p1 := plot3d(U10,r=0..2,z=0..3, color=red):\ndisplay(p1, scalin
g=constrained, labels=[r,z,u], labelfont=[TIMES,ITALIC,12], axes=frame
, tickmarks=[2,3,2], orientation=[-40,60]);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "The surfac
e represents the temperatures on the rectangle " }{TEXT 260 1 "R" }
{TEXT -1 4 " = \{" }{XPPEDIT 18 0 "0<=r" "6#1\"\"!%\"rG" }{XPPEDIT 18 
0 "``<=2" "6#1%!G\"\"#" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "0<=z" "6#1\"
\"!%\"zG" }{XPPEDIT 18 0 "``<=3" "6#1%!G\"\"$" }{TEXT -1 112 "\}, half
 a plane section through the axis of the cylinder.  It is impossible t
o draw a graph of the temperatures " }{TEXT 258 6 "inside" }{TEXT -1 
97 " the cylinder.  Every representation of these temperatures will re
quire some visual compromise.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 62 "In Figure 29.6, the temperatures are grap
hed on the rectangle " }{TEXT 261 1 "R" }{TEXT -1 43 ", shown supporti
ng the temperature surface " }{TEXT 262 6 "inside" }{TEXT -1 14 " the \+
cylinder." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 217 "p2 := cylinderplot(2,theta=Pi/2..2*Pi,z=0..3):\np3
 := display(rotate(p1,-Pi/2,0,-Pi/2)):\ndisplay([p2,p3], orientation=[
30,70], axes=frame, tickmarks=[5,5,4], labels=[`  x`,`    y`,`z   `], \+
labelfont=[TIMES,ITALIC,12]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "The rectangle rotates a
bout the axis of the cylinder, supporting the same temperatures for al
l values of the angle " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 
3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
104 "Figure 29.7 gives another view of the temperatures inside the cyl
inder, the isothermal surface on which " }{XPPEDIT 18 0 "u(r,z)=.1" "6
#/-%\"uG6$%\"rG%\"zG$\"\"\"!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 444 "PP := impl
icitplot3d(U10=.1,r=0..2,theta=0..2*Pi,z=0..3, coords=cylindrical, sca
ling=constrained, orientation=[35,60], color=yellow, style=patchcontou
r, axes=frame, tickmarks=[5,5,4], grid=[20,20,20]):\nPP1 := spacecurve
(\{[[0,0,0],[2,0,0]],[[0,0,0],[0,-2,0]],[[0,0,0], [0,0,1]]\}, color=bl
ack, linestyle=2):\nPP2 := textplot3d([-.1,-.1,.1,`O`]):\ndisplay([PP,
PP1,PP2],labels=[`  x`,`y   `,`z  `], labelfont=[TIMES,ITALIC,12], ori
entation=[-45,65]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "In fact, the level surfaces " }
{XPPEDIT 18 0 "u(r,z)=k/10,k=1,2,`...`,9" "6'/-%\"uG6$%\"rG%\"zG*&%\"k
G\"\"\"\"#5!\"\"/F*F+\"\"#%$...G\"\"*" }{TEXT -1 260 ", are each bowl-
shaped, with the same circle bounding the open top.  This corresponds \+
to the discontinuity in temperatures at the junction of the top of the
 cylinder with the curved lateral wall of the cylinder.  At this junct
ure, all level surfaces intersect." }}{PARA 0 "" 0 "" {TEXT -1 1 " " }
}{PARA 0 "" 0 "" {TEXT -1 138 "Executing the following Maple commands \+
will generate an animation of this sequence of level surfaces through \+
the interior of the cylinder." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 254 "for k from 1 to 9 do\nP||k \+
:= implicitplot3d(U10 = .1*k, r=0..2, theta=0..2*Pi, z=0..3, coords=cy
lindrical, color=COLOR(HUE,k/10)):\nod:\ndisplay([P||(1..9)], insequen
ce=true, orientation=[25,80], style=patchcontour, scaling=constrained,
 lightmodel=light3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 35 "Solution by Separation of Varia
bles" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 41 "Assuming a separated solution of the form" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "U := R(r)*Z
(z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 43 "and substituting it into the boundary data " }
{XPPEDIT 18 0 "u(sigma,z)=0" "6#/-%\"uG6$%&sigmaG%\"zG\"\"!" }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "u[z](r,0) = 0" "6#/-&%\"uG6#%\"zG6$%\"rG
\"\"!F+" }{TEXT -1 10 " leads to " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 274 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(sigma)*Z(z)=0" "6#/
*&-%\"RG6#%&sigmaG\"\"\"-%\"ZG6#%\"zGF)\"\"!" }{TEXT -1 4 " => " }
{XPPEDIT 18 0 "R(sigma)=0" "6#/-%\"RG6#%&sigmaG\"\"!" }}{PARA 0 "" 0 "
" {TEXT -1 3 "and" }}{PARA 275 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
R(r)" "6#-%\"RG6#%\"rG" }{TEXT -1 23 " Z'(0) = 0 => Z'(0) = 0" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Applicati
on of the separation assumption to Laplace's equation" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "q;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 10 "results in" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "q1 := eval(subs(u(r,z)=U,q));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 22 "As usual, division by " }{XPPEDIT 18 0 "u(r,z)" "6#-%\"uG
6$%\"rG%\"zG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "q2 := expand(q1/U);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 58 "leads to a separation of variables if the term containing
 " }{XPPEDIT 18 0 "Z(z)" "6#-%\"ZG6#%\"zG" }{TEXT -1 37 " is brought t
o the righthand side via" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "q3 := q2 - (op(3,lhs(q2))=op(3,lhs(
q2)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 46 "Introducing the Bernoulli separation constant \+
" }{XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT -1 9 " leads to" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 281 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "
`R''`(r)/r/R(r)+`R'`(r)/R(r)=-`Z''`(z)/Z(z)" "6#/,&*(-%$R''G6#%\"rG\"
\"\"F)!\"\"-%\"RG6#F)F+F**&-%#R'G6#F)F*-F-6#F)F+F*,$*&-%$Z''G6#%\"zGF*
-%\"ZG6#F:F+F+" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "mu" "6#%#muG" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 93 "Overlooking the abuse of notation whereby prines denote d
ifferentiation with respect to both " }{XPPEDIT 18 0 "r" "6#%\"rG" }
{TEXT -1 5 " and " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 22 ", we hav
e the two ODEs" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 282 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "r*`R''`(r)+`R'`(r)-mu*r*R(r)=0" "6#/,(*
&%\"rG\"\"\"-%$R''G6#F&F'F'-%#R'G6#F&F'*(%#muGF'F&F'-%\"RG6#F&F'!\"\"
\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 
283 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`Z''`(z)+mu*Z(z)=0" "6#/,&-
%$Z''G6#%\"zG\"\"\"*&%#muGF)-%\"ZG6#F(F)F)\"\"!" }{TEXT -1 1 " " }}
{PARA 0 "" 0 "" {TEXT -1 18 "which follow from " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "q4 := lhs(q
3) = mu;\nq5 := rhs(q3) = mu;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "and the rearrangements" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 88 "q6 := numer(normal(lhs(q4) - rhs(q4))) = 0;\nq7 := -numer(norm
al(lhs(q5) - rhs(q5))) = 0;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{XPPEDIT 18 0 "r" "6
#%\"rG" }{TEXT -1 76 "-equation is a modified form of Bessel's equatio
n.  In fact, the solution is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
284 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(r)=c[1]*J[0](sqrt(-mu)*r)
+c[2]*Y[0](sqrt(-mu)*r)" "6#/-%\"RG6#%\"rG,&*&&%\"cG6#\"\"\"F--&%\"JG6
#\"\"!6#*&-%%sqrtG6#,$%#muG!\"\"F-F'F-F-F-*&&F+6#\"\"#F--&%\"YG6#F26#*
&-F66#,$F9F:F-F'F-F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "
that is, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 16 "dsolve(q6,R(r));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "The appearanc
e of " }{XPPEDIT 18 0 "sqrt(-mu)" "6#-%%sqrtG6#,$%#muG!\"\"" }{TEXT 
-1 46 " suggests renaming the separation constant to " }{XPPEDIT 18 0 
"-lambda^2" "6#,$*$%'lambdaG\"\"#!\"\"" }{TEXT -1 55 " so that the two
 ordinary differential equations become" }}{PARA 285 "" 0 "" {TEXT -1 
1 " " }{XPPEDIT 18 0 "r*`R''`(r)+`R'`(r)+lambda^2*r*R(r)=0" "6#/,(*&%
\"rG\"\"\"-%$R''G6#F&F'F'-%#R'G6#F&F'*(%'lambdaG\"\"#F&F'-%\"RG6#F&F'F
'\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 
286 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`Z''`(z)-lambda^2*Z(z)=0" "
6#/,&-%$Z''G6#%\"zG\"\"\"*&%'lambdaG\"\"#-%\"ZG6#F(F)!\"\"\"\"!" }
{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 0 "" 0 
"" {TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "q8 := s
ubs(mu=-lambda^2,q6);\nq9 := subs(mu=-lambda^2,q7);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "T
hen, the solution of the " }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 12 "
-equation is" }}{PARA 287 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "R(r)=
c[1]*J[0](lambda*r)+c[2]*Y[0](lambda*r)" "6#/-%\"RG6#%\"rG,&*&&%\"cG6#
\"\"\"F--&%\"JG6#\"\"!6#*&%'lambdaGF-F'F-F-F-*&&F+6#\"\"#F--&%\"YG6#F2
6#*&F5F-F'F-F-F-" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 7 "that i
s" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "dsolve(q8,R(r));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 14 "Since we know
 " }{XPPEDIT 18 0 "Y[0](x)" "6#-&%\"YG6#\"\"!6#%\"xG" }{TEXT -1 138 ",
 the zero-order Bessel function of the second kind, is not bounded at \+
the origin, its coefficient is taken as zero, and the solution for " }
{XPPEDIT 18 0 "R(r)" "6#-%\"RG6#%\"rG" }{TEXT -1 18 " is a multiple of
 " }{XPPEDIT 18 0 "J[0](lambda*r)" "6#-&%\"JG6#\"\"!6#*&%'lambdaG\"\"
\"%\"rGF+" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 17 "To transform the " }{XPPEDIT 18 0 "r" "6#%\"rG
" }{TEXT -1 30 "-equation to Bessel's equation" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 288 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2*`
y''`(x)+x*`y'`(x)+(x^2-nu^2)*y(x)=0" "6#/,(*&%\"xG\"\"#-%$y''G6#F&\"\"
\"F+*&F&F+-%#y'G6#F&F+F+*&,&*$F&F'F+*$%#nuGF'!\"\"F+-%\"yG6#F&F+F+\"\"
!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 8 "that is," }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Bess
el_eqn;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 29 "make the change of variables " }{XPPEDIT 
18 0 "r=x/lambda" "6#/%\"rG*&%\"xG\"\"\"%'lambdaG!\"\"" }{TEXT -1 9 " \+
so that " }{XPPEDIT 18 0 "R(r)" "6#-%\"RG6#%\"rG" }{TEXT -1 9 " become
s " }{XPPEDIT 18 0 "R(x/lambda)" "6#-%\"RG6#*&%\"xG\"\"\"%'lambdaG!\"
\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "y(x)" "6#-%\"yG6#%\"xG" }{TEXT 
-1 27 ".  Done in Maple, we obtain" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "q10 := Dchangevar(r=x/lambd
a,R(r)=y(x),q8,r,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "Factoring and \"cancelling\" " }
{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 4 " via" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "expand
(q10/lambda);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "gives precisely Bessel's equation \+
of order zero.  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 121 "Incidentally, the change of variables was accomplished b
y a careful use of the chain rule starting with the definitions  " }
{XPPEDIT 18 0 "x=lambda*r" "6#/%\"xG*&%'lambdaG\"\"\"%\"rGF'" }{TEXT 
-1 4 " and" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 267 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "R(r)=R(x/lambda)" "6#/-%\"RG6#%\"rG-F%6#*&%\"
xG\"\"\"%'lambdaG!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "y(x)=y(lambd
a*r)" "6#/-%\"yG6#%\"xG-F%6#*&%'lambdaG\"\"\"%\"rGF," }}{PARA 0 "" 0 "
" {TEXT -1 5 "Then," }}{PARA 268 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "dR/dr=dy/dx" "6#/*&%#dRG\"\"\"%#drG!\"\"*&%#dyGF&%#dxGF(" }{TEXT 
-1 1 " " }{XPPEDIT 18 0 "dx/dr=dy/dx" "6#/*&%#dxG\"\"\"%#drG!\"\"*&%#d
yGF&F%F(" }{TEXT -1 1 " " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }}
{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 269 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "d^2R/dr^2=d/dx" "6#/*(%\"dG\"\"#%\"RG\"\"\"*$%#drGF&!\"
\"*&F%F(%#dxGF+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "``(dy/dx)" "6#-%!G6#*
&%#dyG\"\"\"%#dxG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "dx/dr" "6#*&%#
dxG\"\"\"%#drG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "lambda" "6#%'lamb
daG" }{TEXT -1 1 " " }}{PARA 270 "" 0 "" {TEXT -1 2 "= " }{XPPEDIT 18 
0 "d^2y/dx^2" "6#*(%\"dG\"\"#%\"yG\"\"\"*$%#dxGF%!\"\"" }{TEXT -1 1 " \+
" }{XPPEDIT 18 0 "lambda^2" "6#*$%'lambdaG\"\"#" }}{PARA 0 "" 0 "" 
{TEXT -1 7 "so that" }}{PARA 271 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "r" "6#%\"rG" }{TEXT -1 11 "R'' + R' + " }{XPPEDIT 18 0 "lambda^2" "
6#*$%'lambdaG\"\"#" }{XPPEDIT 18 0 "r" "6#%\"rG" }{TEXT -1 6 " R = 0" 
}}{PARA 0 "" 0 "" {TEXT -1 7 "becomes" }}{PARA 272 "" 0 "" {TEXT -1 1 
" " }{XPPEDIT 18 0 "x/lambda" "6#*&%\"xG\"\"\"%'lambdaG!\"\"" }{TEXT 
-1 5 " y'' " }{XPPEDIT 18 0 "lambda^2" "6#*$%'lambdaG\"\"#" }{TEXT -1 
6 " + y' " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{TEXT -1 3 " + " }
{XPPEDIT 18 0 "lambda^2" "6#*$%'lambdaG\"\"#" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "x/lambda" "6#*&%\"xG\"\"\"%'lambdaG!\"\"" }{TEXT -1 6 "
 y = 0" }}{PARA 0 "" 0 "" {TEXT -1 9 "and hence" }}{PARA 273 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 12 " y'' + y' + \+
" }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 6 " y = 0" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "when " }{XPPEDIT 18 0 "r" 
"6#%\"rG" }{TEXT -1 16 " is replaced by " }{XPPEDIT 18 0 "x/lambda" "6
#*&%\"xG\"\"\"%'lambdaG!\"\"" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 "R(r)=J[0
](lambda*r)" "6#/-%\"RG6#%\"rG-&%\"JG6#\"\"!6#*&%'lambdaG\"\"\"F'F0" }
{TEXT -1 25 ", the boundary condition " }{XPPEDIT 18 0 "R(sigma)=0" "6
#/-%\"RG6#%&sigmaG\"\"!" }{TEXT -1 26 " requires the solution of " }
{XPPEDIT 18 0 "J[0](lambda*sigma)=0" "6#/-&%\"JG6#\"\"!6#*&%'lambdaG\"
\"\"%&sigmaGF,F(" }{TEXT -1 21 " for the eigenvalues " }{XPPEDIT 18 0 
"lambda[k]" "6#&%'lambdaG6#%\"kG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 "The " }{XPPEDIT 18 0 "z" "
6#%\"zG" }{TEXT -1 9 "-equation" }}{PARA 289 "" 0 "" {TEXT -1 1 " " }
{XPPEDIT 18 0 "`Z''`(z)-lambda^2*Z(z)=0" "6#/,&-%$Z''G6#%\"zG\"\"\"*&%
'lambdaG\"\"#-%\"ZG6#F(F)!\"\"\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "
" {TEXT -1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "q9;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "now inherits \+
the known eigenvalues, and becomes" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 290 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "`Z''`[k](z)-lambda[k]
^2*Z[k](z)=0" "6#/,&-&%$Z''G6#%\"kG6#%\"zG\"\"\"*&&%'lambdaG6#F)\"\"#-
&%\"ZG6#F)6#F+F,!\"\"\"\"!" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 
-1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 48 "unassign('k');\nq11 := subs(lambda=lambda[k],q
9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 101 "which has exponential solutions.  The solution sa
tisfying the Neumann boundary condition Z'(0) = 0 is" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "q12 := ds
olve(\{q11,D(Z)(0)=0\},Z(z));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "which is a multiple of a
 hyperbolic cosine, as seen by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "simplify(convert(q12,trig));
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 39 "Consequently, a single eigensolution is" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 291 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 
0 "u[k](r,z)=A[k]*J[0](lambda[k]*r)*cosh(lambda[k]*z)" "6#/-&%\"uG6#%
\"kG6$%\"rG%\"zG*(&%\"AG6#F(\"\"\"-&%\"JG6#\"\"!6#*&&%'lambdaG6#F(F0F*
F0F0-%%coshG6#*&&F96#F(F0F+F0F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 52 "Uk := A[k]*BesselJ(0,lambda[k]*r)*cosh(la
mbda[k]*z);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 64 "and the general solution, a sum over all \+
such eigensolutions, is" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 292 "
" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "u(r,z)=sum(A[k]*J[0](lambda[k]*r
)*cosh(lambda[k]*z),k=1..infinity)" "6#/-%\"uG6$%\"rG%\"zG-%$sumG6$*(&
%\"AG6#%\"kG\"\"\"-&%\"JG6#\"\"!6#*&&%'lambdaG6#F0F1F'F1F1-%%coshG6#*&
&F:6#F0F1F(F1F1/F0;F1%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" 
{TEXT -1 8 "that is," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 39 "U := Sum(Uk,k=1..infinity):\nu(r,z) = U;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 15 "Application of " }{XPPEDIT 18 0 "u(r,h)=f(r)" "6#/-%
\"uG6$%\"rG%\"hG-%\"fG6#F'" }{TEXT -1 105 ", the final nonhomogeneous \+
Dirichlet boundary condition at the top of the cylinder, leads to the \+
equation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 293 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "f(r)=sum(A[k]*J[0](lambda[k]*r)*cosh(lambda[k
]*h),k=1..infinity)" "6#/-%\"fG6#%\"rG-%$sumG6$*(&%\"AG6#%\"kG\"\"\"-&
%\"JG6#\"\"!6#*&&%'lambdaG6#F/F0F'F0F0-%%coshG6#*&&F96#F/F0%\"hGF0F0/F
/;F0%)infinityG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is
, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "f(r) = subs(z=h,U);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "from which th
e coefficients " }{XPPEDIT 18 0 "A[k]" "6#&%\"AG6#%\"kG" }{TEXT -1 38 
" are determined via multiplication by " }{XPPEDIT 18 0 "r*J[0](lambda
[j]*r)" "6#*&%\"rG\"\"\"-&%\"JG6#\"\"!6#*&&%'lambdaG6#%\"jGF%F$F%F%" }
{TEXT -1 33 " and integrating with respect to " }{XPPEDIT 18 0 "r" "6#
%\"rG" }{TEXT -1 19 " over the interval " }{XPPEDIT 18 0 "0<=r" "6#1\"
\"!%\"rG" }{XPPEDIT 18 0 "``<=sigma" "6#1%!G%&sigmaG" }{TEXT -1 43 ". \+
 The orthogonality of the eigenfunctions " }{XPPEDIT 18 0 "J[0](lambda
[k]*r)" "6#-&%\"JG6#\"\"!6#*&&%'lambdaG6#%\"kG\"\"\"%\"rGF." }{TEXT 
-1 40 " with respect to the weighting function " }{XPPEDIT 18 0 "r" "6
#%\"rG" }{TEXT -1 100 " causes all terms but one on the right side to \+
vanish.  The one surviving term is the one for which " }{XPPEDIT 18 0 
"k=j" "6#/%\"kG%\"jG" }{TEXT -1 38 ".  Hence, what results is the equa
tion" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 111 "q13 := Int(r*f(r)*J[0](lambda[j]*r),r=0..sigma) = A[
j]*cosh(lambda[j]*h)*Int(r*J[0](lambda[j]*r)^2,r=0..sigma);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 11 "from which " }{XPPEDIT 18 0 "A[j]" "6#&%\"AG6#%\"jG" }
{TEXT -1 20 " is determined to be" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "isolate(q13,A[j]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 73 "In Section 16.2, the equivalence of the integral in the d
enominator with " }{XPPEDIT 18 0 "sigma^2/2" "6#*&%&sigmaG\"\"#F%!\"\"
" }{TEXT -1 1 " " }{XPPEDIT 18 0 "J[1]" "6#&%\"JG6#\"\"\"" }{XPPEDIT 
18 0 "``^2" "6#*$%!G\"\"#" }{TEXT -1 1 "(" }{XPPEDIT 18 0 "(lambda[j]*
sigma)" "6#*&&%'lambdaG6#%\"jG\"\"\"%&sigmaGF(" }{TEXT -1 25 ") was ex
plored in Maple.." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "1" 0 }{VIEWOPTS 1 1 0 1 1 1803 
1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }