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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 40 "Partial Differential Equa
tions PowerTool" }}{PARA 19 "" 0 "" {TEXT -1 16 "by Dr. Jim Herod" }}}
{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 256 36 "Sect
ion 6.4: Laplace's Equation on a" }{TEXT 258 1 " " }{TEXT 257 4 "Disk
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 284 
30 "Maple Packages for Section 6.3" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 
"with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 36 "    The form for Laplace's Equation " }
{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT 285 1 "u" }{TEXT -1 209 " =
 0 naturally depends on what coordinate system we are using. So far, w
ith rectangular coordinates, the Laplacian Operator is given by two de
rivatives with respect to x plus two derivatives with respect to y:" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "        \+
       " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT 286 1 "u" }{TEXT 
-1 3 " = " }{XPPEDIT 18 0 "diff(u,`$`(x,2));" "6#-%%diffG6$%\"uG-%\"$G
6$%\"xG\"\"#" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "diff(u,`$`(y,2));" "6#
-%%diffG6$%\"uG-%\"$G6$%\"yG\"\"#" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "It would be no surprise h
ow to write this in three dimensions, instead of two. " }}{PARA 0 "" 
0 "" {TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 285 "     If the origi
ns of the problem to be considered is on a disk, or annulus, or cylind
er, it is often better to think of the problem in polar coordinates. A
 way to think of the change is to remember that in order to change bet
ween rectangular and polar coordinates, use this identity:" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "          " }
{TEXT 287 1 "x" }{TEXT -1 9 " = r cos(" }{XPPEDIT 18 0 "theta;" "6#%&t
hetaG" }{TEXT -1 6 ") and " }{TEXT 288 1 "y" }{TEXT -1 9 " = r sin(" }
{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "In that case, r = " }
{XPPEDIT 18 0 "sqrt(x^2+y^2);" "6#-%%sqrtG6#,&*$%\"xG\"\"#\"\"\"*$%\"y
GF)F*" }{TEXT -1 6 "  and " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }
{TEXT -1 10 " = arctan(" }{TEXT 289 3 "y/x" }{TEXT -1 2 ")." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 75 "     For exampl
e take these examples. We take r = 2 and use five different " }
{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 4 " 's." }}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 43 "r:=2; theta:=[-3*Pi/4,-Pi/2,0,Pi/4,3*Pi/4
];" }}}{PARA 0 "" 0 "" {TEXT -1 85 "We compute the sequence or points \+
[ r cos(t), r sin(t) ] where t is one of the above " }{XPPEDIT 18 0 "t
heta;" "6#%&thetaG" }{TEXT -1 4 " 's." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "map(t->[r*cos(t),r*sin(t)],theta);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 66 "     We now do it the other way. We pick \+
points and make r 's and " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }
{TEXT -1 5 " 's. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 534 "     (A part of the problem here is that arctan(-1/1) an
d arctan(1/(-1)) look the same to the arctangent function. Yet the poi
nts [-1, 1] and [1, -1] are in different quadrants. Thus, we either ha
ve to be careful in our programming, or hope that Maple has a routine \+
to do this. It does. The program computes the angle associated with a \+
complex number a + b I. Thus, we change the point [ x, y] to the compl
ex number x + y I and use Maple's built in program. You might enjoy ma
king a program that does not use Maple's built in scheme.)" }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "    Here, we give
 points and compute r and theta. " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "pts:=[[-1,-1],[0,-1],[1,-1],[1,1],[-1,1]];" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "r:=(x,y)->sqrt(x^2+y^2);\nth
eta:=(x,y)->argument(x+I*y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 75 "seq(r(pts[i][1],pts[i][2]),i=1..5);\nseq(theta(pts[j][1],pts[j][
2]),j=1..5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 297 39 
"Laplace's Equation in Polar Coordinates" }}{PARA 0 "" 0 "" {TEXT -1 
87 "We want to establish Laplace's equation in polar coordinates. To d
o this, suppose that " }{TEXT 260 1 "u" }{TEXT -1 1 "(" }{TEXT 261 1 "
x" }{TEXT -1 2 ", " }{TEXT 262 1 "y" }{TEXT -1 39 ") satisfies Laplace
's equation. Define " }{TEXT 263 1 "v" }{TEXT -1 4 "(r, " }{XPPEDIT 
18 0 "theta;" "6#%&thetaG" }{TEXT -1 5 ") as " }{TEXT 264 1 "u" }
{TEXT -1 7 "(r cos(" }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 9 
"), r sin(" }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 25 ")). We \+
use the fact that " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 1 "
 " }{TEXT 265 1 "u" }{TEXT -1 17 " = 0 to show that" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "                       \+
     " }{XPPEDIT 18 0 "diff(r*diff(v(r,theta),r),r)/r+diff(v(r,theta),
`$`(theta,2))/(r^2);" "6#,&*&-%%diffG6$*&%\"rG\"\"\"-F&6$-%\"vG6$F)%&t
hetaGF)F*F)F*F)!\"\"F**&-F&6$-F.6$F)F0-%\"$G6$F0\"\"#F**$F)F:F1F*" }
{TEXT -1 6 "  =  0" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "v:=(r,
theta)->u(r*cos(theta),r*sin(theta));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 64 "diff(r*diff(v(r,theta),r),r)/r+diff(v(r,theta),theta,
theta)/r^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" 
{TEXT -1 30 "     We see that this last is " }{XPPEDIT 18 0 "Delta;" "
6#%&DeltaG" }{TEXT -1 1 " " }{TEXT 266 1 "u" }{TEXT -1 7 "(r cos(" }
{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 9 "), r sin(" }{XPPEDIT 
18 0 "theta;" "6#%&thetaG" }{TEXT -1 134 ")). The assumption is that t
his is zero. Thus, we know a form for the Laplacian Operator in polar \+
coordinates. Here are some examples." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 71 "LO:=v->diff(r*diff(v(r,theta),r),r)/r+diff(v(r,theta)
,theta,theta)/r^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "v:=(r
,theta)->r*sin(theta);\nLO(v);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 38 "v:=(r,theta)->r^n*cos(n*theta);\nLO(v);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 231 "     What are the type problems we can solve with
 this realization of the Laplacian Operator? You could guess this is a
 typical problem. You can even guess a solution.\n     Suppose we have
 a disk with radius 1. We seek the surface " }{TEXT 267 1 "u" }{TEXT 
-1 16 " which satisfies" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 43 "                                           " }
{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT 268 1 "u" }{TEXT -1 18 "=0 \+
for 0< r < 1, -" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 5 "  <  " }
{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 4 "  < " }{XPPEDIT 18 0 "
Pi;" "6#%#PiG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 52 "       \+
                                    with u(1," }{XPPEDIT 18 0 "theta" 
"6#%&thetaG" }{TEXT -1 8 ") = cos(" }{XPPEDIT 18 0 "theta" "6#%&thetaG
" }{TEXT -1 7 ") for -" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 5 "  \+
<  " }{XPPEDIT 18 0 "theta" "6#%&thetaG" }{TEXT -1 4 "  < " }{XPPEDIT 
18 0 "Pi;" "6#%#PiG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 18 "W
hat do you guess?" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "u:=(r,t
heta)->r*cos(theta);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "LO(u
);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "u(1,theta);" }}}
{PARA 0 "" 0 "" {TEXT -1 31 "Here is a graph of the surface." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "cylinderplot([r,theta,u(r,t
heta)],r=0..1,theta=0..2*Pi,\n        axes=boxed,orientation=[-125,75]
, shading=zhue);" }}}{EXCHG }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 190 "We might be lucky to guess at solutions, but lik
ely will not be. Thus, we need to develop techniques for solving probl
ems on a disk with radius c. We seek bounded solutions for the equatio
ns" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 "   \+
            " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT 269 1 "u" }
{TEXT -1 15 " = 0 with u(c, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }
{TEXT -1 6 ") = f(" }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 4 "
),  " }{XPPEDIT 18 0 "-Pi;" "6#,$%#PiG!\"\"" }{TEXT -1 3 " < " }
{XPPEDIT 18 0 "theta <= Pi;" "6#1%&thetaG%#PiG" }{TEXT -1 2 " ." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Note that
 the ray   " }{XPPEDIT 18 0 "theta = Pi;" "6#/%&thetaG%#PiG" }{TEXT 
-1 12 "   is not a " }{TEXT 259 8 "boundary" }{TEXT -1 111 " in the sa
me sense as the circle r = c is. To be sure that we do not have false \+
boundary problems, we ask that " }}{PARA 0 "" 0 "" {TEXT -1 18 "      \+
            " }{TEXT 270 1 "u" }{TEXT -1 4 "(r, " }{XPPEDIT 18 0 "Pi;
" "6#%#PiG" }{TEXT -1 4 ") = " }{TEXT 271 1 "u" }{TEXT -1 4 "(r, " }
{XPPEDIT 18 0 "-Pi;" "6#,$%#PiG!\"\"" }{TEXT -1 6 ") and " }{XPPEDIT 
18 0 "diff(u,theta);" "6#-%%diffG6$%\"uG%&thetaG" }{TEXT -1 5 " (r, " 
}{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 4 ") = " }{XPPEDIT 18 
0 "diff(u,theta);" "6#-%%diffG6$%\"uG%&thetaG" }{TEXT -1 7 " (r, - " }
{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 3 "). " }}{PARA 0 "" 0 "
" {TEXT -1 71 "These are natural boundary conditions of the problem we
 have described." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 259 
"" 0 "" {TEXT -1 68 "Separation of Variables for Laplace's Equation in
 Polar Coordinates." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 129 "     In these notes to this point, our first guess for
 how to solve such a problem has been to separate variables in the equ
ation" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 "
                     " }{XPPEDIT 18 0 "diff(r*diff(v(r,theta),r),r)/r+
diff(v(r,theta),`$`(theta,2))/(r^2);" "6#,&*&-%%diffG6$*&%\"rG\"\"\"-F
&6$-%\"vG6$F)%&thetaGF)F*F)F*F)!\"\"F**&-F&6$-F.6$F)F0-%\"$G6$F0\"\"#F
**$F)F:F1F*" }{TEXT -1 8 "  =  0, " }}{PARA 0 "" 0 "" {TEXT -1 6 "to g
et" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 9 "    \+
     " }{XPPEDIT 18 0 "1/r;" "6#*&\"\"\"F$%\"rG!\"\"" }{TEXT -1 11 " (
r R ') ' " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 5 "  +  " }
{XPPEDIT 18 0 "1/(r^2);" "6#*&\"\"\"F$*$%\"rG\"\"#!\"\"" }{TEXT -1 3 "
 R " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 8 " '' = 0." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "Multiply \+
by " }{XPPEDIT 18 0 "r^2;" "6#*$%\"rG\"\"#" }{TEXT -1 14 ", divide by \+
R " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 8 "  to get" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "         \+
 " }{XPPEDIT 18 0 "1/R;" "6#*&\"\"\"F$%\"RG!\"\"" }{TEXT -1 16 " r (r \+
R ') ' +  " }{XPPEDIT 18 0 "1/theta;" "6#*&\"\"\"F$%&thetaG!\"\"" }
{TEXT -1 1 " " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 10 " '' \+
 =  0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 
"Boundary conditions are  " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }
{TEXT -1 4 " (- " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 4 ") = " }
{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 3 " ( " }{XPPEDIT 18 0 "
pi;" "6#%#piG" }{TEXT -1 2 ")," }}{PARA 0 "" 0 "" {TEXT -1 40 "       \+
                                 " }{XPPEDIT 18 0 "theta;" "6#%&thetaG
" }{TEXT -1 5 " '(- " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 4 ") = \+
" }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 4 " '( " }{XPPEDIT 
18 0 "pi;" "6#%#piG" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "" {TEXT -1 34 "T
he two differential equations are" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "theta;" "6#%&thetaG
" }{TEXT -1 6 " '' + " }{XPPEDIT 18 0 "lambda^2*theta;" "6#*&%'lambdaG
\"\"#%&thetaG\"\"\"" }{TEXT -1 6 " = 0, " }{XPPEDIT 18 0 "theta;" "6#%
&thetaG" }{TEXT -1 4 " (- " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 
4 ") = " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 3 " ( " }
{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 7 "), and " }{XPPEDIT 18 0 "th
eta;" "6#%&thetaG" }{TEXT -1 5 " '(- " }{XPPEDIT 18 0 "pi;" "6#%#piG" 
}{TEXT -1 4 ") = " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 4 " \+
'( " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "
" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 18 "     r( rR ') ' - " 
}{XPPEDIT 18 0 "lambda^2;" "6#*$%'lambdaG\"\"#" }{TEXT -1 7 " R = 0." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "The sol
utions for the first equation, with eigenvalues and eigenfunctions are
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "     \+
" }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 9 " = 0,    " }
{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 5 " = 1." }}{PARA 0 "" 
0 "" {TEXT -1 5 "     " }{XPPEDIT 18 0 "-lambda^2;" "6#,$*$%'lambdaG\"
\"#!\"\"" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "-n^2;" "6#,$*$%\"nG\"\"#!
\"\"" }{TEXT -1 5 ",    " }{XPPEDIT 18 0 "Theta;" "6#%&ThetaG" }{TEXT 
-1 1 "(" }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 10 ") = cos(n \+
" }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 13 ")  or  sin(n " }
{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 2 ")." }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "The second equation is \+
now  " }{XPPEDIT 18 0 "r^2;" "6#*$%\"rG\"\"#" }{TEXT -1 16 " R '' + r \+
R ' - " }{XPPEDIT 18 0 "n^2;" "6#*$%\"nG\"\"#" }{TEXT -1 6 "R = 0." }}
{PARA 0 "" 0 "" {TEXT -1 32 "Solutions for this equation are " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "     R(r)
 = 1 or ln(r) in case n = 0," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 15 "and, otherwise," }}{PARA 0 "" 0 "" {TEXT -1 17 
"          R(r) = " }{XPPEDIT 18 0 "r^n;" "6#)%\"rG%\"nG" }{TEXT -1 6 
"  or  " }{XPPEDIT 18 0 "r^(-n);" "6#)%\"rG,$%\"nG!\"\"" }{TEXT -1 2 "
.\n" }}{PARA 0 "" 0 "" {TEXT -1 69 "Here's how you might remember the \+
solution for this equation in case " }{TEXT 298 5 "n = 0" }{TEXT -1 4 
" or " }{XPPEDIT 18 0 "n <> 0;" "6#0%\"nG\"\"!" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG }{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "unassign('r');" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 91 "n:=0;\nde := r^2*diff(R(r),r,r)+r*diff(R(r),r)-n^2*R(r)=0;\nds
olve(de, R(r)); \nunassign('n');" }}}{EXCHG }{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "de := r^2*diff(R(r),r,r)+r*diff(R(r),r)-n^2*R(r);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "dsolve(de);" }}}{EXCHG }
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 68 "As a resu
lt, we can make the general solution for this problem with " }
{XPPEDIT 18 0 "0 <= r;" "6#1\"\"!%\"rG" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "     v(r, " }
{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 
"a[0];" "6#&%\"aG6#\"\"!" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "sum(a[n]*r
^n*cos(n*theta),n);" "6#-%$sumG6$*(&%\"aG6#%\"nG\"\"\")%\"rGF*F+-%$cos
G6#*&F*F+%&thetaGF+F+F*" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "sum(b[n]*r^
n*sin(n*theta),n);" "6#-%$sumG6$*(&%\"bG6#%\"nG\"\"\")%\"rGF*F+-%$sinG
6#*&F*F+%&thetaGF+F+F*" }{TEXT -1 1 "." }}}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 103 "We have changed the problem to one \+
that can be solved by Fourier Series Techniques. Here is an example." 
}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 0 {PARA 3 "" 0 "" {TEXT 272 11 "Example 1: " }{XPPEDIT 273 0 "De
lta;" "6#%&DeltaG" }{TEXT 274 12 "u = 0, u(1, " }{XPPEDIT 275 0 "theta
;" "6#%&thetaG" }{TEXT 276 5 " ) = " }{XPPEDIT 277 0 "sin(theta)^2;" "
6#*$-%$sinG6#%&thetaG\"\"#" }}{PARA 0 "" 0 "" {TEXT -1 8 "We solve" }}
{PARA 0 "" 0 "" {TEXT -1 15 "               " }{XPPEDIT 18 0 "Delta;" 
"6#%&DeltaG" }{TEXT -1 12 "u = 0, u(1, " }{XPPEDIT 18 0 "theta;" "6#%&
thetaG" }{TEXT -1 5 " ) = " }{XPPEDIT 18 0 "sin(theta)^2;" "6#*$-%$sin
G6#%&thetaG\"\"#" }{TEXT -1 5 " for " }{XPPEDIT 18 0 "theta;" "6#%&the
taG" }{TEXT -1 23 " in the interval [0, 2 " }{XPPEDIT 18 0 "Pi;" "6#%#
PiG" }{TEXT -1 3 " ]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 39 "First, we draw the boundary conditions:" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 120 "spacecurve([cos(theta),sin(theta),
sin(theta)^2],theta=-Pi..Pi,\n            color=BLACK,orientation=[35,
65],axes=NORMAL);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "We have t
he general solution " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "     " }{TEXT 290 1 "u" 
}{TEXT -1 4 "(r, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 4 ")
 = " }{XPPEDIT 18 0 "a[0];" "6#&%\"aG6#\"\"!" }{TEXT -1 3 " + " }
{XPPEDIT 18 0 "sum(a[n]*r^n*cos(n*theta),n);" "6#-%$sumG6$*(&%\"aG6#%
\"nG\"\"\")%\"rGF*F+-%$cosG6#*&F*F+%&thetaGF+F+F*" }{TEXT -1 3 " + " }
{XPPEDIT 18 0 "sum(b[n]*r^n*sin(n*theta),n);" "6#-%$sumG6$*(&%\"bG6#%
\"nG\"\"\")%\"rGF*F+-%$sinG6#*&F*F+%&thetaGF+F+F*" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 ". If r = \+
1, we have" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 11 "          (" }{XPPEDIT 18 0 "sin(theta)^2;" "6#*$-%$sinG6#%&the
taG\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "a[0];" "6#&%\"aG6#\"\"!" }
{TEXT -1 3 " + " }{XPPEDIT 18 0 "sum(a[n]*1^n*cos(n*theta),n);" "6#-%$
sumG6$*(&%\"aG6#%\"nG\"\"\")F+F*F+-%$cosG6#*&F*F+%&thetaGF+F+F*" }
{TEXT -1 3 " + " }{XPPEDIT 18 0 "sum(b[n]*1^n*sin(n*theta),n);" "6#-%$
sumG6$*(&%\"bG6#%\"nG\"\"\")F+F*F+-%$sinG6#*&F*F+%&thetaGF+F+F*" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 110 "We have only to compute the coefficients. This is a job \+
for Fourier Series. Or, maybe you remember some trig. " }}{PARA 0 "" 
0 "" {TEXT -1 15 "               " }{XPPEDIT 18 0 "sin(theta)^2;" "6#*
$-%$sinG6#%&thetaG\"\"#" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "(1-cos(2*th
eta))/2;" "6#*&,&\"\"\"F%-%$cosG6#*&\"\"#F%%&thetaGF%!\"\"F%F*F," }
{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 6 "Thus, " }{XPPEDIT 18 0 "a[0];" "6#&%\"aG6#\"\"!" }{TEXT 
-1 9 " = 1/2,  " }{XPPEDIT 18 0 "a[2];" "6#&%\"aG6#\"\"#" }{TEXT -1 
27 " = -1/2, and all the other " }{TEXT 291 1 "a" }{TEXT -1 38 " 's ar
e zero. The implication is that " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
35 "u:=(r,theta)->1/2-r^2*cos(2*theta);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 129 "cylinderplot([r,theta,u(r,theta)],r=0..1,theta=0..2*
Pi,\n          orientation=[40,40],axes=NORMAL, shading=zhue, numpoint
s=1000);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 278 12 "Example 2
,  " }{XPPEDIT 279 0 "Delta;" "6#%&DeltaG" }{TEXT 280 12 "u = 0, u(1, \+
" }{XPPEDIT 281 0 "theta;" "6#%&thetaG" }{TEXT 282 5 " ) = " }
{XPPEDIT 283 0 "theta^3*(Pi^2-theta^2);" "6#*&%&thetaG\"\"$,&*$%#PiG\"
\"#\"\"\"*$F$F)!\"\"F*" }}{PARA 0 "" 0 "" {TEXT -1 8 "We solve" }}
{PARA 0 "" 0 "" {TEXT -1 15 "               " }{XPPEDIT 18 0 "Delta;" 
"6#%&DeltaG" }{TEXT 292 1 "u" }{TEXT -1 11 " = 0, u(1, " }{XPPEDIT 18 
0 "theta;" "6#%&thetaG" }{TEXT -1 5 " ) = " }{XPPEDIT 18 0 "theta^3*(P
i^2-theta^2);" "6#*&%&thetaG\"\"$,&*$%#PiG\"\"#\"\"\"*$F$F)!\"\"F*" }
{TEXT -1 5 " for " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 23 "
 in the interval [0, 2 " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 3 " \+
]." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "Fir
st, we draw the boundary conditions:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 130 "spacecurve([cos(theta),sin(theta),theta^3*(Pi^2-thet
a^2)],theta=-Pi..Pi,\n            color=BLACK,orientation=[10,55],axes
=NORMAL);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 29 "We have the genera
l solution " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "     " }{TEXT 293 1 "u" }{TEXT 
-1 4 "(r, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 4 ") = " }
{XPPEDIT 18 0 "a[0];" "6#&%\"aG6#\"\"!" }{TEXT -1 3 " + " }{XPPEDIT 
18 0 "sum(a[n]*r^n*cos(n*theta),n);" "6#-%$sumG6$*(&%\"aG6#%\"nG\"\"\"
)%\"rGF*F+-%$cosG6#*&F*F+%&thetaGF+F+F*" }{TEXT -1 3 " + " }{XPPEDIT 
18 0 "sum(b[n]*r^n*sin(n*theta),n);" "6#-%$sumG6$*(&%\"bG6#%\"nG\"\"\"
)%\"rGF*F+-%$sinG6#*&F*F+%&thetaGF+F+F*" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 19 ". If r = 1, we have
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "     \+
     " }{XPPEDIT 18 0 "theta^3*(Pi^2-theta^2);" "6#*&%&thetaG\"\"$,&*$
%#PiG\"\"#\"\"\"*$F$F)!\"\"F*" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "a[0];
" "6#&%\"aG6#\"\"!" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "sum(a[n]*1^n*cos
(n*theta),n);" "6#-%$sumG6$*(&%\"aG6#%\"nG\"\"\")F+F*F+-%$cosG6#*&F*F+
%&thetaGF+F+F*" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "sum(b[n]*1^n*sin(n*t
heta),n);" "6#-%$sumG6$*(&%\"bG6#%\"nG\"\"\")F+F*F+-%$sinG6#*&F*F+%&th
etaGF+F+F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 123 "We compute coefficients. It seems to be of no va
lue to list the numerical output of these integrals, so we suppress ou
tput." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 80 "a[0]:=int(theta^3*(Pi^2-theta^2),theta=-Pi..Pi)/\n   \+
       int(1,theta=-Pi..Pi):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 147 "for n from 1 to 10 do\n   a[n]:=int(theta^3*(Pi^2-theta^2)*cos(
n*theta),theta=-Pi..Pi)/\n          int(cos(n*theta)^2,theta=-Pi..Pi):
\nend do:\nn:='n':" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 147 "for \+
n from 1 to 10 do\n   b[n]:=int(theta^3*(Pi^2-theta^2)*sin(n*theta),th
eta=-Pi..Pi)/\n          int(sin(n*theta)^2,theta=-Pi..Pi):\nend do:\n
n:='n':" }}}{PARA 0 "" 0 "" {TEXT -1 16 "Here, we define " }{TEXT 294 
1 "u" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "u:=(
r,theta)->a[0]+sum(a[n]*r^n*cos(n*theta)+b[n]*r^n*sin(n*theta),\n     \+
             n=1..10);" }}}{PARA 0 "" 0 "" {TEXT -1 116 "Here is a gra
ph of the surface. Compare this \"rubber sheet\" with the space curve \+
given at the start of this problem." }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 129 "cylinderplot([r,theta,u(r,theta)],r=0..1,theta=0..2*
Pi,\n          orientation=[10,55],axes=NORMAL, shading=zhue, numpoint
s=1000);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 104 "In summary, Laplace's Equation on a disk can be solved i
n a manner consistent with what has come before:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "                " }{TEXT 
295 1 "u" }{TEXT -1 4 "(r, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }
{TEXT -1 4 ") = " }{XPPEDIT 18 0 "a[0];" "6#&%\"aG6#\"\"!" }{TEXT -1 
3 " + " }{XPPEDIT 18 0 "sum(a[n]*r^n*cos(n*theta),n);" "6#-%$sumG6$*(&
%\"aG6#%\"nG\"\"\")%\"rGF*F+-%$cosG6#*&F*F+%&thetaGF+F+F*" }{TEXT -1 
3 " + " }{XPPEDIT 18 0 "sum(b[n]*r^n*sin(n*theta),n);" "6#-%$sumG6$*(&
%\"bG6#%\"nG\"\"\")%\"rGF*F+-%$sinG6#*&F*F+%&thetaGF+F+F*" }{TEXT -1 
2 " ," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 "
where     " }{XPPEDIT 18 0 "a[0];" "6#&%\"aG6#\"\"!" }{TEXT -1 3 " = \+
" }{XPPEDIT 18 0 "int(f(t),t = -Pi .. Pi)/(2*Pi);" "6#*&-%$intG6$-%\"f
G6#%\"tG/F*;,$%#PiG!\"\"F.\"\"\"*&\"\"#F0F.F0F/" }{TEXT -1 2 " ," }}
{PARA 0 "" 0 "" {TEXT -1 15 "               " }{XPPEDIT 18 0 "a[n];" "
6#&%\"aG6#%\"nG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "int(f(t)*cos(n*t),t
 = -Pi .. Pi)/(Pi*c^n);" "6#*&-%$intG6$*&-%\"fG6#%\"tG\"\"\"-%$cosG6#*
&%\"nGF,F+F,F,/F+;,$%#PiG!\"\"F5F,*&F5F,)%\"cGF1F,F6" }{TEXT -1 2 " ,
" }}{PARA 0 "" 0 "" {TEXT -1 3 "and" }}{PARA 0 "" 0 "" {TEXT -1 15 "  \+
             " }{XPPEDIT 18 0 "b[n];" "6#&%\"bG6#%\"nG" }{TEXT -1 3 " \+
= " }{XPPEDIT 18 0 "int(f(t)*sin(n*t),t = -Pi .. Pi)/(Pi*c^n);" "6#*&-
%$intG6$*&-%\"fG6#%\"tG\"\"\"-%$sinG6#*&%\"nGF,F+F,F,/F+;,$%#PiG!\"\"F
5F,*&F5F,)%\"cGF1F,F6" }{TEXT -1 2 " ," }}{PARA 0 "" 0 "" {TEXT -1 6 "
where " }{TEXT 296 1 "c" }{TEXT -1 27 " is the radius of the disk." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "EMAIL: he
rod@math.gatech.edu   or   jherod@tds.net" }}{PARA 0 "" 0 "" {TEXT -1 
38 "URL: http://www.math.gatech.edu/~herod" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 36 "Copyright \251  2003  by Jame
s V. Herod" }}{PARA 258 "" 0 "" {TEXT -1 19 "All rights reserved" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 1 1 2 
33 1 1 }