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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 24 "POTENTIAL IN A RECTANGLE
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undary conditions " }}{PARA 0 "" 0 "" {TEXT -1 54 "u(x,0)=s(x);  u(x,b
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{PARA 0 "" 0 "" {TEXT -1 79 "We need to expand these function in Fouri
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{MPLTEXT 1 0 54 "sn:=unapply(2/a*int(south(x)*sin(n*Pi*x/a),x=0..a),n)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#snGR6#%\"nG6\"6$%)operatorG%&a
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" 0 "" {MPLTEXT 1 0 54 "nn:=unapply(2/a*int(north(x)*sin(n*Pi*x/a),x=0
..a),n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#nnGR6#%\"nG6\"6$%)opera
torG%&arrowGF(,$*&,(*(9$\"\"\"%#PiGF1-%$sinG6#*&F0\"\"\"F2F7F7F7-%$cos
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0 "> " 0 "" {MPLTEXT 1 0 53 "wn:=unapply(2/b*int(west(y)*sin(n*Pi*y/b)
,y=0..b),n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#wnGR6#%\"nG6\"6$%)o
peratorG%&arrowGF(,$*&,(-%$sinG6#,$*&9$\"\"\"%#PiGF5#F5\"\"$!\"\"-F06#
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*sin(n*Pi*y/b),y=0..b),n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#enGR6
#%\"nG6\"6$%)operatorG%&arrowGF(,$*&,(-%$sinG6#,$*&9$\"\"\"%#PiGF5#F5
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\"!#7F(F(F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "Have a look at the
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" {XPPMATH 20 "6)$\"+.#)4k?!\"*\"\"!$\"+e\"3[k(!#6F&$\"+i&y7l\"F)F&$\"
+=!zx,'!#7" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)$\"+,\"\\?.\"!\"*\"\"!$
\"+zSSAQ!#6F&$\"+7GRc#)!#7F&$\"+4&*))3IF," }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 50 "seq(evalf(wn(n)),n=1..7);seq(evalf(en(n)),n=1..7);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)$\"+F1'H0\"!\"*\"\"!F&F&$!+5D%=@%!
#6F&$\"+S#**)[@F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6)$\"+b7#f5#!\"*\"
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{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "We notice tha
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{TEXT -1 31 "We first find the coefficients " }{XPPEDIT 18 0 "alpha" "
6#%&alphaG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "beta" "6#%%betaG" }{TEXT 
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{XPPEDIT 18 0 "delta" "6#%&deltaG" }{TEXT -1 21 " corresponding to n=1
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "sol1:=evalf(solve(\{sn(
1)=alpha+beta,nn(1)=alpha*exp(b*Pi/a)\n+beta*exp(-b*Pi/a)\},\{alpha,be
ta\}));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol1G<$/%&alphaG$\"+u)fg
%=!#6/%%betaG$\"+.&)e85!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "We
 can manipulate the solutions to \"solve\" by using the substitute com
mand \"subs\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "alpha1:=su
bs(sol1,alpha);beta1:=subs(sol1,beta);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%'alpha1G$\"+u)fg%=!#6" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&be
ta1G$\"+.&)e85!\"*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "sol2:
=evalf(solve(\{wn(1)=g+delta,en(1)=g*exp(a*Pi/b)\n+delta*exp(-a*Pi/b)
\},\{g,delta\}));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%sol2G<$/%\"gG$
\"+mo7rC!#5/%&deltaG$\"+7%z%e!)F*" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "g1:=subs(sol2,g);delta1:=subs(sol2,delta);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%#g1G$\"+mo7rC!#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'delta1G$\"+7%z%e!)!#5" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 28 "We have to use g instead of " }{XPPEDIT 18 0 "gamma" "6#%
&gammaG" }{TEXT -1 53 " because the latter is a protected symbol. The \+
first " }}{PARA 0 "" 0 "" {TEXT -1 40 "approximation to the desired fu
nction is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 118 "ufirst:=(x,y)
->(alpha1*exp(Pi*y/a)\n+beta1*exp(-Pi*y/a))*sin(Pi*x/a)+(g1*exp(Pi*x/b
)+\ndelta1*exp(-Pi*x/b))*sin(Pi*y/b);" }}{PARA 12 "" 1 "" {XPPMATH 20 
"6#>%'ufirstGR6$%\"xG%\"yG6\"6$%)operatorG%&arrowGF),&*&,&*&%'alpha1G
\"\"\"-%$expG6#*&*&%#PiGF29%F2\"\"\"%\"aG!\"\"F2F2*&%&beta1GF2-F46#,$F
6!\"\"F2F2F2-%$sinG6#*&*&F8F:9$F2F:F;F<F2F2*&,&*&%#g1GF2-F46#*&*&F8F:F
HF:F:%\"bGF<F2F2*&%'delta1GF2-F46#,$FOFBF2F2F2-FD6#*&*&F8F:F9F:F:FQF<F
2F2F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "plot3d(ufirst(
x,y),x=0..a,y=0..b,axes=framed);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 
300 300 {PLOTDATA 3 "6%-%%GRIDG6%;\"\"!$\"\"#F';F'$\"\"$F'W(\\bm\":\":
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ELSG6%%\"xG%\"yG%!G-%*AXESSTYLEG6#%&FRAMEG" 1 2 0 1 0 2 1 1 3 2 
1.000000 45.000000 45.000000 0 }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 
"The next approximation involves putting n=3" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "sol13:=evalf(solv
e(\{sn(3)=alpha+beta,nn(3)=alpha*exp(3*b*Pi/a)\n+beta*exp(-b*Pi/a)\},
\{alpha,beta\}));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%&sol13G<$/%%bet
aG$\"+i&)RAQ!#6/%&alphaG$\"+?'*=<b!#<" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "alpha3:=subs(sol13,alpha);beta3:=subs(sol13,beta);" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'alpha3G$\"+?'*=<b!#<" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%&beta3G$\"+i&)RAQ!#6" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 91 "sol23:=evalf(solve(\{wn(3)=g+delta,en(3)=g*exp(a
*3*Pi/b)\n+delta*exp(-a*3*Pi/b)\},\{g,delta\}));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&sol23G<$/%\"gG\"\"!/%&deltaGF(" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 28 "We have to use g instead of " }{XPPEDIT 18 0 "gamm
a" "6#%&gammaG" }{TEXT -1 53 " because the latter is a protected symbo
l. The first " }}{PARA 0 "" 0 "" {TEXT -1 40 "approximation to the des
ired function is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "uthird:
=(x,y)->(alpha3*exp(3*Pi*y/a)\n+beta3*exp(-3*Pi*y/a))*sin(3*Pi*x/a);" 
}}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'uthirdGR6$%\"xG%\"yG6\"6$%)operat
orG%&arrowGF)*&,&*&%'alpha3G\"\"\"-%$expG6#,$*&*&%#PiGF19%F1\"\"\"%\"a
G!\"\"\"\"$F1F1*&%&beta3GF1-F36#,$F6!\"$F1F1F1-%$sinG6#,$*&*&F8F:9$F1F
:F;F<F=F1F)F)F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "plot3d(ufirst(x,y)+uthird(x,y),x=0.
.a,y=0..b,axes=framed);" }}{PARA 13 "" 1 "" {GLPLOT3D 400 300 300 
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{TEXT -1 92 "The figure above represents the correction due to the thi
rd order term in the series. To get" }}{PARA 0 "" 0 "" {TEXT -1 70 "th
e trapezoidal shape near the east and west edges we need more terms." 
}}}}{MARK "34 1 0" 9 }{VIEWOPTS 1 1 0 1 1 1803 }