{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Cour ier" 1 10 0 0 255 1 0 0 0 0 0 1 3 0 3 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" 2 7 1 {CSTYLE "" -1 -1 "" 0 1 0 0 255 1 0 0 0 0 0 0 1 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Title" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 }{PSTYLE " Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 8 8 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 57 "Ferramenta de Resolu\347 \343o de Equa\347\365es Diferenciais Parciais" }}{PARA 19 "" 0 "" {TEXT -1 0 "" }}}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT 256 39 "Se\347\343o 7.3: Equa\347\343o de Calor em um Disco" }} {PARA 257 "" 0 "" {TEXT 258 0 "" }{TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT 257 40 "Pacotes do Maple utilizados na Se\347\343o 7.3" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 298 "Nesta se\347\343o, \+ consideraremos a equa\347\343o de difus\343o em um disco. Fisicamente, tem-se a seguinte situa\347\343o: uma viga circular longa, ou um disc o isolado por cima e por baixo, com uma temperatura fixa dos lados, e \+ uma temperatura inicial ao longo do disco. Este problema pode ser des crito pelas equa\347\365es" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 " 1/k " }{XPPEDIT 18 0 "diff(u,t);" "6#-% %diffG6$%\"uG%\"tG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Delta;" "6#%&Del taG" }{TEXT -1 3 " u," }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "com u(t, a, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" } {TEXT -1 8 " ) = f( " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 15 " ), 0 < t, " }{XPPEDIT 18 0 "-pi;" "6#,$%#piG!\"\"" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 3 " < " } {XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 15 "e u(0, r, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 11 " ) = g( r, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 6 " ), " }{XPPEDIT 18 0 "0 <= r;" "6#1\"\"!%\"rG" }{TEXT -1 9 " < a, \+ " }{XPPEDIT 18 0 "-pi;" "6#,$%#piG!\"\"" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "Pi;" "6#% #PiG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "Primeiramente, deve-se resolver o problema de estado e stacion\341rio" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 " 0 = " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" } {TEXT -1 10 " v, v(a, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 8 " ) = f( " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 5 " ), \+ " }{XPPEDIT 18 0 "-pi;" "6#,$%#piG!\"\"" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "Pi;" "6#% #PiG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 50 "Em seguida, deve-se resolver o problema transiente" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 14 " \+ 1/k " }{XPPEDIT 18 0 "diff(u,t);" "6#-%%diffG6$%\"uG%\"tG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" }{TEXT -1 3 " u," }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 12 "com u(t, \+ a, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 17 " ) = 0, 0 < t , " }{XPPEDIT 18 0 "-pi;" "6#,$%#piG!\"\"" }{TEXT -1 3 " < " } {XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 3 " < " }{XPPEDIT 18 0 " Pi;" "6#%#PiG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 15 "e u (0, r, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 11 " ) = g( r, " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 10 " ) - v(r, " } {XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 5 ") , " }{XPPEDIT 18 0 "0 <= r;" "6#1\"\"!%\"rG" }{TEXT -1 7 " < a, " }{XPPEDIT 18 0 "-pi; " "6#,$%#piG!\"\"" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "theta;" "6#%&thet aG" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "A solu \347\343o para o problema original \351 dada pela soma destas duas sol u\347\365es." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 259 "O primeiro problema j\341 foi considerado nas Se\347\365es 6.4 e 6.5. Portanto, nesta se\347\343o, concentraremo-nos na resolu\347 \343o do segundo problema. Descrevemos, a seguir, a equa\347\343o dif erencial parcial de forma n\343o abreviada, a fim de facilitar a separ a\347\343o de vari\341veis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 21 " " }{XPPEDIT 18 0 "diff(r*d iff(u(r,theta),r),r)/r+diff(u(r,theta),`$`(theta,2))/(r^2);" "6#,&*&-% %diffG6$*&%\"rG\"\"\"-F&6$-%\"uG6$F)%&thetaGF)F*F)F*F)!\"\"F**&-F&6$-F .6$F)F0-%\"$G6$F0\"\"#F**$F)F:F1F*" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "diff(u,t);" "6#-%%diffG6$%\"uG%\"tG" }{TEXT -1 1 "." }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 166 "(Omitiu-se a constante de difus\343o para a ilustra\347\343o do problema.) A separa\347\343o de vari\341veis resulta nas tr\352s equa\347\365es diferenciais a seg uir, com condi\347\365es de contorno:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 5 "(1) " }{XPPEDIT 18 0 "Theta;" "6#%&Th etaG" }{TEXT -1 6 " '' + " }{XPPEDIT 18 0 "mu^2*Theta;" "6#*&%#muG\"\" #%&ThetaG\"\"\"" }{TEXT -1 7 " = 0, " }{XPPEDIT 18 0 "Theta(Pi);" "6# -%&ThetaG6#%#PiG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Theta(-Pi);" "6#-% &ThetaG6#,$%#PiG!\"\"" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "Theta;" "6#%& ThetaG" }{TEXT -1 3 " '(" }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 5 " ) = " }{XPPEDIT 18 0 "Theta;" "6#%&ThetaG" }{TEXT -1 3 " '(" } {XPPEDIT 18 0 "-Pi;" "6#,$%#PiG!\"\"" }{TEXT -1 2 ")," }}{PARA 0 "" 0 "" {TEXT -1 14 "(2) T ' = - " }{XPPEDIT 18 0 "lambda^2;" "6#*$%'lamb daG\"\"#" }{TEXT -1 7 " T, e " }}{PARA 0 "" 0 "" {TEXT -1 22 "(3) r ( r R ' ) ' - " }{XPPEDIT 18 0 "mu^2;" "6#*$%#muG\"\"#" }{TEXT -1 7 " R = - " }{XPPEDIT 18 0 "lambda^2;" "6#*$%'lambdaG\"\"#" }{TEXT -1 1 " " }{XPPEDIT 18 0 "r^2;" "6#*$%\"rG\"\"#" }{TEXT -1 39 " R, R(a) = 0, R \351 limitado em [0, a)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 118 "J\341 sabemos como resolver as equa\347\365es \+ (1) e (2). Por outro lado, a equa\347\343o 3 exige conhecimento das f un\347\365es de Bessel." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 56 "Iniciaremos este estudo com um exemplo independente \+ de " }{XPPEDIT 18 0 "Theta;" "6#%&ThetaG" }{TEXT -1 27 ". Analisarem os o problema:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 15 " " }{XPPEDIT 18 0 "diff(u,t);" "6#-%%diffG6 $%\"uG%\"tG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Delta;" "6#%&DeltaG" } {TEXT -1 3 " u," }}{PARA 0 "" 0 "" {TEXT -1 14 "com u(t, a, " } {XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 17 " ) = 0, 0 < t, " } {XPPEDIT 18 0 "-pi;" "6#,$%#piG!\"\"" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "Pi;" "6#%#Pi G" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 13 "e u(0, r, " } {XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 14 " ) = g( r), " } {XPPEDIT 18 0 "0 <= r;" "6#1\"\"!%\"rG" }{TEXT -1 7 " < a, " } {XPPEDIT 18 0 "-pi;" "6#,$%#piG!\"\"" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 3 " < " }{XPPEDIT 18 0 "Pi;" "6#%#Pi G" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "\311 evidente que a solu\347\343o para esta equa\347\343o ser\341 independente de " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" } {TEXT -1 58 " . Sendo assim, a equa\347\343o diferencial parcial redu z-se a " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 " " }{XPPEDIT 18 0 "diff(r*diff(u(r,theta),r),r)/r; " "6#*&-%%diffG6$*&%\"rG\"\"\"-F%6$-%\"uG6$F(%&thetaGF(F)F(F)F(!\"\"" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "diff(u,t);" "6#-%%diffG6$%\"uG%\"tG " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "Tem-se agora as duas equa\347\365es diferenciais" }} {PARA 0 "" 0 "" {TEXT -1 10 "(1) T ' + " }{XPPEDIT 18 0 "lambda^2;" "6 #*$%'lambdaG\"\"#" }{TEXT -1 9 " T = 0, e" }}{PARA 0 "" 0 "" {TEXT -1 20 "(2) r ( r R ' ) ' + " }{XPPEDIT 18 0 "lambda^2;" "6#*$%'lambdaG\" \"#" }{TEXT -1 1 " " }{XPPEDIT 18 0 "r^2;" "6#*$%\"rG\"\"#" }{TEXT -1 45 " R = 0, com R(a) = 0, e R limitado em [0, a)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Examinemos a segunda equa \347\343o diferencial. Expandida, esta torna-se" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 10 " " }{XPPEDIT 18 0 "r^2;" "6#*$%\"rG\"\"#" }{TEXT -1 16 " R '' + r R ' + " }{XPPEDIT 18 0 "lambda^2;" "6#*$%'lambdaG\"\"#" }{TEXT -1 1 " " }{XPPEDIT 18 0 " r^2;" "6#*$%\"rG\"\"#" }{TEXT -1 7 " R = 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 156 "O Maple \351 perfeitamente cap az de resolver esta equa\347\343o. Comumente, as solu\347\365es de eq ua\347\365es diferenciais elementares s\343o introduzidas no estudo de s\351ries. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Neste caso, assumimos " }{XPPEDIT 18 0 "lambda;" "6#%'la mbdaG" }{TEXT -1 101 " > 0. Por outro lado, o Maple assume que este va lor \351 complexo, caso n\343o seja informado do contr\341rio." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "assume(lambda>0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "ds olve(r^2*diff(R(r),r,r)+ r*diff(R(r),r)+lambda^2*r^2*R(r)=0,R(r),serie s);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"rG,&*&%$_C1G\"\"\"+ +F'F+\"\"!,$*&\"\"%!\"\"%(lambda|irG\"\"#F1F3,$*&\"#kF1F2F0F+F0-%\"OG6 #F+\"\"'F+F+*&%$_C2GF+,&*&-%#lnGF&F+F,F+F++)F',$*&F0F1F2F3F+F3,$*(\"\" $F+\"$G\"F1F2F0F1F0F7F:F+F+F+" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 145 "Obviamente, o Maple conhece estas solu \347\365es em detalhes, sem o uso de s\351ries. Sendo assim, calcular emos novamente a solu\347\343o, sem o uso de s\351ries." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "dsolve( r^2*diff(R(r),r,r)+ r*diff(R(r),r)+lambda^2*r^2*R(r)=0,R(r));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"RG6#%\"rG,&*&%$_C1G\"\"\"-%(Besse lJG6$\"\"!*&%(lambda|irGF+F'F+F+F+*&%$_C2GF+-%(BesselYGF.F+F+" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 134 "Verifica -se que existem duas solu\347\365es. As fun\347\365es de Bessel econt radas na solu\347\343o do problema podem ser observadas no gr\341fico \+ abaixo." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "plot([BesselY(0,r ),BesselJ(0,r)],r=0..10,color=[BLACK,RED]);" }}{PARA 13 "" 1 "" {GLPLOT2D 384 384 384 {PLOTDATA 2 "6&-%'CURVESG6$7ap7$$\"+v1h6o!#7$!+) *e$*\\K!\"*7$$\"+N@Ki8!#6$!+E*\\&3GF-7$$\"+-K[V?F1$!+b*\\-b#F-7$$\"+qU kCFF1$!+z5)oO#F-7$$\"+P`!eS$F1$!+06bCAF-7$$\"+0k'p3%F1$!+c\\;3@F-7$$\" +su7oZF1$!++9n4?F-7$$\"+S&)G\\aF1$!+W_EC>F-7$$F)F1$!+wFI\"y\"F-7$$\"+5 G$R<)F1$!+iD>k;F-7$$\"+X\\DO&*F1$!+B;*[c\"F-7$$\"+3x&)*3\"!#5$!+oF]o$!+\"\\BE6\" F-7$$\"+;arz@F]o$!+\"fPP-\"F-7$$\"+)4bQl#F]o$!+@Z#4\"*)F]o7$$\"+!y%*z7 $F]o$!+8Yn%y(F]o7$$\"+iW8-OF]o$!+&)HR,oF]o7$$\"+XTFwSF]o$!+oh^DfF]o7$$ \"+1Q\\4YF]o$!+&H_)R]F]o7$$\"+oMrU^F]o$!+:\")oPUF]o7$$\"+IJ$fn&F]o$!+0 0F.NF]o7$$\"+\"z_\"4iF]o$!+&oE`#GF]o7$$\"+m6m#G(F]o$!+,TA+;F]o7$$\"+S& phN)F]o$!+9f*pE&F17$$\"+:ddC%*F]o$\"+M;a$=%F17$$\"+*=)H\\5F-$\"+YQHd7F ]o7$$\"+=JN[6F-$\"+W())*\\>F]o7$$\"+[!3uC\"F-$\"+$p(*pc#F]o7$$\"+J$RDX \"F-$\"+6PO@OF]o7$$\"+)R'ok;F-$\"+%)p%eT%F]o7$$\"+_(>/x\"F-$\"+Ek^0ZF] o7$$\"+1J:w=F-$\"+E6LF\\F]o7$$\"+dG\"\\)>F-$\"+pc'p3&F]o7$$\"+3En$4#F- $\"+(pe&z^F]o7$$\"+K/dT@F-$\"+Uev*>&F]o7$$\"+c#o%*=#F-$\"+P1r2_F]o7$$ \"+!3mtB#F-$\"+4on._F]o7$$\"+/RE&G#F-$\"+*R=z=&F]o7$$\"+9r5$R#F-$\"+;E '46&F]o7$$\"+D.&4]#F-$\"+:\\Jz\\F]o7$$\"+]jB4EF-$\"+)*Rt&z%F]o7$$\"+vB _IDF]o7$$\"+LY.KNF-$\"+#Hq%e#F]o7$$\"+R\"3Gy%F-$!+Q^&eo# F]o7$$\"+.T1&*\\F-$!+5n$y2$F]o7$$\"+]7I0^F-$!+5c+AKF]o7$$\"+(RQb@&F-$! +KfRCLF]o7$$\"+d,]6`F-$!+@uEzLF]o7$$\"+=>Y2aF-$!+$GRBS$F]o7$$\"+[K56bF -$!+*eh>R$F]o7$$\"+yXu9cF-$!+oIrXLF]o7$$\"+8i\"=s&F-$!+7RZhKF]o7$$\"+ \\y))GeF-$!+xoyTJF]o7$$\"+i_QQgF-$!+u1)G\"GF]o7$$\"+!y%3TiF-$!+O3,!R#F ]o7$$\"+O![hY'F-$!+T+WC=F]o7$$\"+#Qx$omF-$!+N-la7F]o7$$\"+u.I%)oF-$!+$ 4n25'F17$$\"+(pe*zqF-$!+\"zy!H=F*7$$\"+C\\'QH(F-$\"+Utf-hF17$$\"+8S8& \\(F-$\"+od]g6F]o7$$\"+0#=bq(F-$\"+W\\Oq;F]o7$$\"+2s?6zF-$\"+Y3V&3#F]o 7$$\"+IXaE\")F-$\"+KsN:CF]o7$$\"+l*RRL)F-$\"+b3&4i#F]o7$$\"+ee)*R%)F-$ \"+mQU\"o#F]o7$$\"+`<.Y&)F-$\"+YJ?6FF]o7$$\"+K&*>^')F-$\"+JQZ5FF]o7$$ \"+8tOc()F-$\"+&)H)*zEF]o7$$\"+\\Qk\\*)F-$\"+$*o%)[DF]o7$$\"+p0;r\"*F- $\"+Qs1(G#F]o7$$\"+lxGp$*F-$\"+(G(4k>F]o7$$\"+!oK0e*F-$\"+*Q7Qa\"F]o7$ $\"+<5s#y*F-$\"+yWD'3\"F]o7$$\"#5\"\"!$\"+Gn6nbF1-%'COLOURG6&%$RGBG$Fj elFjelFaflFafl-F$6$7hn7$Fafl$\"\"\"Fjel7$FN$\"+\"pxD***F]o7$F[o$\"+#HF .(**F]o7$Feo$\"+S$)HL**F]o7$F_p$\"+kKd\"))*F]o7$Fip$\"+%*H)ov*F]o7$Fcq $\"+-Q*))e*F]o7$Fgr$\"+vz8f!*F]o7$Fas$\"+p03H$)F]o7$F[t$\"+QV:JuF]o7$F et$\"+xzGskF]o7$Fjt$\"+[,V\"Q&F]o7$F_u$\"+kc[$=%F]o7$Fiu$\"+!oKo&HF]o7 $Fcv$\"+I&[?q\"F]o7$Fgw$\"+6hY^jF17$Fax$!+ulg&)[F17$F[y$!+\"f+9]\"F]o7 $F`y$!+*ol.M#F]o7$Fey$!+eBdnHF]o7$Fjy$!+-o@INF]o7$$\"+qxdOMF-$!+X+s0PF ]o7$F_z$!+BP8VQF]o7$$\"+cO2VOF-$!+XU$\\&RF]o7$Fdz$!+zCR:SF]o7$$\"+K5S_ QF-$!+aItESF]o7$Fiz$!+EhS**RF]o7$$\"+#GF&eSF-$!+q$zj#RF]o7$F^[l$!+zvC5 QF]o7$Fc[l$!+)y!oyMF]o7$Fh[l$!+hC*p*HF]o7$F]\\l$!+D[IbCF]o7$Fb\\l$!+hH 7#z\"F]o7$F\\]l$!+-<_\\5F]o7$Ff]l$!+oAQjQF17$F`^l$\"+R&*3*=$F17$Fj^l$ \"+[uD15F]o7$F__l$\"+!y'=6;F]o7$Fd_l$\"+pU76@F]o7$Fi_l$\"+.*[sa#F]o7$F ^`l$\"+CR+>GF]o7$$\"+y)Qjx'F-$\"+Mxn9HF]o7$Fc`l$\"+)=n^(HF]o7$$\"+N&H@ )pF-$\"+&Gv%**HF]o7$Fh`l$\"+<0'\\*HF]o7$$\"+5=\"p=(F-$\"+z2ddHF]o7$F]a l$\"+xI=()GF]o7$Fbal$\"+@$[*pEF]o7$Fgal$\"+>TbNBF]o7$F\\bl$\"+z/&*=>F] o7$Fabl$\"+tU/49F]o7$Ffbl$\"+A@=&p)F17$F`cl$\"+Zr4OHF17$Fjcl$!+!*f\"\\ w#F17$F_dl$!+@zk$y(F17$Fddl$!+#GDTI\"F]o7$Fidl$!+Q`,6 " 0 "" {MPLTEXT 1 0 83 "R:=r->BesselY(0,lambda*r); \n1/r*(diff(r*diff(R(r),r),r))+lambda^2*R(r);\nsimplify(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RGf*6#%\"rG6\"6$%)operatorG%&arrowGF(-%( BesselYG6$\"\"!*&%'lambdaG\"\"\"9$F2F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&*&%\"rG!\"\",&*&-%(BesselYG6$\"\"\"*&%(lambda|irGF,F% F,F,F.F,F&*(F%F,,&-F*6$\"\"!F-F,*(F.F&F%F&F)F,F&F,)F.\"\"#F,F&F,F,*&F5 F,F1F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "R:=r->Besse lJ(0,lambda*r);\n1/r*(diff(r*diff(R(r),r),r))+lambda^2*R(r);\nsimplify (%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"RGf*6#%\"rG6\"6$%)operator G%&arrowGF(-%(BesselJG6$\"\"!*&%'lambdaG\"\"\"9$F2F(F(F(" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#,&*&%\"rG!\"\",&*&-%(BesselJG6$\"\"\"*&%(lambda| irGF,F%F,F,F.F,F&*(F%F,,&-F*6$\"\"!F-F,*(F.F&F%F&F)F,F&F,)F.\"\"#F,F&F ,F,*&F5F,F1F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#\"\"!" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 129 "Pode ser bastante \372til a compara\347\343o destas duas fun\347\365es com as fun\347 \365es coseno e seno, que definem as autofun\347\365es para o problema " }{XPPEDIT 18 0 "Theta;" "6#%&ThetaG" }{TEXT -1 6 " '' + " } {XPPEDIT 18 0 "lambda^2;" "6#*$%'lambdaG\"\"#" }{TEXT -1 1 " " } {XPPEDIT 18 0 "Theta;" "6#%&ThetaG" }{TEXT -1 191 " = 0. Esta compara \347\343o pode ajudar na memoriza\347\343o da estrutura de resolu\347 \343o do problema. Como estamos interessados apenas em solu\347\365es limitadas no disco, deixaremos de lado a fun\347\343o BesselY." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Seja " } {TEXT 261 1 "a" }{TEXT -1 22 " > 0. Tomamos, ainda, " }{TEXT 262 1 "J " }{TEXT -1 1 "(" }{TEXT 263 1 "r" }{TEXT -1 30 ") como sendo a fun \347\343o BesselJ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 13 "Compara\347\343o 1:" }{TEXT -1 39 " H \341 uma sequ\352ncia crescente infinita " }{XPPEDIT 18 0 "lambda[1]; " "6#&%'lambdaG6#\"\"\"" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "lambda[2]; " "6#&%'lambdaG6#\"\"#" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "lambda[3]; " "6#&%'lambdaG6#\"\"$" }{TEXT -1 20 " , ... tal que J( " }{XPPEDIT 18 0 "lambda[p];" "6#&%'lambdaG6#%\"pG" }{TEXT -1 50 " a) = 0. Para a \+ fun\347\343o coseno, estes n\372meros s\343o " }{XPPEDIT 18 0 "n*Pi/a ;" "6#*(%\"nG\"\"\"%#PiGF%%\"aG!\"\"" }{TEXT -1 2 " ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 265 14 "Compar a\347\343o 2: " }{TEXT -1 14 " Os n\372meros - " }{XPPEDIT 18 0 "lambd a[j]^2;" "6#*$&%'lambdaG6#%\"jG\"\"#" }{TEXT -1 30 " s\343o autovalor es e R(r) = J( " }{XPPEDIT 18 0 "lambda[j];" "6#&%'lambdaG6#%\"jG" } {TEXT -1 35 " r) \351 uma autofun\347\343o para operador " }{TEXT 266 1 "L" }{TEXT -1 1 "(" }{TEXT 267 1 "R" }{TEXT -1 22 "), definido acima com " }{TEXT 268 1 "R" }{TEXT -1 1 "(" }{TEXT 269 1 "a" }{TEXT -1 22 ") = 0. A fun\347\343o cos( " }{XPPEDIT 18 0 "n*Pi/a;" "6#*(%\"nG\"\" \"%#PiGF%%\"aG!\"\"" }{TEXT -1 27 " x ) \351 uma autonfu\347\343o para " }{XPPEDIT 18 0 "Theta;" "6#%&ThetaG" }{TEXT -1 9 " '', com " } {XPPEDIT 18 0 "Theta;" "6#%&ThetaG" }{TEXT -1 9 " (a) = 0." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 270 13 "C ompara\347\343o 3:" }{TEXT -1 51 " A ortogonalidade \351 preservada, \+ no sentido em que " }}{PARA 0 "" 0 "" {TEXT -1 10 " " } {XPPEDIT 18 0 "int(J(lambda[j]*x)*J(lambda[k]*x)*x,x = 0 .. a);" "6#-% $intG6$*(-%\"JG6#*&&%'lambdaG6#%\"jG\"\"\"%\"xGF/F/-F(6#*&&F,6#%\"kGF/ F0F/F/F0F//F0;\"\"!%\"aG" }{TEXT -1 8 " = 0 if " }{XPPEDIT 18 0 "j <> \+ k;" "6#0%\"jG%\"kG" }{TEXT -1 6 " e " }{XPPEDIT 18 0 "int(J(lambda[ j]*x)^2*x,x = 0 .. a);" "6#-%$intG6$*&-%\"JG6#*&&%'lambdaG6#%\"jG\"\" \"%\"xGF/\"\"#F0F//F0;\"\"!%\"aG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "a^ 2/2;" "6#*&%\"aG\"\"#F%!\"\"" }{TEXT -1 4 " J '" }{XPPEDIT 18 0 "(lamb da[j]*a)^2" "6#*$*&&%'lambdaG6#%\"jG\"\"\"%\"aGF)\"\"#" }{TEXT -1 2 " \+ ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 271 13 "Compara\347\343o 4:" }{TEXT -1 5 " Se " }{TEXT 272 1 "f " }{TEXT -1 1 "(" }{TEXT 273 1 "r" }{TEXT -1 40 ") \351 seccionalmente diferenci\341vel em (0, " }{TEXT 274 1 "a" }{TEXT -1 8 "), ent\343o" }}{PARA 0 "" 0 "" {TEXT -1 17 " f(r) = " }{XPPEDIT 18 0 "sum( a[n]*J(lambda[n]*r),n);" "6#-%$sumG6$*&&%\"aG6#%\"nG\"\"\"-%\"JG6#*&&% 'lambdaG6#F*F+%\"rGF+F+F*" }{TEXT -1 6 " onde " }{XPPEDIT 18 0 "a[n]; " "6#&%\"aG6#%\"nG" }{TEXT -1 12 " = /" }{XPPEDIT 18 0 "abs(J (lambda[n]^2*r))^2;" "6#*$-%$absG6#-%\"JG6#*&&%'lambdaG6#%\"nG\"\"#%\" rG\"\"\"F/" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 46 "Neste caso, o produto interno \351 definido por " }{XPPEDIT 18 0 ";" "6#-%$<,>G6$%\"fG6#%\"gG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "int(f(x)*g(x)*x,x = 0 .. a);" "6#-%$intG6$*(-%\"fG6#%\"xG\"\"\"- %\"gG6#F*F+F*F+/F*;\"\"!%\"aG" }{TEXT -1 27 ", com uma norma associada . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 195 "A \+ apresenta\347\343o de um exemplo \351 bastante apropriada. Considerar emos condi\347\365es de contorno nulas, de forma a eliminar o problema de estado estacion\341rio. Al\351m disso, tomaremos a condi\347\343o inicial " }{XPPEDIT 18 0 "1-r^2;" "6#,&\"\"\"F$*$%\"rG\"\"#!\"\"" } {TEXT -1 25 " , que \351 independente de " }{XPPEDIT 18 0 "theta;" "6# %&thetaG" }{TEXT -1 6 ". Com " }{TEXT 275 1 "a" }{TEXT -1 51 " = 1, te m-se uma fun\347\343o cont\355nua no disco unit\341rio." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 49 "O gr\341fico da cond i\347\343o inicial \351 exibido a seguir." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f:=r->1-r^2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"f Gf*6#%\"rG6\"6$%)operatorG%&arrowGF(,&\"\"\"F-*$)9$\"\"#F-!\"\"F(F(F( " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "cylinderplot([r,theta,f (r)],theta=-Pi..Pi,r=0..1,axes=NORMAL, shading=zhue);" }}{PARA 13 "" 1 "" {GLPLOT3D 384 384 384 {PLOTDATA 3 "6&-%%MESHG6#X.%)anythingG6\"6 \"[gl'!%\"!!$^`r\":\":\"$800000000000000080000000000000003FF0000000000 000BFA5555555555555BCA366EC4220D92A3FEFF1C71C71C71CBFB5555555555555BCB 366EC4220D92A3FEFC71C71C71C72BFC0000000000000BCBD1A62633145C03FEF80000 0000000BFC5555555555555BCC366EC4220D92A3FEF1C71C71C71C7BFCAAAAAAAAAAAA ABCC840A752A90F753FEE9C71C71C71C7BFCFFFFFFFFFFFFFBCCD1A62633145BF3FEE0 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Abaixo, s\343o apresentados os detalhes. Em primeiro lugar, obt\351m-se as r a\355zes de R(r) = BesselJ(r). Pode-se criar um procedimento para tal. No gr\341fico, deve-se localizar as dist\342ncias entre as ra\355zes da equa\347\343o. Por fim, deve-se resolver a equa\347\343o para uma raiz em determinado intervalo." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "zero[0]:=0;\nfor n from 1 t o 4 do\n zero[n]:=fsolve(BesselJ(0,x)=0,x,zero[n-1]+1..zero[n-1]+4 );\nend do;\nn:='n';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%zeroG6#\" \"!F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%zeroG6#\"\"\"$\"+eb#[S#! \"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%zeroG6#\"\"#$\"+5\"y+_&!\"* " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%zeroG6#\"\"$$\"+8zs`')!\"*" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%zeroG6#\"\"%$\"+WM:z6!\")" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nGF$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "At\351 aqui, n\343o h\341 nenhuma novidade. Estas ra\355zes s\343o t\343o importantes, que j\341 s\343 o conhecidas pelo Maple." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "for n from 1 to 10 do\n zero[n]: =evalf(BesselJZeros(0,n));\nend do;\nn:='n';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%zeroG6#\"\"\"$\"+eb#[S#!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%zeroG6#\"\"#$\"+5\"y+_&!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%zeroG6#\"\"$$\"+8zs`')!\"*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%zeroG6#\"\"%$\"+WM:z6!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%zeroG6#\"\"&$\"+r<4$\\\"!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%zeroG6#\"\"'$\"+(R1r!=!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%zeroG6#\"\"($\"+jO;@@!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%zeroG6#\"\")$\"+`rCNC!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%zeroG6#\"\"*$\"+8zM\\F!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>&%%zeroG6#\"#5$\"+Z1YjI!\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"nGF$" }}}{PARA 0 "" 0 "" {TEXT -1 87 "Para maiores \+ compara\347\365es, deve-se tra\347ar o gr\341fico das primeiras seis a utofun\347\365es cos( " }{XPPEDIT 18 0 "n*Pi/a;" "6#*(%\"nG\"\"\"%#PiG F%%\"aG!\"\"" }{TEXT -1 49 " x), e das primeiras seis autofun\347\365e s BesselJ(0, " }{XPPEDIT 18 0 "lambda[n]/a;" "6#*&&%'lambdaG6#%\"nG\" \"\"%\"aG!\"\"" }{TEXT -1 4 " x)." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "plot([seq(cos(n*Pi*x),n=0..5)],x=0..1,color=[black,re d,green,blue,yellow, brown]);" }}{PARA 13 "" 1 "" {GLPLOT2D 384 384 384 {PLOTDATA 2 "6*-%'CURVESG6$7S7$$\"\"!F)$\"\"\"F)7$$\"3emmm;arz@!#> F*7$$\"3[LL$e9ui2%F/F*7$$\"3nmmm\"z_\"4iF/F*7$$\"3[mmmT&phN)F/F*7$$\"3 CLLe*=)H\\5!#=F*7$$\"3gmm\"z/3uC\"FY2aF$\"3zW%H&)R3>C*F<7$FA$\"3'eWRhEjn(*)F<7$FD$\"3/S-ig HOj')F<7$FG$\"3qa[DuHo7$)F<7$FJ$\"3IOYbCzr8zF<7$FM$\"3DI7E7bkJvF<7$FP$ \"3!3)fR0k&*oqF<7$FS$\"3#e1XdUEF<7$F ho$\"3QQIC5T(oH\"F<7$F[p$\"33tg,9-*z\"oF/7$F^p$\"3eLEX(yc1b\"!#?7$Fap$ !3M5&yR6lhw'F/7$Fdp$!3=)[n+L'ew7F<7$Fgp$!3m?[LhQH>>F<7$Fjp$!3KK9(*4zpu DF<7$F]q$!3A%e&3+7j/KF<7$F`q$!3+f(zHMV4!QF<7$Fcq$!3=kpLIB*[W%F<7$Ffq$! 3k\"*=)Gi`Y+&F<7$Fiq$!3w*z9?Xs*zbF<7$F\\r$!3My([F&**>zgF<7$F_r$!3'Q_7# 3)[')f'F<7$Fbr$!3mYA2_+DgqF<7$Fer$!3!yihB*Qc7vF<7$Fhr$!3^UduJ13BzF<7$F [s$!3Q?^(3$4R<$)F<7$F^s$!3G\"yxzx17m)F<7$Fas$!3=')3qHUyu*)F<7$Fds$!3+- ]JZ=VY#*F<7$Fgs$!3#=36\\5\"\\g%*F<7$Fjs$!3I*GTok-Hm*F<7$F]t$!3F!*p'y,O V!)*F<7$F`t$!3kLecSjH8**F<7$Fct$!3eSwim42`')F<7$F:$\"3Z#Q-M._U!zF<7$F>$\"3;<=OpTd#3(F <7$FA$\"3R3@EZdX;hF<7$FD$\"3cQ$*[d:x5]F<7$FG$\"3Z3L4Y'R,#QF<7$FJ$\"3[o M)yF'QDDF<7$FM$\"3qYHZCo8X8F<7$FP$!3)3CEAsn5(f!#@7$FS$!3_'zKqN#[i8F<7$ FV$!3C7?/[@=YEF<7$FY$!3Y1rpG*3Jx$F<7$Ffn$!3ro`Dr_JU]F<7$Fin$!3Q/l/u%*[ RgF<7$F\\o$!3iU?fWjJ*3(F<7$F_o$!3gNrc.U?/zF<7$Fbo$!3AZ15QnIf')F<7$Feo$ !3%37d &*****F<7$$\"3@m;/^7I0^F<$!3?\"y#z]/7y**F<7$Fap$!3Sg2I>!Q%3**F<7$$\"3A LLLe,]6`F<$!3\"y)Q8jk24)*F<7$Fdp$!3YkpPoa1u'*F<7$Fgp$!35&\\oZ@iKE*F<7$ Fjp$!37ZxZMh=u')F<7$F]q$!390wGux1YzF<7$F`q$!36r:pSfc5rF<7$Fcq$!3yEBMMk e[gF<7$Ffq$!3cJ]*HU)o!*\\F<7$Fiq$!3k'=^p[\"ysPF<7$F\\r$!3%)\\)**oel'3E F<7$F_r$!3Q'3v6ym:H\"F<7$Fbr$!3Q#owU/%QdIFey7$Fer$\"3kQJOZKs(G\"F<7$Fh r$\"3K-*G%R8/bDF<7$F[s$\"3UR2Sz$)zNQF<7$F^s$\"3G_wvr'))oa'*F<7$$\"3C+++]oi\"o*F<$\" 3a&H_**z'e+)*F<7$Fct$\"3&HGt[`bp!**F<7$$\"35+]P40O\"*)*F<$\"3eSwim$\"3i\"4x7%> R\\QF<7$$\"33LLe*ot*\\8F<$\"397`Df-kSHF<7$FA$\"3\\]&e!)yJW+#F<7$$\"3/L LekGhe:F<$\"3wW;&GsBm,\"F<7$FD$\"3s?kiV#ej'=Fey7$$\"39LL3_(>/x\"F<$!3a !p1*p=#Hw*F/7$FG$!3w$p$=:5ch>F<7$$\"3gm;HdG\"\\)>F<$!3CRD.d&GY&HF<7$FJ $!3^$\\7Q**ym\"RF<7$$\"33++Dc#o%*=#F<$!3PF,/@#>.t%F<7$FM$!3PIuxmnU0bF< 7$FP$!3g'R!))[#)RxqF<7$FS$!3=O)4yo)\\i$)F<7$FV$!3%f\")Htd5HF*F<7$$\"3- L$eky#*4-$F<$!3m&Hj?\\R)p&*F<7$FY$!3?Kbo9Q[!z*F<7$$\"3SL3_]r4sJF<$!3c[ #Qs\\gZ))*F<7$$\"3>+]7.%Q%GKF<$!3g+N;g@<^**F<7$$\"3Um\"HdlzZG$F<$!3]o6 z*eJ&*)**F<7$Ffn$!33YcN^1t****F<7$$\"3+LLeR$\\))Q$F<$!3Mn=\\[[J')**F<7 $$\"3QLL$3xxlV$F<$!3oQGJ`ap_**F<7$$\"3?LL3-iI%[$F<$!3Ed`:#[S*)*)*F<7$F in$!3K???)oe^#)*F<7$$\"3mm;HdO2VOF<$!3=CV7#4@pd*F<7$F\\o$!3Z$)zsDF*QA* F<7$F_o$!3G(oH@p/XN)F<7$Fbo$!3En#4IswI2(F<7$Feo$!31h2]_jA\"e&F<7$Fho$! 3_K%)3QZP.QF<7$Fbcl$!3V3Xv;'Q1$HF<7$F[p$!3g#yvdp>F.#F<7$Fjcl$!3gO(p)4> %[/\"F<7$F^p$!3E;\\\"zWb>l%Fey7$Fbdl$\"3./B%)fb73**F/7$Fap$\"3Tv?gk!fu ,#F<7$Fjdl$\"3-fU&G'z#Q*GF<7$Fdp$\"39x!*owAaYPF<7$Fgp$\"3)e\"GGA$y]Z&F <7$Fjp$\"3XS+?#pz8/(F<7$F]q$\"37=4eTXZ(H)F<7$F`q$\"3a1yR)*\\J1#*F<7$$ \"3;**\\P4kh`jF<$\"31:`()Qm*yc*F<7$Fcq$\"3WYYh?a&>#)*F<7$$\"3\\K$eRP0n ^'F<$\"3xB@UszG+**F<7$$\"3)em;/risc'F<$\"3[()R7On9c**F<7$$\"3D**\\(o/? 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Neste caso, mostraremos que a solu \347\343o da forma" }}{PARA 0 "" 0 "" {TEXT -1 15 " u(t, r) = " } {XPPEDIT 18 0 "sum(A[n]*BesselJ(0,zero[n]*r)*exp(-zero[n]^2*t),n = 1 . . infinity);" "6#-%$sumG6$*(&%\"AG6#%\"nG\"\"\"-%(BesselJG6$\"\"!*&&%% zeroG6#F*F+%\"rGF+F+-%$expG6#,$*&&F26#F*\"\"#%\"tGF+!\"\"F+/F*;F+%)inf inityG" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 87 "\351 uma solu \347\343o da equa\347\343o para 0 < r < 1, e t > 0. Se o intervalo for 0 < r < a, tem-se" }}{PARA 0 "" 0 "" {TEXT -1 14 " u(t, r) = " } {XPPEDIT 18 0 "sum(A[n]*BesselJ(0, zero[n]*r/a)*exp(-zero[n]^2*t/(a^2) ),n = 1 .. infinity);" "6#-%$sumG6$*(&%\"AG6#%\"nG\"\"\"-%(BesselJG6$ \"\"!*(&%%zeroG6#F*F+%\"rGF+%\"aG!\"\"F+-%$expG6#,$*(&F26#F*\"\"#%\"tG F+*$F5F>F6F6F+/F*;F+%)infinityG" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 95 "Considera-se a soma com a penas tr\352s termos na verifica\347\343o abaixo. 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