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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" 
{TEXT 256 48 "MAT1154 - Equa\347\365es Diferenciais e de Diferen\347as
 " }}{PARA 256 "" 0 "" {TEXT 257 78 "No\347\365es Elementares de Utili
za\347\343o do MAPLE para Estudo de Equa\347\365es Diferenciais" }}
{PARA 256 "" 0 "" {TEXT -1 3 "por" }}{PARA 256 "" 0 "" {TEXT -1 34 "Ma
rco Antonio Grivet Mattoso Maia\n" }}{PARA 257 "" 0 "" {TEXT 263 57 "
\005\005\005M\311TODOS NUM\311RICOS DE SOLU\307\303O DE ODE DE \010PRI
MEIRA ORDEM" }}{PARA 258 "" 0 "" {TEXT 264 40 "Baseado nos scripts do \+
site da Maplesoft" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 8 "restart;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "with( DEtools ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with( plots
 ):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with( linalg ):" }}}{SECT 0 
{PARA 3 "" 0 "" {TEXT -1 18 "  M\351todo de Euler " }}{PARA 0 "" 0 "" 
{TEXT -1 65 "Considere o seguinte problema de valor inicial de primeir
a ordem:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff( y, \+
x ) = F(x,y(x))" "6#/-%%DiffG6$%\"yG%\"xG-%\"FG6$F(-F'6#F(" }{TEXT -1 
6 ",     " }{XPPEDIT 18 0 "y(x[0]) = y[0]" "6#/-%\"yG6#&%\"xG6#\"\"!&F
%6#F*" }}{PARA 0 "" 0 "" {TEXT -1 74 "O m\351todo de Euler para obten
\347\343o de sua solu\347\343o pode ser sumarizado abaixo:" }}{PARA 
259 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "x[n+1] = x[n] + h" "6#/&%
\"xG6#,&%\"nG\"\"\"F)F),&&F%6#F(F)%\"hGF)" }{TEXT -1 19 "             \+
      " }}{PARA 260 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "y[n+1] = y
[h] + h*F(x[n],y[n])" "6#/&%\"yG6#,&%\"nG\"\"\"F)F),&&F%6#%\"hGF)*&F-F
)-%\"FG6$&%\"xG6#F(&F%6#F(F)F)" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" 
{TEXT -1 17 "onde h \351 o passo." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 71 "Uma implementa\347\343o simples deste m
\351todo pode ser feito por meio de uma  " }{TEXT 266 9 "procedure" }
{TEXT -1 35 " que tem como argumentos de entrada" }}{PARA 0 "" 0 "" 
{TEXT -1 30 "a fun\347\343o F, o momento inicial " }{XPPEDIT 18 0 "x[0
]" "6#&%\"xG6#\"\"!" }{TEXT -1 12 ", o valor   " }{XPPEDIT 18 0 "y[0]
" "6#&%\"yG6#\"\"!" }{TEXT -1 34 " da solu\347\343o neste ponto, o pas
so  " }{XPPEDIT 18 0 "h" "6#%\"hG" }{TEXT -1 41 ",e o no. de de passos
 para a evolu\347\343o do " }}{PARA 0 "" 0 "" {TEXT -1 7 "m\351todo." 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 32 "Euler := proc( F, x0, y0, h, N )" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 16 "  local i, L, X;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "  X := \+
evalf( [ x0, y0 ] );" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "  L := X;" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "  for i from 1 to N do" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 31 "    X := X + [ h, h*F(op(X)) ];" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 14 "    L := L, X;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 9 "  end do;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "  retu
rn L;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end proc;" }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "Como exemplo, vamos ca
lcular pelo M\351todo de Euler no intervalo  [ 0, 1 ] com  " }
{XPPEDIT 18 0 "h" "6#%\"hG" }{TEXT -1 23 "=0.1 a a fun\347\343o abaixo
:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "func:=(x,y)->sin(3*x^2)-y;" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "a := Euler( func, \+
0, 1, 0.05, 20 );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 20 "Agora para um passo " }{TEXT 267 1 "h" }{TEXT -1 6 "=0.2 \+
:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "b := Euler( func, 0, 1, 0.2, 5 );" }}}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Finalmente para um pass
o " }{TEXT 268 1 "h" }{TEXT -1 7 "=0.25 :" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "c := Euler( func, 0, \+
1, 0.25, 4 );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 47 "O gr\341fico abaixo ilustra os resultados obtidos:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
44 "plot([[a],[b],[c]],colour=[red,green,blue]);" }}}}{SECT 0 {PARA 3 
"" 0 "" {TEXT -1 16 "  M\351todo de Heun" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 68 "Considere o mesmo problema de valor inicial acima descrit
o, ou seja:" }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "Diff( \+
y, x ) = F(x,y(x))" "6#/-%%DiffG6$%\"yG%\"xG-%\"FG6$F(-F'6#F(" }{TEXT 
-1 6 ",     " }{XPPEDIT 18 0 "y(x[0]) = y[0]" "6#/-%\"yG6#&%\"xG6#\"\"
!&F%6#F*" }}{PARA 0 "" 0 "" {TEXT -1 98 "O m\351todo de Heun (ou M\351
todo de Euler Melhorado ou M\351todo dos Trap\351zios) \351 descrito p
elas equa\347\365es:" }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 
18 0 "x[n+1] = x[n] + h" "6#/&%\"xG6#,&%\"nG\"\"\"F)F),&&F%6#F(F)%\"hG
F)" }{TEXT -1 54 "                                                    \+
  " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "y[n+1]^`*` = y
[n] + h*F(x[n],y[n])" "6#/)&%\"yG6#,&%\"nG\"\"\"F*F*%\"*G,&&F&6#F)F**&
%\"hGF*-%\"FG6$&%\"xG6#F)&F&6#F)F*F*" }{TEXT -1 41 "                  \+
                       " }}{PARA 256 "" 0 "" {TEXT -1 2 "  " }
{XPPEDIT 18 0 "y[n+1] = y[n] + h/2" "6#/&%\"yG6#,&%\"nG\"\"\"F)F),&&F%
6#F(F)*&%\"hGF)\"\"#!\"\"F)" }{TEXT -1 3 " ( " }{XPPEDIT 18 0 "F(x[n],
y[n] ) + F(x[n+1],y[n+1]^`*`)" "6#,&-%\"FG6$&%\"xG6#%\"nG&%\"yG6#F*\"
\"\"-F%6$&F(6#,&F*F.F.F.)&F,6#,&F*F.F.F.%\"*GF." }{TEXT -1 4 " )  " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "Sua imple
mental\347\343o \351 mostrada abaixo:" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "Heun := proc( F, x0, y0
, h, N )" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "  local i, L, X, X1, X2
;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "  X := evalf( [ x0, y0 ] );" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "  L := X;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "  for i from 1 to N do" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 32 "    X1 := X + [ h, h*F(op(X)) ];" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 33 "    X2 := X + [ h, h*F(op(X1)) ];" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 19 "    X := (X1+X2)/2;" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 14 "    L := L, X;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "  end do;
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "  return L;" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 9 "end proc;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Como exemplo, vamos calcul
ar pelo M\351todo de Heun no intervalo  [ 0, 1 ] com  " }{XPPEDIT 18 
0 "h" "6#%\"hG" }{TEXT -1 23 "=0.1 a a fun\347\343o abaixo:" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "a
h:= Heun( func, 0, 1, 0.05, 20 );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 20 "Agora para um passo " }{TEXT 269 1 "h" }
{TEXT -1 6 "=0.2 :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 32 "bh:= Heun( func, 0, 1, 0.2, 5 );" }}}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 25 "Finalmente para
 um passo " }{TEXT 270 1 "h" }{TEXT -1 7 "=0.25 :" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "ch:= Heun( f
unc, 0, 1, 0.25, 4 );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 47 "O gr\341fico abaixo ilustra os resultados obtidos:" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 47 "plot([[ah],[bh],[ch]],colour=[red,green,blue]);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT 
-1 23 "  M\351todo de Runge-Kutta" }}{PARA 0 "" 0 "" {TEXT -1 50 "O m
\351todo de Runge-Kutta \351 descrito pelas equa\347\365es:" }}{PARA 
262 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "x[n+1] = x[n]+h;" "6#/&%\"
xG6#,&%\"nG\"\"\"F)F),&&F%6#F(F)%\"hGF)" }{TEXT -1 55 "               \+
                                        " }}{PARA 263 "" 0 "" {TEXT 
-1 1 " " }{XPPEDIT 18 0 "y[n+1] = y[n]+(k[1]+2*k[2]+2*k[3]+k[4])/6;" "
6#/&%\"yG6#,&%\"nG\"\"\"F)F),&&F%6#F(F)*&,*&%\"kG6#F)F)*&\"\"#F)&F06#F
3F)F)*&F3F)&F06#\"\"$F)F)&F06#\"\"%F)F)\"\"'!\"\"F)" }{TEXT -1 1 " " }
{XPPEDIT 18 0 "h" "6#%\"hG" }{TEXT -1 23 "                       " }
{XPPEDIT 18 0 "k[1] = F(x[n],y(n));" "6#/&%\"kG6#\"\"\"-%\"FG6$&%\"xG6
#%\"nG-%\"yG6#F." }{TEXT -1 8 "        " }}}{PARA 264 "" 0 "" 
{XPPEDIT 18 0 "k[2] = F(x[n]+h/2,y[n]+h*k[1]/2);" "6#/&%\"kG6#\"\"#-%
\"FG6$,&&%\"xG6#%\"nG\"\"\"*&%\"hGF0F'!\"\"F0,&&%\"yG6#F/F0*&*&F2F0&F%
6#F0F0F0F'F3F0" }{TEXT -1 2 "  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 264 "" 0 "" {XPPEDIT 18 0 "k[3] = F(x[n]+h/2,y[n]+h*k[2]/2);" "6
#/&%\"kG6#\"\"$-%\"FG6$,&&%\"xG6#%\"nG\"\"\"*&%\"hGF0\"\"#!\"\"F0,&&%
\"yG6#F/F0*(F2F0&F%6#F3F0F3F4F0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 264 "" 0 "" {XPPEDIT 18 0 "k[3] = F(x[n]+h,y[n]+h*k[3]);" "6#/&%
\"kG6#\"\"$-%\"FG6$,&&%\"xG6#%\"nG\"\"\"%\"hGF0,&&%\"yG6#F/F0*&F1F0&F%
6#F'F0F0" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" 
{TEXT -1 21 "M\351todo de Runge-Kutta" }}{PARA 0 "" 0 "" {TEXT -1 50 "
O m\351todo de Runge-Kutta \351 descrito pelas equa\347\365es:" }}
{PARA 256 "" 0 "" {TEXT -1 2 "  " }{XPPEDIT 18 0 "x[n+1] = x[n]+h;" "6
#/&%\"xG6#,&%\"nG\"\"\"F)F),&&F%6#F(F)%\"hGF)" }{TEXT -1 55 "         \+
                                              " }}{PARA 256 "" 0 "" 
{TEXT -1 1 " " }{XPPEDIT 18 0 "y[n+1] = y[n]+(k[1]+2*k[2]+2*k[3]+k[4])
/6;" "6#/&%\"yG6#,&%\"nG\"\"\"F)F),&&F%6#F(F)*&,*&%\"kG6#F)F)*&\"\"#F)
&F06#F3F)F)*&F3F)&F06#\"\"$F)F)&F06#\"\"%F)F)\"\"'!\"\"F)" }{TEXT -1 
1 " " }{XPPEDIT 18 0 "h" "6#%\"hG" }{TEXT -1 22 "                     \+
 " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "k[1] = F(x[n],y(
n));" "6#/&%\"kG6#\"\"\"-%\"FG6$&%\"xG6#%\"nG-%\"yG6#F." }{TEXT -1 20 
"                    " }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "k[2] = F(x[n
]+h/2,y[n]+h*k[1]/2);" "6#/&%\"kG6#\"\"#-%\"FG6$,&&%\"xG6#%\"nG\"\"\"*
&%\"hGF0F'!\"\"F0,&&%\"yG6#F/F0*(F2F0&F%6#F0F0F'F3F0" }{TEXT -1 2 "  \+
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "k
[3] = F(x[n]+h/2,y[n]+h*k[2]/2);" "6#/&%\"kG6#\"\"$-%\"FG6$,&&%\"xG6#%
\"nG\"\"\"*&%\"hGF0\"\"#!\"\"F0,&&%\"yG6#F/F0*(F2F0&F%6#F3F0F3F4F0" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {XPPEDIT 18 0 "k[4] \+
= F(x[n]+h,y[n]+h*k[3]);" "6#/&%\"kG6#\"\"%-%\"FG6$,&&%\"xG6#%\"nG\"\"
\"%\"hGF0,&&%\"yG6#F/F0*&F1F0&F%6#\"\"$F0F0" }}{PARA 3 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 
"Sua implemental\347\343o \351 mostrada abaixo:" }}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 38 "RungeKutta := proc ( F, x0, y0, h, N )" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "  local f, i, L, X, k1, k2, k3, k4;
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "  X := evalf( [ x0, y0 ] );" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "  L := X;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "  for i from 1 to N do" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 19 "    k1 := F(op(X));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "    \+
k2 := F(op(X + [ h/2, h/2*k1 ]));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
37 "    k3 := F(op(X + [ h/2, h/2*k2 ]));" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 37 "    k4 := F(op(X + [ h  , h  *k3 ]));" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 41 "    X := X+ [ h ,h*(k1+2*k2+2*k3+k4)/6 ];" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "    L := L, X;" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 9 "  end do;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "  re
turn L;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "end proc;" }}}{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 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 "Como exemplo, vamo
s calcular pelo M\351todo de Heun no intervalo  [ 0, 1 ] com  " }
{XPPEDIT 18 0 "h" "6#%\"hG" }{TEXT -1 23 "=0.1 a a fun\347\343o abaixo
:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "ar:= RungeKutta( func, 0, 1, 0.05, 20 );" }}}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 20 "Agora para um pas
so " }{TEXT 271 1 "h" }{TEXT -1 6 "=0.2 :" }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "br:= RungeKutta( func
, 0, 1, 0.2, 5 );" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 25 "Finalmente para um passo " }{TEXT 272 1 "h" }{TEXT -1 7 "
=0.25 :" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 39 "cr:= RungeKutta( func, 0, 1, 0.25, 4 );" }}}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 47 "O gr\341fico abaix
o ilustra os resultados obtidos:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "plot([[ar],[br],[cr]],colour
=[red,green,blue]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}}{MARK "0 5 0" 57 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }
{PAGENUMBERS 0 1 2 33 1 1 }