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{SECT 0 {PARA 0 "" 0 "" {TEXT -1 3 "   " }}{PARA 256 "" 0 "" {TEXT 
258 0 "" }{TEXT 256 0 "" }{TEXT 259 0 "" }{MPLTEXT 0 21 34 "Korteweg -
 de Vries\n (KdV)egyenlet" }}{PARA 257 "" 0 "" {TEXT 257 13 "Bartha Fe
renc" }}{PARA 258 "" 0 "" {TEXT -1 17 "2002. m\341rcius 27." }}{PARA 
260 "" 0 "" {TEXT -1 24 "rev\355zi\363: 2010. m\341jus 21." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "A Koteweg - de Vr
ies egyenlet:          " }{XPPEDIT 18 0 "diff(u,t)+u*diff(u,x)+diff(u,
`$`(x,3)) = 0;" "6#/,(-%%diffG6$%\"uG%\"tG\"\"\"*&F(F*-F&6$F(%\"xGF*F*
-F&6$F(-%\"$G6$F.\"\"$F*\"\"!" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 
-1 39 "     Soliton megol\341sai:                " }{XPPEDIT 18 0 "u(x
,t)=12*alpha^2*sech^2*[alpha*(x-4*alpha^2*t)]:" "6#/-%\"uG6$%\"xG%\"tG
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$-%\"uG6$%\"xG%\"tGF-\"\"\"*&-F*6$F,F-F.-F'6$-F*6$F,F-F,F.F.-F'6$-F*6$
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\373s\355tj\374k" }{MPLTEXT 1 0 19 "\nKdV=simplify(KdV);" }}{PARA 12 "
" 1 "" {XPPMATH 20 "6#/,**()%&alphaG\"\"&\"\"\")-%%sechG6#*&F'F),&%\"x
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"> " 0 "" {MPLTEXT 1 0 37 "plot(u(x,0),x=xL..xR,title=`U(x,0)`);" }}
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%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "C
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y az al\341bb adott beoszt\341sban mekkora k\374l\366nbs\351g van az i
d\366 \351s a t\351r k\366z\366tt. Pr\363b\341ljuk majd ki, hogy finom
itjuk az x tengelyen a beoszt\341st, \351s nem v\341lasztunk kell\366e
n finom beoszt\341st az id\366ben. Ekkor a megold\341sunk 'elsz\341ll'
... INSTABILIT\301S!!! \nGrid megad\341sa: " }{MPLTEXT 1 0 21 "dt:=0.0
003:dx:=0.15:\n" }{TEXT -1 18 "Szimul\341ci\363 ideje: " }{MPLTEXT 1 
0 7 "T:=1/5:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Nn:=ceil(T/dt);Nj:=
 ceil((xR-xL)/dx);N2:=Nj+2:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NnG
\"$n'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#NjG\"#n" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 26 "Kezd\365felt\351tel \341tad\341sa    \n" }
{MPLTEXT 1 0 85 "t:=0:\nfor j from -1 by 1 to N2 do\n  x[j] := xL+j*dx
:\n  U0[j] := evalf(u(x[j],t)):\nod:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 11 " Leap Frog " }}{PARA 0 "" 0 "" {TEXT -1 3 "   " }{XPPEDIT 18 0 
"diff(U,t)+diff((U^2/2),x)+diff(U,`$`(x,3)) = 0;" "6#/,(-%%diffG6$%\"U
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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "0 = U[j]^[n+1]-U[j]^
[n-1]\n+c1*((U^2)[j+1]^[n]-(U^2)[j-1]^[n])\n+c2*(U[j+2]^[n]-2*U[j+1]^[
n]+2*U[j-1]^[n]-U[j-2]^[n]);" "6#/\"\"!,*)&%\"UG6#%\"jG7#,&%\"nG\"\"\"
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\351ges, hogy c2 kicsi maradjon....." }}{PARA 11 "" 1 "" {XPPMATH 20 "
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{TEXT -1 26 "Ciklikus hat\341rfelt\351tel...." }{MPLTEXT 1 0 65 "\nU1[
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{TEXT -1 71 "\351s jobb lesz ellen\366rizni, hogy az els\366 l\351p
\351s nem lett-e nagyon durva..." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
{MPLTEXT 1 0 73 "listplot([seq([x[j],U1[j]],j=-1..N2)],view=[x[-1]..x[
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{PARA 0 "" 0 "" {TEXT -1 9 "Leap frog" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }{TEXT -1 0 "" }{MPLTEXT 1 0 35 "for n from 1 by 1 to Nn do\nt:
=t+dt:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "  for j from 1 by 1 to Nj
 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 171 "   U2[j] := U0[j]-c1*(U1[j+
1]^2-U1[j-1]^2)\n                 -c2*(U1[j+2]-2.*U1[j+1]+2.*U1[j-1]-U
1[j-2])\nod:\nU2[-1]:=U2[Nj-1]:U2[0]:=U2[Nj]:U2[Nj+1]:=U2[1]:U2[Nj+2]:
=U2[2]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "  for j from -1 by 1 to \+
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{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "    U1[j] := U2[j]:" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 45 "  od:\nif((n mod Ng)=0 or n=Nn) then\n  m:=m+1
:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 218 "  nr[m] := listplot([seq([x[j
],U2[j]],j=1..Nj)],color=blue,style=point):\n  ar[m] := listplot([seq(
[x[j],u(x[j],t)],j=1..Nj)],color=green):\n  dr[m] := listplot([seq([x[
j],u(x[j],t)-U2[j]],j=1..Nj)],color=red):\n  end if:" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 93 "Hason
l\355tsuk \366ssze a numerikus \351s az egzakt megold\341st!\nKattitnt
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isplay(seq(nr[n],n=1..m),insequence=true):\nansol:=display(seq(ar[n],n
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