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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 26 "Korteweg-d
e Vries egyenlet" }}{PARA 257 "" 0 "" {TEXT 257 13 "Bartha Ferenc" }}
{PARA 258 "" 0 "" {TEXT -1 17 "2002. m\341rcius 27." }}{PARA 0 "" 0 "
" {TEXT -1 40 "A Koteweg - de Vries 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.\"\"$F*\"\"!" }{TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 39 "     Soliton megol\341sai:     \+
           " }{XPPEDIT 18 0 "u(x,t)=12*alpha^2*sech^2*[alpha*(x-4*alph
a^2*t)]:" "6#/-%\"uG6$%\"xG%\"tG**\"#7\"\"\"*$%&alphaG\"\"#F+%%sechGF.
7#*&F-F+,&F'F+*(\"\"%F+*$F-F.F+F(F+!\"\"F+F+" }{TEXT -1 0 "" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "restart:with(plots):\nu:=(x,t) -> 1
2*alpha^2*sech(alpha*(x-4*alpha^2*t))^2:" }}}{EXCHG {PARA 0 "" 0 "" 
{MPLTEXT 0 21 69 "Vizsg\341ljuk ennek a mozg\341s\341t analitikusan \+
\351s numerikusan!\nParam\351terek:" }{XPPEDIT 19 1 "alpha:=2.0:xL:=-5
.0:xR:=5.0:" "6#C%>%&alphaG-%&FloatG6$\"#?!\"\">%#xLG,$-F'6$\"#]F*F*>%
#xRG-F'6$F0F*" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "plot(u(x,0),x=xL..
xR,title=`U(x,0)`);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 274 "Figyelj
\374k meg, hogy az al\341bb adott beoszt\341sban mekkora k\374l\366nbs
\351g van az id\366 \351s a t\351r k\366z\366tt. Pr\363b\341ljuk majd \+
ki, hogy finomitjuk az x tengelyen a beoszt\341st, \351s nem v\341lasz
tunk kell\366en finom beoszt\341st az id\366ben. Ekkor a megold\341sun
k 'elsz\341ll'... INSTABILIT\301S!!! \nGrid megad\341sa: " }{MPLTEXT 
1 0 21 "dt:=0.0003: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:=c
eil(T/dt);Nj:= ceil((xR-xL)/dx);N2:=Nj+2:" }}}{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)+di
ff((U^2/2),x)+diff(U,`$`(x,3)) = 0;" "6#/,(-%%diffG6$%\"UG%\"tG\"\"\"-
F&6$*&F(\"\"#F.!\"\"%\"xGF*-F&6$F(-%\"$G6$F0\"\"$F*\"\"!" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "0 = U[j]^[n+1]-U[j]^[n-1]\n+c1*((
U^2)[j+1]^[n]-(U^2)[j-1]^[n])\n+c2*(U[j+2]^[n]-2*U[j+1]^[n]+2*U[j-1]^[
n]-U[j-2]^[n]);" "6#/\"\"!,*)&%\"UG6#%\"jG7#,&%\"nG\"\"\"F.F.F.)&F(6#F
*7#,&F-F.F.!\"\"F4*&%#c1GF.,&)&*$F(\"\"#6#,&F*F.F.F.7#F-F.)&*$F(F;6#,&
F*F.F.F47#F-F4F.F.*&%#c2GF.,*)&F(6#,&F*F.F;F.7#F-F.*&F;F.)&F(6#,&F*F.F
.F.7#F-F.F4*&F;F.)&F(6#,&F*F.F.F47#F-F.F.)&F(6#,&F*F.F;F47#F-F4F.F." }
}{PARA 259 "> " 0 "" {MPLTEXT 1 0 29 "c1 := dt/2/dx;c2 := dt/dx^3;\n" 
}{TEXT -1 64 "a megold\341s stabilit\341s\341hoz sz\374ks\351ges, hogy
 c2 kicsi maradjon....." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "Els
\365 l\351p\351s: FTCS" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "for j fro
m 1 by 1 to Nj do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "     U1[j]:= \+
U0[j]-c1/2*(U0[j+1]^2-U0[j-1]^2)\n                  -c2/2*(U0[j+2]-2.*
U0[j+1]+2.*U0[j-1]-U0[j-2]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "od:
\nt:=t+dt:\n" }{TEXT -1 26 "Ciklikus hat\341rfelt\351tel...." }
{MPLTEXT 1 0 65 "\nU1[-1]:=U1[Nj-1]:U1[0]:=U1[Nj]:U1[Nj+1]:=U1[1]:U1[N
j+2]:=U1[2]:\n" }{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[N2],0..12*alpha^2]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 29 "Anim\341c\363hoz a frame-ek sz\341ma: " }{MPLTEXT 1 0 29 "Nf:=3
0:\nNg:=ceil(Nn/Nf):m:=0:" }}{PARA 0 "" 0 "" {TEXT -1 9 "Leap frog" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }{TEXT -1 0 "" }{MPLTEXT 1 0 35 "f
or 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]-U1[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 N2 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 19 "    U0[j] := U1[j]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "    U1
[j] := U2[j]:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "  od:\nif((n mod N
g)=0 or n=Nn) then\n  m:=m+1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 113 " \+
 nr[m] := listplot([seq([x[j],U2[j]],j=1..Nj)]):\n  ar[m] := listplot(
[seq([x[j],u(x[j],t)],j=1..Nj)]):\n  end if:" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "od:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 130 "Kattitnts
 a grafikonra az anim\341ci\363\351rt!\n(nem siker\374lt megoldani, ho
gy az analitikus \351s a numerikus megold\341s k\374l\366nb\366z\366 s
zin\374 legyen)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "nusol:=display(
seq(nr[n],n=1..m),insequence=true,color=green):\nansol:=display(seq(ar
[n],n=1..m),insequence=true,color=black):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "display([ansol,nusol]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
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