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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 51 "Az FTCS m\363dszer vizsg
\341lata egy diff\372zi\363s probl\351m\341ra" }}{PARA 19 "" 0 "" 
{TEXT -1 44 "Bartha Ferenc ir\341ny\355t\341s\341val k\366z\366s \363r
ai munka" }}{PARA 256 "" 0 "" {TEXT -1 30 "SZTE, Elm. Fiz., 2003. \341
prilis" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "f:=t->1
/(1+4*a*t):\ng:=(x,t,b)->exp(-a*(x-b)^2*f(t))*sqrt(f(t)):" }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 16 "print(g(x,t,b));" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 31 "Al\341bb demonstr\341ljuk, hogy ez a " }{TEXT 258 6 "g(x,
t)" }{TEXT -1 17 " f\374ggv\351ny minden " }{TEXT 256 1 "a" }{TEXT -1 
4 " \351s " }{TEXT 257 1 "b" }{TEXT -1 50 "  param\351ter mellett diff
\372zi\363s egyenlet megold\341sa: " }{XPPEDIT 18 0 "diff(g(x,t),t)-di
ff(g(x,t),x$2)=0" "6#/,&-%%diffG6$-%\"gG6$%\"xG%\"tGF,\"\"\"-F&6$-F)6$
F+F,-%\"$G6$F+\"\"#!\"\"\"\"!" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "si
mplify(diff(g(x,t,b),t)-diff(g(x,t,b),x$2));" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 127 "H\341rom k\374l\366nb\366z\365 helyen centr\341lt ilye
n megold\341s line\341ris kombin\341ci\363ja is megold\341s, hiszen a \+
diff.egyenlet line\341ris. Al\341bb olyan " }{TEXT 260 6 "v(x,t)" }
{TEXT -1 102 " megold\341st k\351sz\355t\374nk, hogy az j\363 k\366zel
\355t\351ssel (a vizsg\341lni k\355v\341nt id\365tartam allatt v\351gi
g) teljes\355ti a " }{TEXT 259 17 "v(-1,t)=v(+1,t)=0" }{TEXT -1 16 " p
eremfelt\351telt." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "a:=20:v:=(x,t)
->g(x,t,0)-g(x,t,-2)-g(x,t,2);\nplot(v(x,1/100),x=-1..1);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 146 "\311s most j\366het a numerikus megold
\341s el\365\341ll\355t\341sa. \nBeoszt\341s, l\351p\351shossz, l\351p
\351ssz\341m:  Ezeket vari\341ljuk a tanulm\341nyoz\341s sor\341n a st
abilit\341st figyelve! " }{TEXT 263 4 "n=25" }{TEXT -1 4 " \351s " }
{TEXT 264 8 "dt=0.001" }{TEXT -1 41 " mellett t\373rhet\365 az eredm
\351ny, de p\351ld\341ul " }{TEXT 265 4 "n>51" }{TEXT -1 16 " \351s v
\341ltozatlan " }{TEXT 266 2 "dt" }{TEXT -1 17 " m\341r katasztr\363fa
." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "n:=25;dx:=2./n;dt:=1/1000.;m:=
50;" }}{PARA 0 "" 0 "" {TEXT -1 16 "Kezdzeti \351rt\351k: " }{TEXT 
261 3 "t=0" }{MPLTEXT 1 0 43 "\nfor j from 0 to n do\n s[j]:=v(-1+j*dx
,0):\n" }{TEXT -1 0 "" }{MPLTEXT 1 0 4 "od:\n" }{TEXT -1 21 "FTCS form
ula alapj\341n " }{TEXT 262 4 "m*dt" }{TEXT -1 23 " l\351p\351s el\365
re az id\365ben:" }{MPLTEXT 1 0 147 "\nalpha:=dt/dx/dx:\nfor i from 1 \+
to m do\n  tmp:=0:\n  for j from 1 to n-1 do\n  nw:=s[j]+alpha*(tmp-2*
s[j]+s[j+1]):\n  tmp:=s[j]: \n  s[j]:=nw:\n  od:\nod:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "T:=dt*m:print(`A numerikus megold
\341s`,T,`id\365n\351l tart`);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "p
l1:=plot(v(x,T),x=-1..1):" }{TEXT -1 25 "Ez az analitikus megold\341s
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "pl2:=plot([seq([-1+k*dx,s[k]],k
=0..n)]):" }{TEXT -1 20 "ez pedig a numerikus" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "with(plots):display(pl1,pl2);" }{TEXT -1 21 "Hasonl
\355tsuk \366ssze...." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
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