teorie

+:::+anx
n
i ¡yi)1 = 0P
k
i=1(a0 +a1xi +:::+anx
n
i ¡yi)xi = 0
... ... ...
Pk
i=1(a0 +a1xi +:::+anx
n
i ¡yi)x
n
i = 0
4. Soustava norm¶aln¶‡ch rovnic V¶y•seuveden¶e podm¶‡nky tvo•r¶‡ soustavu n+1 line¶arn¶‡ch algebraick¶ych
rovnic pro n+1 nezn¶am¶ych koeflcient”u a0;a1;:::;an.
a0
kX
i=1
1 +a1
kX
i=1
xi + ::: +an
kX
i=1
xni =
kX
i=1
yi
a0
kX
i=1
xi +a1
kX
i=1
x2i +:::+an
kX
i=1
xn+1i =
kX
i=1
xiyi
... ... ...
a0
kX
i=1
xni +a1
kX
i=1
xn+1i +:::+an
kX
i=1
x2ni =
kX
i=1
xni yi
5. Z¶apis polynomu nejlep•s¶‡ aproximace
p⁄n(x) = a⁄0 +a⁄1x1 +:::+a⁄nxn
6. V¶ypo•cet kvadratick¶e odchylky
–2(p⁄n(x)) =Pki=1(a⁄0 +a⁄1xi +:::+a⁄nxni ¡yi)2
Preliminary version { June 1, 2001 { 11:17
7
Deflnice: Cauchyova ¶uloha pro ODR 1.•r¶adu
Nalezn•ete funkci y = y(x) pro kterou plat¶‡
y0 = f(x;y) x 2ha;bi‰R
(y :R7¡!R, f :R£R7¡!R) a pro kterou je spln•ena po•c¶ate•cn¶‡ podm¶‡nka y(a) = 0y.
Deflnice: Cauchyova ¶uloha pro syst¶em ODR 1.•r¶adu
Nalezn•ete funkci ~y = ~y(x) pro kterou plat¶‡
~y0 = ~f(x;~y) x 2ha;bi‰R
(~y :R7¡!Rm, ~f :R£Rm 7¡!Rm) a pro kterou je spln•ena po•c¶ate•cn¶‡ podm¶‡nka ~y(a) = 0~y.
Deflnice: Cauchyova ¶uloha pro ODR m.•r¶adu
Nalezn•ete funkci y = y(x) pro kterou plat¶‡
y(m) = g(x;y;y0;:::;y(m¡1)) x 2ha;bi‰R
(y : R 7¡! R, g : R£Rm 7¡! R) a jsou spln•eny po•c¶ate•cn¶‡ podm¶‡nky y(a) = 0y, y0(a) = 0y0, :::,
y(m¡1)(a) = 0y(m¡1).
Pozn¶amka: P•revod ODR m.•r¶adu na soustavu ODR 1.•r¶adu
M¶am: y(m) = g(x;y ;y0 ;:::;y(m¡1)) y(a) = 0y , y0(a) = 0y0, :::, y(m¡1)(a) = 0y(m¡1)
Chci : ~y0 = ~f(x;y1;y2;:::;ym) y1(a) = 0y1, y2(a) = 0y2, :::, ym(a) = 0ym
Zavedu substituci:
y1 = y
y2 = y0
...
ym = y(m¡1)
9
>>
=
>>
;
=)
8
>>
<
>>
:
y01 = y2
y02 = y3
...
y0m = g(x;y1;y2;:::;ym)
9
>>
=
>>
;
=)
8
>>
<
>>
:
0y1(a) = 0y
0y2(a) = 0y0
...
0ym(a) = 0y(m¡1)
V•eta: Existence a jednozna•cnost •re•sen¶‡
Necht’ je d¶ana sostava m diferenci¶aln¶‡ch rovnic v norm¶aln¶‡m tvaru ~y 0 = ~f(x;~y) a necht’ funkce fi(x;~y),
@fi
@yj ; i;j = 1;2;:::;m jsou spojit¶e v oblasti ›‰R£R
m. Potom pro ka•zd¶y bod [x0;~y0]2› existuje
pr¶av•e jedno maxim¶aln¶‡ •re•sen¶‡ dan¶e soustavy, spl•nuj¶‡c¶‡ po•c¶ate•cn¶‡ podm¶‡nku ~y(x0) = ~y0. Toto •re•sen¶‡ je
deflnov¶ano v intervalu I takov¶em, •ze x0 2 I a pro ka•zd¶e x 2 I levz¶‡ bod [x;~y(x)] v oblasti ›.
Deflnice: Obecn¶a jednokrokov¶a metoda
yi+1 = yi +h`(xi;yi;h) y(x0) = y0
P•r¶‡r”ustkov¶a funkce ` mus¶‡ spl•novat podm¶‡nku konzistence `(x;y;0) = f(x;y).
V•eta: Konvergence obecn¶e jednokrokov¶e metody
Necht’ jsou spln•eny posta•cuj¶‡c¶‡ podm¶‡nky existence ajednozna•cnosti •re•sen¶‡ Cauchyovy ¶ulohy
y0 = f(x;y); y(x0) = y0 a funkce ` a @`@y jsou spojit¶e na mno•zin•e
D ·
n
[x;y;h] : x 2hx0;bi;y 2R;0• h • h0;h0 > 0
o
Potom
Obecn¶a jednokrokov¶a metoda je konvergentn¶‡ () `(x;y;0) = f(x;y) 8[x;y]2fhx0;bi£Rg
Preliminary version { June 1, 2001 { 11:17
8
Obecn¶a jednokrokov¶a metoda
x x
1
y
y(x)
00y(x )=y
0 1
y1
xx 2
y(x )
y2
y(x )2
h
`(x;y)… f(x;y)
`(x;y;h) h!0¡! f(x;y)
yi+1 = yi +h`(xi;yi;h)
Eulerova metoda
ix xi+1
i+1
x
y
ii
y(x )
y(x )=y i ii+1y =y +hf(x )
y(x)
`(xi;yi;h) = f(xi;yi|{z}
vlevo
)
yi+1 = yi +hf(xi;yi)
1. Modiflkace Eulerovy metody
ix xi+1 x
y
y(x)
xi+1/2
y(x )i+1
i+1 i 2
h
ii
h
2 i i
iiy(x )=y
y =y +hf(x + ,y + f(x ,y ))
`(xi;yi;h) = f
‡
xi + h2;yi + h2f(xi;yi)
| {z }
vprost•red
·
yi+1 = yi +hf
‡
xi + h2;yi + h2f(xi;yi)
·
2. Modiflkace Eulerovy metody
ix xi+1 x
y
y(x)
xi+1/2
iiy(x )=y
y =y + (f(x ,y )+f(x +h,y +hf(x ,y )))
y(x )i+1
ii
h
2i+1 i i iii
`(xi;yi;h) = 12

f(xi;yi|{z}
vlevo
)+f
‡
xi +h;yi +hf(xi;yi)| {z }
vpravo
·¶
yi+1 = yi+h2

f(xi;yi)+f
‡
xi+h;yi+hf(xi;yi)
·¶
Preliminary version { June 1, 2001 { 11:17
9
Deflnice: Okrajov¶a ¶uloha pro line¶arn¶‡ ODR 2.•r¶adu
Nalezn•ete funkci y = y(x) pro kterou plat¶‡
y00 +f1(x)y0 +f2(x) = f3(x) x 2ha;bi‰R ;fi 2 C(ha;bi) i = 1;2;3
a z¶arove•n jsou spln•eny okrajov¶e podm¶‡nky typu
fi1y0(a)¡fi2y(a) = fi3
fl1y0(b) + fl2y(b) = fl3
kde koeflcienty fii;fli 2R spl•nuj¶‡ n¶asleduj¶‡c¶‡ relace:
i = 1;2 fii ‚0; fli ‚0 fii +fli 6= 0
Pozn¶amka: Okrajov¶e podm¶‡nky
V¶y•seuveden¶e obecn¶e okrajov¶e podm¶‡nky se naz¶yvaj¶‡ Sturmovy. V jednotliv¶ych konkr¶etn¶‡ch p•r¶‡padech
rozli•sujeme jejich n¶asleduj¶‡c¶‡ varianty:
a) Dirichletovy y(a) = fi, y(b) = fl
b) Neumannovy y0(a) = fi, y0(b) = fl
c) Newtonovy y(a) = fi, y0(b) = fl
V•eta: P•revod na samoadjungovan¶y tvar
Necht’ je d¶ana rovnice
y00 +f1(x)y0 +f2(x) = f3(x) (1)
kde fi 2 C(ha;bi) i = 1;2;3. Potom ji lze p•rev¶est do samoadjungovan¶eho tvaru:
¡
‡
p(x)y0
·0
+q(x)y = f(x) (2)
kde p(x) = e
R f
1(x)dx, q(x) =¡f2(x)p(x), f(x) =¡f3(x)p(x).
N¶avod: P•revod na samoadjungovan¶y tvar
1. N¶asob (1) ¡p(x)
¡p(x)y00 ¡p(x)f1(x)y0 ¡p(x)f2(x)y =¡p(x)f3(x)
2. \Rozderivuj" (2)
¡p(x)y00 ¡p0(x)y0 +q(x) = f(x)
3. Srovnej koeflcienty u stejn¶ych derivac¶‡ y
¡p(x)f1(x) =¡p0(x) ) p(x) = e
R f
1(x)dx
¡p(x)f2(x) = q(x) ) q(x) =¡p(x)f2(x)
¡p(x)f3(x) = f(x) ) f(x) =¡p(x)f3(x)
V•eta: Existence a jednozna•cnost •re•sen¶‡
Uva•zujme line¶arn¶‡ diferenci¶aln¶‡ rovnici 2. •r¶adu v samoadjungovan¶em tvaru
¡
‡
p(x)y0
·0
+q(x)y = f(x)
Necht’ nav¶‡c plat¶‡:
a) q;f 2 C(ha;bi), p 2 C1(ha;bi)
b) p(x) > 0; q(x)‚0 8x 2ha;bi
Potom existuje pr¶av•e jedno •re•sen¶‡ spl•nuj¶‡c¶‡ Sturmovy okrajov¶e podm¶‡nky (s vyj¶‡mkou p•r¶‡padu
fi2 = fl2 = 0; q(x)·0 v ha;bi).
Preliminary version { June 1, 2001 { 11:17
10
Deflnice: Metoda s¶‡t¶‡ pro Dirichletovu ¶ulohu
i) ¶Uloha
¡
‡
p(x)y0
·0
+q(x)y = f(x) x 2(a;b) y(a) = fi; y(b) = fl
ii) S¶‡t’ Ekvidistantn¶‡ d•elen¶‡ intervalu ha;bi·hx0;xni s krokem h = b¡an
T ·
n
xi = x0 +ih; i = 0;:::;n
o
iii) Zna•cen¶‡
Yi … y(xi) fi = f(xi) qi = q(xi) pi = p(xi)
iv) Diskretizace
(a) Aproximace
‡
p(x)y0
·
v bodech xi¡1=2;xi+1=2 centr¶aln•e, u•zit¶‡m hodnot v bodech
xi¡1;xi;xi+1.
‡
p(x)y0
·
i¡1=2
… pi¡1=2Yi ¡Yi¡1h
‡
p(x)y0
·
i+1=2
… pi+1=2Yi+1 ¡Yih
(b) Nahrazen¶‡ vn•ej•s¶‡ derivace ve v¶yrazu
‡
p(x)y0
·0
centr¶aln¶‡m diferen•cn¶‡m pod¶‡lem
v¶y•seuveden¶ych aproximac¶‡
‡
p(x)y0
·0
…
‡
p(x)y0
·
i+1=2
¡
‡
p(x)y0
·
i¡1=2
h …
pi+1=2 Yi+1¡Yih ¡pi¡1=2 Yi¡Yi¡1h
h
(c) Sestaven¶‡ diskr¶etn¶‡ aproximace
¡
ˆ
pi+1=2 Yi+1¡Yih ¡pi¡1=2 Yi¡Yi¡1h
h
!
+qiYi = fi i = 1;:::;n¡1
v) Soustava ¶Upravou a p•reskupen¶‡m •clen”v p•redchoz¶‡ rovnici dostaneme
¡pi¡1=2Yi¡1 +(pi¡1=2 +pi+1=2 +h2qi)Yi ¡pi+1=2Yi+1 = h2fi i = 1;:::;n¡1
Pozn¶amka: Vlastnosti soustavy
V¶yslednou soustavu line¶arn¶‡ch algebraick¶ych rovnic lze zapsat v maticov¶em tvaru, kde matice je
t•r¶‡diagon¶aln¶‡, symetrick¶a a pozitivn•e-deflnitn¶‡. To zaru•cuje jednozna•cnou •re•sitelnost diskr¶etn¶‡ ¶ulohy.
Preliminary version { June 1, 2001 { 11:17
11
Deflnice: Line¶arn¶‡ PDE 2.•r¶adu (ve 2 prom•enn¶ych)
a(x;y)@
2u
@x2 +b(x;y)
@2u
@x@y +c(x;y)
@2u
@y2 +d(x;y)
@u
@x +e(x;y)
@u
@y +g(x;y)u = f(x;y) (1)
Deflnice: Klasiflkace line¶arn¶‡ch PDE 2.•r¶adu
Pro rovnici ve tvaru (1) deflnujme
r(x;y) = b2(x;y)¡4a(x;y)c(x;y)
Rovnice je v bod•e [x0;y0]
a) Eliptick¶a () r(x0;y0) < 0
b) Parabolick¶a () r(x0;y0) = 0
c) Hyperbolick¶a () r(x0;y0) > 0
Pozn¶amka: P•r¶‡klady rovnic
a) Eliptick¶a =) ¢u = f...Poissonova rovnice
b) Parabolick¶a =) @u@t = p@2u@x2 ...Rovnice veden¶‡ tepla
c) Hyperbolick¶a =) @2u@t2 = c2 @2u@x2 ...Vlnov¶a rovnice
Preliminary version { June 1, 2001 { 11:17
12
¶Uloha:
D¶ana oblast ›‰R2 s hranic¶‡ @› a funkce f(x;y) deflnovan¶a na oblasti ›.
Nalezn•ete funkci u = u(x;y) takovou, aby spl•novala Poissonovu rovnici
¢u = f 8x 2›
a okrajov¶e podm¶‡nky dle typu ¶ulohy.
Dirichletova ¶uloha
›‰R2 @›·¡D
¢u = f 8x 2›
u
flfl
fl
¡D
= `D(x;y)
a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0
a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0
a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0
a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0
a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0
a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0
a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0
a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0a1a0
a2a1a2a1a2a1a2a1a2a1a2a1a2a1a2a1a2
a2a1a2a1a2a1a2a1a2a1a2a1a2a1a2a1a2
a2a1a2a1a2a1a2a1a2a1a2a1a2a1a2a1a2
a2a1a2a1a2a1a2a1a2a1a2a1a2a1a2a1a2
a2a1a2a1a2a1a2a1a2a1a2a1a2a1a2a1a2
a2a1a2a1a2a1a2a1a2a1a2a1a2a1a2a1a2
a2a1a2a1a2a1a2a1a2a1a2a1a2a1a2a1a2
a2a1a2a1a2a1a2a1a2a1a2a1a2a1a2a1a2
Ω
ΓD
Neumannova ¶uloha
›‰R2 @›·¡N
¢u = f 8x 2›
@u
@~n
flfl
flfl
¡N
= `N(x;y) a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3
a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3
a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3
a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3
a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3
a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3
a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3
a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3a1a3
a4a1a4a1a4a1a4a1a4a1a4a1a4a1a4a1a4
a4a1a4a1a4a1a4a1a4a1a4a1a4a1a4a1a4
a4a1a4a1a4a1a4a1a4a1