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Giperbolik tipdagi ushbu — + — = 0 tenglama uchun (bunda F=F(u), masalan
dt dx
F=vu, a=aAt/Ax; g - o‘tish ko‘paytuvchishi):
■ Birinchi tartibli aniqlikka ega oshkor usul:
<sup>A
un+1 = un -

2Ax

t</sup> .(f+ - FA); g=1+ia sin(kAx).
Bu ayirmali sxema doimo noustivor.
L
■

■
aks usuli:
U+ = 1 fci+uh )^-^- (fa ^- FA⁾; g=^cos(kAx)+ia^sin(kAx).
Bu ayirmali sxema AtLelevye usuli:
uⁿ+¹ = u - — Vau) A - (au)ⁿ ] , agar aⁿ < 0 ; uⁿ+¹ = uⁿ -—[(an)ⁿ - (an)ⁿ₁ ] , agar aⁿ > 0; ^A” Ax
g=1-|a|+|a|cos(kAx)+iasin(kAx). Bu ayirmali sxema AtЛ
g=1-iasin(kAx)+a [cos(kAx)-1]. Bu ayirmali sxema At da ustivor.

va c:„2 =¹ c(u"i: <);

L
■

■


Laks-Vendroffning ikki qadamli usuli:
At


<,n = \ (>c, + u )—A k:, - F,h);


u"^:1 = u" —


At
Ax


(jg n:1/2 rgn:1/2 ).
^Fi:1/2 ^{— F}i-1/2 /;


At


2


u"^:1 = u ——(f”i — F—1 ):-drr [Ci':1/2 (F"i — F,")— СП—1/2 (f" — F—1 )],


2Ax


2Ax


bu yerda C - yakobian; C


JF,


,^V


Qu„


2

aks-Vendroffning bir qadamli usuli:
Л
g=1-iasin(kAx)+a [cos(kAx)-1]. Bu ayirmali sxema At
"^—1 — — (F"i — F”i); g=ia sin(kAx)±J 1—a²sin²(kAx).


" —⁽¹’⁵ :^E)ftt (F "1 — F— ): (0,5 :e)-A(FI? — F-¹).

■
■


u"^:1 = u"^—1 —


Ax


Bu ayirmali sxema At
Kvaziikkinchi tartibli aniqlikka ega usul:
At


u"^:1 = u" —(1,5 :

«Sakrab qadamlash» usuli:
Bu ayirmali sxema a<0,5 da ustivor, agar s >0,25a²+0,5a⁴, ya’ni aynan AtA
agar s >
t ²a² At ⁴ a⁴
4 Ax⁴ 2 Ax

3. 
   Parabolik tipdagi tenglama uchun oshkor konservativ usullar
   

Qu Q ²u
Parabolik tipdagi ushbu —:a—- = 0 (g - o‘tish ko‘paytuvchishi; p = aAt/ Ax²)
Qt Qx
tenglama uchun:

(u"+i — 2u" : u"__x); g=1 -4/kin²(kAx/2).

■
aAt


u"^:1 = u" :^A (u"₊, — 2u" : uⁿ


Ax

Birinchi tartibli oshkor usul:
Л
Bu ayirmali sxema At<0,5Ax /a da ustivor.
■ Krank-Nikolson usuli:
"
aAt
2 Ax^{2 u :1}
:1 " . ^aAt / ":1 — ":1 ":1 \ ^aAt t " " " V
^ui = ^ui ^: 2Ax2 ^(ui:1 ^{— 2u}i ^{: u}i—1 ^):^dJ Vi:1 ^{— 2u}i ^{: u}i—1)^;
Л Л

«Sakrab qadamlash» usuli:
(u"_+x — 2u" :u*__x); g=-4^sin²(kAx/2)±^1 :16fi² sin⁴(kAx/2).


Ax¹

g
■


u"^:1 = u"^—1: ^2aAt \yh — 2u" : u"


■


Bu ayirmali sxema doimo noustivor.
Dyufor-Frankelning oshkor usuli:


1 — 2p


\


2P


,,":1 ,,"—1 i ^2aAt i^,":1 \ .."—A i ,1* , ,w:1 ^{1 2}P ,,w—1 . ²P V
^ui = ^u' ^:~A)F ^[ui:1 ^{— (Л}’ ^{: u}’ ^{): u}i—1^{]; u}i = ¹ . о n> ^ui ^: T~WT> :1 ^{: u}i—1 /^;


1: 2P.


1: 2p

=[1-2/fein (kAx/2)][ 1+2/kin (kAx/2)]. Bu ayirmali sxema doimo ustivor.
g = Y1jj 2ficos(kAx) ±y] 1—4^²sin²(kA)] . Bu ayirmali sxema doimo ustivor.
Mustaqil ish topshiriqlari
Quyida keltirilgan parabolik (1-jadval) va giperbolik (2-jadval) tipdagi xususiy hosilali differensial tenglamalar uchun boshlang‘ich chegaraviy masalalarni: oshkor ayirmali sxema bo‘yicha yeching; oshkormas ayirmali sxema bo‘yicha yeching.
Hosilalarni o‘z ichiga olgan chegaraviy shartlarni uch xil approksimatsiyalash: birinchi tartibli ikki nuqtali approksimatsiya; ikkinchi tartibli uch nuqtali approksimatsiya; ikkinchi tartibli ikki nuqtali approksimatsiya yo‘li bilan bajargan holda chegaraviy masalalarni yeching. Har xil vaqt momentlarida sonli yechim xatoligini berilgan analitik yechim bilan taqqoslagan holda aniqlang. Xatolikning to‘r parametrlari z, h lardan bog‘liqligini tadqiq qiling [2,4,5].
(1-jadval)

№	Тенглама	Boshlang‘ich shart	Chegaraviy shartlar	Analitik yechim ^uanal^(x,t^)
1.	Cu C^2 u - a 2 > ^a > ^0, Ct Cx	u( x,0) — sin(2^x)	u(0, t) — 0; u(1, t) — 0	e~^4^^at sin(2^x)
2.	cu C^2 u ^a 2 > ^a > ^0,Ct Cx	u( x,0) — x + sin(^x)	u(0, t) — 0; u(1, t) — 1	x + e^- at sin(^r)
3.	cu d^2 u — ^a 2 > ^a > ^0, Ct Cx	u(x,0) — cos x	u(0, t) — exp (-at); u(n, t) — - exp (-at)	e~^at cosx
4.	Cu C^2 u — ^a 2 > ^a > ^0,Ct Cx	u( x,0) — sin x	ux (0, t) — exp (-at); ux (ж, t) — - exp (-at)	e~^at sin x
5.	Ck C^2u . . , ~ = 2 ^+ ^Ч^ж) Ct Cx	u(x,0) — 0	u(0, t) — 0; u(1, t) — 0	i 2 — (1 -e ^ж ‘)sin(^x) ж
6.	Ci C^2u — 9 + Ct Cx^2 + cos x(cost + sin t)	u(x,0) — 0	u(0, t) — sin t; u (ж/2, t) — - sin t	sin t cos x
7.	Ci C^2u — 9 + Ct Cx^2 + 0,5e~^0,5t cos x	u( x,0) — sin x	ux (0, t) — exp(-0,5t); ux (ж, t) — - exp(-0,5t)	e~^0,5t sin x
8.	Cu C^2u — — a—- + cu, Ct Cx^2 a > 0, c < 0	u( x,0) — sin x	ux (0, t) — exp((c - a)t); u(ж / 2, t) — exp ((c - a)t)	e^(-a)t sin x
9.	Ch C^2 u Ci Ct - ^a & 2 + ^b C ’ a > 0, b > 0	u( x,0) — cos x	ux (0, t) - u(0, t) — -e~^at (cos(bt) + sin(bt)), ux (ж, t) - u^, t) — e^-t (cos(bt) + sin(bt))	e^-t cos(x + bt)
10.	Cu C^2 u Ci _ — a . + b + cu, Ct Cx^2 Cc a > 0, b > 0, c < 0	u( x,0) — sin x	ux (0, t) + u(0, t) — e^(c-a )l (cos(bt) + sin(bt)), ux (ж, t) + u (ж, t) — -e^(c-a)t (cos(bt) + sin(bt))	e^(c-a)t sin(x + bt)

(2-jadval)

№	Тенглама	Boshlang‘ich shartlar	Chegaraviy shartlar	Analitik yechim ^uanal^(x,^t)
1.	d'^2 u 9 d'^2 u a^2^a ^^2 ’ a > 0,	u(x,0) = sin x , ut (x,0) = —a cos x	u(0, t) = — sin( at), u(n, t) = sin( at)	sin( x — at)
2.	a^2 u 9 а^2 u a^2^a ax^2 ’ a > 0,	u( x,0) = sin x + cos x, ut (x,0) = —a(sin x + cos x )	ux (0, t) — u(0, t) = 0, ux (ж, t) — u(n, t) = 0	sin( x — at) + + cos(x + at)
3.	a^2u a^2u . = , — 3u at^2 ax^2	u( x,0) = 0, ut (x,0) = 2cos x.	u(0, t) = sin(2t)), м(ж, t) = — sin(2t),	cos x sin(2t)
4.	a^2u a^2u —^ ^ - 5u at ^2 ax ^2	u( x,0) = exp(2x ), ut (x,0) = 0.	ux (0, t) — 2u(0, t) = 0, ^ux ^(1, t) — 2u(1, t) = ^0,	e^2x cost
5.	a^2 u a^2 u au at^2 ax^2 ax	u( x,0) = 0, ut (x,0) = exp(—x ) sin x	u(0, t) = 0, w^, t) = 0,