2021 Mundarija

Birinchi chegaraviy masalani yechish. Biz endi tebranish tenglamasi uchun G = {0 < x < 1,0 < t < T} sohada ushbu birinchi chegaraviy masalani ko’rib chiqamiz. Yani G sohada ikki marta uzluksiz differensiallanuvchi u(x, t)
funksiyanitopishkerakki , bu sohadau
= (1.5.10)
tenglamani qanoatlantirib, t = 0 to’g’ri chiziqda

u(x,0) = Q(x), (x,0) =Ψ (x)
dastlabki shartlarniva
u(0, t) = μ₁ (t), u(1, t) = μ₂ (t), 0 < t < T


(1.5.11)
(1.5.12)

^{Chegaraviy shartlarni qanoatlantirsin. Bu maslani to’r metodi bilan yechish uchun}
ushbu
G_hτ = {x_i = ih, i = 0, M, hM = 1;t_k = kτ, k = 0, N, Nτ = T}
To’rni kiritamiz va 1.3-chizmadagidek uch qatlamli andaza bo’yicha (1.5.1) differensial tenglamani (1.5.3) dagi ayirmali sxema bilan almashtiramiz, bu yerda
i va k quyidagi qiymatlarni qabul qiladi:
i = 1,2,......, M - 1; k = 1,2,....., N - 1
Dastlabki shartlar uchun (1.5.7) formuladan foydalanamiz. Chegaraviy
shartlar quyidagichayoziladi:
^y₀^{+1 = μ}₁ ^(t_k+1^{), y}_M^{1 = μ}₂ ^(t_k+1^{), k = 0,1,...., N - 1}
Bularning hammasini birlashtirib, ayirmali sxemaning quyidagi hisoblash
algoritmiga egabo’lamiz:

y_i = Q(x_i ), y_i = Q(x_i )+τΨ(x_i )+ Δ ₂Q_i ,
^y_i^{+1 = 2y}_i ^{+τ2 Δ} ₂ ^y_i ^{- y}_i ^{-1 , i = 1,2,....., M - 1,
y}₀^{+1 = μ}₁ ^(t_k+1^{), y}_M^{1 = μ}₂ ^(t_k+1^{), k = 0,1,......, N - 1}


(1.5.13)
(1.5.14)
(1.5.15)

Yuqorida ko’rdikki, bu sxema (1.5.1), (1.5.3) chegaraviy masalani 0(τ² + h² ) aniqlikda approksimatsiya qiladi. Ko’rsatish mumkinki, agar ixtiyoriy ε > 0
uchun τ va h qadamlar quyidagi
^{< (1.5.16)}
Shartni qanoatlantirsa , (1.5.7), (1.5.10), (1.5.11) sxema turg’un bo’ladi. Biz buning
isbotiga to’xtalib o’tirmaymiz.
Misol. To’r metodi bilan G = {0 < x <1, 0 < t < T} sohada
δ ² u δ ² u
₌
δt ² δx²
To’rtenglamasining
u(0, t) = u(1, t) = 0(0 < t < 1),
u(x,0) = sinπx, (x,0) = 0
Chegaraviy va dastlabki shartlarni qanoatlantiradigantaqribiy yechimi topilsin.

Yechish. Bu yerda h = 0,1 va Q(x) = sinτx, Q^" (x) = 一τ² sinπx hamda Ψ(x) = 0
quyidagichayozamiz:


τ = 0.08 deb olamiz. Keyin liginihisobga olib, (6.5) formulani

^{yi = yi + Q}" ^{(xi ) = (1 一 0.0032π}2 ^{)sinπxi
Endi Δ} 2 ^{operatorning ko’rinishini e’tiborga olsak, hisoblash uchun quyidagi}
algoritm hosilbo’ladi:
^y_i ^{= sinπx}_i ^{, y}_i ^{= (1 一 0.0032π2 )sinπx}_i ^{, i = 1,2,......, M 一 1,}
y ₀⁺¹ = y_M¹ = 0, k = 0,1,........, N 一 1,
^y_i^{+1 = 0.64y} ₁ ^{+ 0.72y}_i ^{+ 0.64y}_i^k_一1 ^{一 y}_i ^一1
Hisoblashni faqat 0 < x < uchun bajarsa yetarli bo’ladi, chunki u = u⁽x, t⁾
yechimning grafigi x = tekislikkanisbatan simmetrik ravishda joylashgan.
I bobning qisqacha xulosasi

^{I bobdaxususiy hosilali differensial tenglamalarnitaqribiy yechish metodlari}
haqida gapirilgan.
II bob. Xususiy hosilali differensial tenglamalarni Mathcad muhitida taqribiy
yechish
2.1. Elliptik tenglamalarni Mathcad dasturiyordamida taqribiy yechish.

Faraz qilamiz:
chegaraviy shartlar
u(A, y) = u(y)₁
u(x, C) = u(x)3
bilan
^u_{x ,x} ^{+ u}_{y ,y} ^{= f (x,y)
u(B,y) = u(y)}2
u(x,D) = u(x)₄
^{A < x < B C < y < D}

tenglamani sonli yechish talab qilingan bo’lsin.
^u_i-1,j ^{- 2u}_{i , j} ^{+ u}_{i+1, j} ^u_{i ,j-1} ^{- 2u}_{i , j} ^{+ u}_{i , j+1}

h k ^,
₂ ⁺ ₂ ^{= y}_{i j}

^{i = 1,..N - 1}
ayirmali sxema bilan almashtiramiz.


j = 1,..M - 1

^{Bu sistemani yechib:} yi ,j i = 1,..N
taqribiy yechiminitopamiz .
Konkret bir masalani qaraymiz.


j := 0,1..M

^u(x₁ ^{, x}₂ ^{) + u(x}₁ ^{, x}₂ ^{) = 2x}₂
f (x₁ , x₂ ) := -2(x₁ + x₂ )
u(0, x2 ) = 0 μ ^{chap- μ}left (x2 ) := 0
u(x₁ ,0) = 0 μ quyi- μ_bottom (x₁ ) := 0

u(1, x2 ) = x2 + (x1 )2
^u(x₁ ^{,1) = (x ) + x}₁₁


^μ_right ^(x₂ ^{) := x}₂ ^{+ x}₂
μ_top (x₁ ) := x₁² + x₁

tenglamaning aniq yechimi:


u(x₁ , x₂ ) := x₁²x₂ + x₁x₂²

G = {(x₁ , x₂ ) : 0 < x₁ < 1,0 < x₂ < 1}
ekanligima’lum.


_{N := 5 M := 5 a:= 0 b:= 1 c := 0 d := 1 h := l :=}
h
^y_{j ,0} ^{:= μ}_left ^(x_2,j ⁾
= 0.2 l = 0.2 i := 0...N j := 0..M x_1,,i := ih x_2,j := jl

^y_{j ,N} ^{:= μ}_right ^(x_2,j ^{) y}_0,i ^{:= μ}_bottom ^(x_1,i <sup>)


y</sup>_{M ,i} ^{:= μ}_top ^(x_1,i ^{) j := 0..M - 2 i := 0...N}

(
^|
|
|
|

| | | |

_「- 0.4 - 0.8 - 1.2 - 1.6 - 2 -
|

2.4] | 2.8 | 3.2| | ^3.6」
^{- 0.}8)
-
_{F = |}^{- 0.8 - 1.2 - 1.6 - 2 - 2.4 -} |- 1.2 - 1.6 - 2 - 2.4 - 2.8 - |
1.2
-
F_{j ,i} := f (x_1,i , x_2,j+1) F^<1> =
1.6
^（
_L<sup>- 1.6 - 2 - 2.4 - 2.8 - 3.2 -
- 2 )</sup>

^F_0,2 ^{:= F}_0,i ^+
y0,i
l²


_「- 0.4 - 0.8 - 1.2 - 1.6 - 2 -
^F
|
^{M -2,i := FM -2,i + F =} - _1.2 - _1.6 - 1₂6 _--₂2₄ - _2.8 -
|
_L<sup>- 1.6 - 2 - 2.4 - 2.8 - 3.2 -


2.4]
|</sup>
_2.8
|
3.2|
|
_3.6」

A := identity(M - 1) B:= A m := 0,...M - 2 n := 0,..N - 2
h
Λ_m,n := if (m = n - 1) + (m - 1 = n),1,0 C := identity(M - 1) - Λ a₁ := C^{- 1}B β₁ := C^-1F^<0> i := 1,..N a_i := (C - Aa_i )^-1 B C^-1A + C ^-1B = 0.01 C^-1B = 0.005 a_i+1 := (C - Aa_i )^-1B
β_i+1 := (C - Aa_i )^-1 (Aβ_i + F^<i>) i := 0,..N j := 0,..M - 2 W_{j ,i} := y_j+1,i i := N - 1,...0 W^<i> := a_i+1W^<i+1> + β_i+1 i := 0,..N j := 0,...M - 2 y_j+1,i := W_{j ,i}

2.1 chima. Taqribiy yechim grafigi



2.2. chizma. To’rni ichkinuqtalaridagi yechim grafigi

2.2. Chebishev parametrlar majmuasi bilan oshkor iteratsion metodi Mathcad