Differensial tenglamalar kafedrasi

> with(plots): T0:=1; a:=1;L:=12;

u(t,x):=1/2*T0+sum(2*(-1)^An*cos(Pi*(2*n+1)/L*x)*exp(-

Pi^A2*(2*n+1)^A2/L^A2*a^A2*t)*T0/Pi/(2*n+1),n=0..1000):

T0 := l a := l

L := l2

Olingan natijalami ikki o’lchamli animasiyali grafik ko’rinishda tasvirlaymiz: > animate(plot,[u(t,x),x=0..12, y=-0.1..1.1], t=0.0001..30, frames=30,thickness=3);

t
-. Iй-З

bir jinsli tenglamani ikkinchi tipli chegaraviy va quyidagi boshlang’ich shartlar bilan yeching:

u(0, x )=f( x),

bu yerda f( x) funksiya quyidagi ko’rinishda berilgan: > a:=1;L:=12;T0:=1;

f(x):=x->piecewise(x

a := 1 L := 12 TO := 1

„ _л • . ( L 2 TOx 2 TO (L - x) Л

f(x) := x ^ piecewise! x < — ———, x < L, — I

>
plot(f(x),0..12,-0.1..1.1, numpoints=400,color=blue,thickness=3);

> restart;

f

f(£ ) := £ ^ piecewise! £ <


L 2 TO £ 2 L


, £ < L,


2 TO (L - £ )

L

(xi):=xi->piecewise(xi

Yechish uchun yuqorida olingan formuladan (2- chegaraviy shardan) foydalanamiz:

u(t,x):=simplify(sum(2/L*cos(Pi*n/L*x)*exp(-

Pi^A2*n^A2/L^A2*a^A2*t)*(int(2*T0*xi/L*cos(Pi*n/L*xi),xi=

0. 
   .L/2)+int(2*T0*(L-xi)/L*cos(Pi*n/L*xi),xi=L/2..L)),n=1..infinity));
   

C2 0 := TO

\
u( t, x )


I

П = 1


(к nx 4 cos


L


e


L


2


, (к n

T0 \ -1 + 2 cos! —


(-1)ⁿ


_к² _n²



( к n ^

cos! —^— J = 0 n --> 2(n — 1)

к n

cosl J = (—1)ⁿ n --> 2 n

n _--> 4 n + 2

Tenglamaning yechimi: > u(t,x):=C2_0/2-sum(16*cos(Pi*(4*n+2)/L*x)*exp(- Pi^A2*(4*n+2)^A2/(L^A2)*a^A2*t)*T0/(Pi^A2)/((4*n+2)^A2),n =0..infinity);

2
W


u( t, x ) :=


(к (4 n +2)x 16 cos! ——-—— | e


L


2


TO


к² ( 4 n + 2 )²

2 2 n² (4 n + 2)² a² t

> with(plots): a:=1;L:=12;T0:=1;

u(t,x):=1/2*T0-sum(16*cos(Pi*(4*n+2)/L*x)*exp(-

PiA2*(4*n+2)A2/LA2*aA2*t)*T0/PiA2/(4*n+2)A2,n=0..222):

a := 1 L := 12 TO := 1

Olingan natijalami ikki o’lchamli animasiyali grafik ko’rinishda tasvirlaymiz:

3. 
   -
   
   1
   
   dt
   
   
   u( t, x ) = a²
   
   
   fd²
   
   Misol. > restart;
   

bir jinsli tenglamani ikkinchi tipli chegaraviy va quyidagi boshlang’ich shartlar bilan yeching:

u(0, x )=f( x),

bu yerda f( x) funksiya quyidagi ko’rinishda berilgan: > T0:=1; a:=1;L:=12; f(x):=x->piecewise(x

TO := 1 a := 1

L := 12

f(x) :=x ^ piecewisi(x < L, TO)

> plot(f(x),0..13,-0.1..1.1, numpoints=400,color=blue,thickness=3);

>
restart;

f(x):=x->piecewise(x

f(x) :=x ^piecewisex < L, TO)

Yechish uchun yuqorida olingan formuladan (3- chegaraviy shardan) foydalanamiz:

> u(t,x):=sum(2/L*cos(1/2*Pi*(1+2*n)/L*x)*exp(-

1
u( t, x ) :=


Z


у П = 1 V


( n (1 + 2 n ) x

^4co{ 2jL <sup>le


(</sup> n²(1 + 2 n )² a² t ^
4 L²


w


cos(n n ) (-TO + T2 )


n (1 + 2 n )


+ T2

/4*Pi^A2*(1+2*n)^A2/L^A2*a^A2*t)*int((T0-T2)*cos(1/2*Pi*(1+2*n)/L*xi),xi = 0 .. L),n = 1 .. infinity)+T2;

cos(n n ) = (-1 )ⁿ

Tenglamani yechimi:

> u(t,x):=sum(4*cos(1/2*Pi*(1+2*n)/L*x)*exp(-

1/4*Pi^A2*(1+2*n)^A2/L^A2*a^A2*t)*(-1)^An*(T0-

T2)/Pi/(1+2*n),n=0..infinity)+T2;

> with(plots):

T0:=1; a:=1.5;L:=12;T2:=T0/4;

u(t,x):=sum(4*cos(1/2*Pi*(1+2*n)/L*x)*exp(-

1/4*Pi^A2*(1+2*n)^A2/L^A2*a^A2*t)*(-1)^An*(T0-T2)/Pi/(1+2*n),n=0..1000)+T2:

Warning, the name changecoords has been redefined

TO := 1 a := 1.5

L := 12

T
1

4


u( t, x ) :=


Z


. n (1+2 n) x ^4coS 2L ^{1 e}


. n²(1 + 2 n )² a² t ^
4 L²


w


(-1)ⁿ (TO - T2)


V n = 0 V


n (1 + 2 n )


2 T2

2 : =

Olingan natijalami ikki o’lchamli animasiyali grafik ko’rinishda tasvirlaymiz: > animate(plot,[u(t,x),x=0..12, y=-0.1..1.1], t=0.0001..30, frames=30,thickness=3);

1.8

1.8

1.1

1.2

3. 
   Bob. Bir jinsli silindrda issiqlik o’tkazuvchanlik tenglamasi
   

3.1-§. Fazoda issiqlik o’tkazuvchanlik tenglamasini chiqarilishi

Uch o'lchovli fazoda biror issiqlik o’tkazuvchi va koordinatalari (x,y, z) bo’lgan ixtiyoriy M nuqtaning temperaturasi t vakt momentida u(x, y, z, t) funksiya

ko’rinishida beriluvchi jismni qaraymiz [1,2,4]. Ma’lumki, issiqlik potoki vektori

—^

uchun W quyidagi Fur’ye qonuni deb ataluvchi formula o’rinlidir.

——

W = -k grad u

Bu yerda k (x, y, z) - issiqlik o’tkazuvchanlik koeffitsiyenti.

Agar jism E³fazoda berilgan bo’lsa Q soxaning chegarasi ъ bo’ladi. Shunda jismning issiqlik miqdori t vaqt momentida quyidagi formula bilan hisoblanadi:

^Q2 ^-Q1 = JJJc⁽M)p⁽M)^u(M^,t₂^)dT_M ^- JJJc⁽M)p⁽M)^u(M^,t1⁾dr_M =

Q Q

=^(t2 ^-11 ⁾JJJ^c(M)P^(M)ut^{(M, t}3 ^)dTM

Q

[t\; t₂ ] (Q(t\) = Q, Q(h) = Q) vaqt oralig’ini qaraymiz. Shunda

^Q2 ^-Q1 = JJJ^c(M)p^(M)^u(M,12^M ^—JJJ^c(m)p^(M)u(m^t1^)dTm

Q Q

bo’ladi. Issiqlik miqdorining o’zgarishi tashqaridan issiqlik oqib kelish natijasida va ba’zi ichki manbaning (stoklarning) harakati tufayli ro’y beradi:

2 — — ‘"2

Q2 - Q =J-JJ (^W , n )dv dt + Jl JJJ F(M, t)dT dt

t
^t1 -

ъ
1 — Q

B
2 —

^Q2 ^{- Q}1 =^-J JJJ^(divW)dT

Q


dt + (t_x -t₂)JJJ F(M, t₄ )dT

Q

irinchi integral uchun Ostogradskiy-Gauss formulasini qo’llaymiz va o’rta qiymat haqidagi formulani esa ikkinchi integral uchun qo’llaymiz:

Bu yerda 1₄ e t; 1₂ ] ga qarashli.

Lagranj formulasidan quyidagi silliq (buni faraz qilamiz) u funksiya uchun foydalanamiz:

u(M, 12 ) - u(M, tj = u_t (M, 13 )(t2 -11), t3 e t; 12 ]

Bundan quyidagini hosil qilamiz:

Q
Q


Q

2 - Q1 = HIc(M)p(M)u(M, t₂)dTM -j]Jc(M)p(M)u(M, Wtm =

= (t2 - t_x){И c(M)p(M)u_t (M, t₃ )dz_M

Q

Demak,

^t2 —

(t₂ - p )j||c(M)p(M)u_t (M,t₃ )dr_M = -{ {И(divW)drM dt + (p - t₂ )jjjF(M,t₄ )dr.

Q tj L Q J Q

E
^c(M1⁾P^(M1^)ut ^(MV ^t3^)VQ ^(t 2 ^{- t}1⁾ = <sup>-divW


V</sup>Q ^(t2 ^{- t}1⁾ + ^F(M3^,14^)VQ^(t2 ^{- t}1^),


M =M

ndi hamma integral uchun umumlashtirilgan o’rat qiymat formulani qo’llaymiz:

Bunda t₅ e[tj;t₂JM_t,M₂ eQ, V_q-Q ning hajmi bo’ladi. V_Q(t₂tj) ga qisqartirib, Q

dan olingan biror bir M₁, M ₂ nuqtalar uchun quyidagini hosil qilamiz:

—^

^c(M 1 ⁾P^(M 1 ^)ut ^(M 1^{, t}3 ^{) V}Q ^(t2 ^{- t}1⁾ = ^{-div W} t=t₃ + ^F(M3^,14 ⁾ .

M =M 2

Endi biror M₀ nuqtagacha Q ni qissak, [t₁, 1₂ ] kesma ham t₀ nuqtagacha qisiladi. Bundan ko’rinadiki M₁,M₂ nuqtalar M₀ ga o’tadi, t₃,t₄,t₅ lar esa t₀ ga. Bundan limitga o’tganda quyidagi hosil bo’ladi:

——

^c(M0⁾P^(M0^)ut ^(M0^,10⁾ = ^{-div W} t=t₀ + ^F(M0^,10⁾

M =M ₀

——

W uchun Fur’ye qonunini qo’llab quyidagini hosil qilamiz:

.. — /7 7 4 d , du d , du d , du

div W = div(-kgradu) = - — k — - — k — - — k — ^

dx dx dy dy dz dz

d Pz d О/ d О/

^ c(M 0)p(M 0)ut (M 0, t0) = - k — + - k — +- k — + F (M 0, t0)

ox ox dy py dz dz

M₀, t₀ nuqtalarni ixtiyoriy olganimiz sababli, hosil qilingan formulani butun [t₁.t ₂ ] va Q ni soha uchun yoyish mumkin:

o
c( x, y, z )p( x, y, z)u_t (x, y, z, t)


^(k(x, y^{, z})^ux ^(x, y^{, z,} t⁾⁾ +

ox


^(k(^ y^{, z)u}y ⁽^ y^{, z,} t⁾⁾ +

Oy ^y



+ — (k(x, y, z)u_z (x, y, z, t)) + F(x, y, z, t) oz

B
^ut = ^{a 2(u}xx + ^uyy + ^uzz ⁾ + ^f (^x, y^{, z, t} X


a


k

<sup>cp


c</sup>P


(3.1.1)

u ifoda fazoda issiqlik o’tkazuvchanlik tenglamasi deb nomlanadi. c, p, k larni konstanta da deb olib, quyidagi tenglik hosil qilamiz:

Agar u, f faqat x va t o’zgaruvchilari bilan bog’liq bo’lsa, u holda bu tenglik quyidagicha yoziladi:

^ut = ^{a 2u}xx + ^{f (x, t)} (3.1.2)

Fizik interpretasiyada bir jinsli yupqa sterjinda issiqlik o’tkazuvchanlik (yoyilish) tenglamasidir. (3.1.2) tenglamani biz keyinchalik issiqlik o’tkazuvchi tenglamasi deb yuritamiz.

Analogik fikrlashni boshqa bir fizik prosesslar uchun ham o’tkazishimiz mumkin, masalan diffuziya uchun. Agar u( x. y.z.t)- fazoda gazning konsentrasiyasi bo’lsa, u holda diffuziya tenglamasi quyidagicha bo’ladi:

cu_t = div( Dgradu) + F (x, y, z, t)

D - diffuziya koeffitsiy enti F - biror bir funksiya

2. 
   §
   Bir jinsli silindrda issiqlik tarqalish jarayonini qaraymiz [6-8]. Buning uchun quyidagi:
   
   (o \\
   
   \ ¹ Ifr ^{u( t}’ ^r )
   
   
   o
   
   Ot
   
   
   u( t, r ) =a
   
   
   (o²

   vv

   
   
   
   or.2^u( '■^r)
   
   
   + -
   
   
   r
   
   . Maple paketi orqali bir jinsli silindrda issiqlik o’tkazuvchanlik tenglamasini Fur’ye usuli (o’zgaruvchilarni ajratish usuli) yordamida yechish
   

bir jinsli tenglamani

u(0, r ) =F( r )

Boshlang’ich shart va quyidagi bir jinsli chegaraviy shart bilan yeching:

u( t, r0) = 0 .

> restart;

Bir jinsli tenglama va uning yechimini o’zgaruvchilarni almashtirish usuli bilan

s
yechamiz:

> PDE:=diff(u(t,r),t)=a^A2*(diff(u(t,r),r,r)+(1/r)*diff(u(t,r),r)); struc:=pdsolve(PDE,HINT=T(t)*R(r));


O ~

PDE := — u( t, r ) = a² ot


f

(O² vV


д ₂^u( t Г ⁾

_or²


o

u( t, r )


r


d


dd^{4 r )} =


^{R(r )} _^ci dr


R( r )


(u(t, r) = T(t) R(r)) &where

> dsolve(diff(T(t),t) = _c[1]*T(t));

dsolve(diff(R(r),'$'(r,2)) = R(r)/a^A2*_c[1]-diff(R(r),r)/r);

(

T( t) = C1 e


a


^(t)


d


r dt


T( t) = _ci T( t)


R( r) = _C1 BesselJ


0


-_C r


a


+ C2 BesselY


0


-_C r


a


Quyidagicha almashtirish olamiz:


c


22


l


= =- X²a

truc :=

>
, ,2 2 ₄ (-X a t)
dsolve(diff(T(t),t) = -lambda^A2*a^A2*T(t)); dsolve(diff(R(r),'$'(r,2)) = -lambdaA2*R(r)-diff(R(r),r)/r);

T( t) = C1 e⁽

R(r) = _C1 BesselJ0, X r) + _C2BesselXO, X r)

Ikkinchi tenglamaning yechimi

R(r) = _C1 BesselJ 0, X r) + _C2 BesselY0, X r) silindrning markazida regulyarlik shartini qanoatlantirishi shart. BesselJ funksiyasi nolda regulyar:

> BesselJ(0,lambda*r)=series(BesselJ(0,lambda*r), r=0,4);

X²

B
BesselY funksiya nolda singulyar:

> BesselY(0,lambda*r)=series(BesselY(0, lambda*r), r=0,4);


BesselY(0, X r) =

² i ^ln(y]^{+ln( r)}


v


₊ 2y * J


+

ⁱv


2


+ ln( r) *


X²


(2 - 2 у) X²

+ “ —

4 * J


r² + O( r⁴)

esselJ0, X r) = l - — r² + O(r⁴)

Natija, radial tenglamaning umumiy yechimida quyidagini qo’yishga olib keladi:

_C2 = 0

Shuning uchun radial tenglamaning yechimi quyidagi ko’rinishga ega bo’ladi:

R( r) = BesselJ 0, X r)

(bunda o’zgarmas umumiy yechimda hisobga olinadi).

Silindming (г = r0) uchi chegaraviy shartni hisobga olib (nol temperatura):

R( r0) =0

Bu shartni quyidagi funksiya qanoatlantiradi:

> R[n](r):=BesselJ(0,lambda[n]*r);

R_n(r) := BesselJ(0, X_n r)

Bu yerda \ - Bessel funksiyasining р_и nollarini tanlab olish orqali aniqlanadigan

o’garmaslarni ajratishni o’zgarmas qiymatidir:

> BesselJ(0,mu[n])=0; mu:=BesselJZeros :mu(0,n);

BesselJ(0, р_и ) = 0

B
bunda:

> lambda[n]:=mu(0,n)/r0;


X :

n


BesselJZeros(0, n )
r0

esselJZero(S0, n)

Shuning uchun radial tenglamani yechimi quyidagi ko’rinishga ega bo’ladi: > R[n](r):=BesselJ(0,r*lambda[n]);

R
BesselJZeros( 0, n ) r
r0

„( r ) := BesselJl 0,

Vaqt tenglamasi uchun quyidagiga ega bo’lamiz: > T[n](t):= C[n]*exp(-lambda[n]^A2*a^A2*t);

T
9 9 л

BesselJZeros0, n )² a² t r0² ,


> u[n](t,r):=T[n](t)*R[n](r);

/ 2 2 Л

BesselJZerosQ, n )² a² t


u (t, r) := C e^v

nn


r0


⁷ BesselJl 0,


BesselJZeros(0, n) r
r0


Natijada tenglamaning umumiy yechimi quyidagi ko’rinishga ega bo’ladi: > u(t,r):=Sum(u[n](t,r), n=1..infinity);

/ 9 9 \

BesselJZerosQ, n)² a² t


to


^{u( t, r) :}= E ^C_n

n = 1


e^v


r0


⁷ BesselJl 0,


BesselJZeros(0, n) r
r0

(t) := C e

Quyidagicha almashtirish olamiz:

r

r --> p = ■ , r = p r0

Интервал изменения переменной p - o’zgaruvchining o’zgarish intervali [0, 1] birlik kesma.

> u(t,rho):=subs(r=rho*r0,u(t,r));

n
BesselJ(0, BesselJZeros(0, n) p)


^u(1, p) := Z ^C_n ^e


9 9 \

BesselJZeroSO, n )² a² t

rO²

= 1

C_n koeffisiyentni aniqlash boshlang’ich shartdan foydalanamiz:

u(0, r ) = F( r )

> simplify(subs(t=0,u(t,rho))=F(rho*r0));

Z C BesselJ0, BesselJZero(s0, n) p) = F( p rO)

П = 1

Oxirgi munosabatlar F(p rO) funksiyalarni Bessel funksiyasi bo’yicha qatorga yoyishni bildiradi.

Shuning uchun yoyilmaning koeffisiyentlari quyidagiga teng:

>

C
C :

n


2

BesselJ( 1, BesselJZeros(0, n ))²


p BesselJ(0, BesselJZeros(0, n) p) F(p rO) dp

0

[n]:=2/BesselJ(1,BesselJZeros(0,n))^A2*int(rho*BesselJ(0,BesselJZeros(0,n)* rho)*F(rho*r0),rho = 0 .. 1);

Natijada quyidagiga ega bo’lamiz:

> u(t,rho) := Sum(C[n]*exp(-

B
2 2 Л

' BesselJZero^0, n) a t


^{u( t,} p^{) :}= z


n = 1


2 e^v


rO


BesselJ(0, BesselJZeros(0, n ) p )


BesselJ( 1, BesselJZeros(0, n ))²

esselJZeros(0,n)^A2/r0^A2*a^A2*t)*BesselJ(0,BesselJZeros(0,n)*rho),n = 1 .. infinity);

p BesselJ(0, BesselJZeros(0, n ) p) F( p rO ) dp

0

3. 
   §. Tenglamani yechishga doir misollar
   

1
d_

dt


u( t, r ) = a¹


f

fd²

vv


я ₂^u( ^ ^{r )}

dr


л i (a.^u(t,^r)

+ -


r

- Misol. > restart;

bir jinsli tenglamani bir jinsli chegaraviy shartlar va quyidagi boshlang’ich shartlar bilan yeching:

u(0, r ) = F( r )

bu yerda F( r) funksiya quyidagi ko’rinishda berilgan [2,5]: > r0:=1; H:=1;a:=0.25;

r0 := 1

H := 1
a := 0.25

^F( P) ^:= -P² + ¹

> addcoords(M_cylindrical,[z,rho,theta],[rho*cos(theta),rho*sin(theta),z]); plot3d(F (rho),rho=0..r0,theta=0..2* Pi,coords=M_cylindrical,axes=BOXED,lig htmodel=light3,numpoints=1000,orientation=[45,65],shading=ZHUE,style=P

A



1 1

TCHCONTOUR);

Yoyilmaning koeffisiyentlari:

>

C[n]:=simplify(2/BesselJ(1,BesselJZeros(0,n))^A2*int(rho*BesselJ(0,BesselJZer os(0,n)*rho)*F(rho*r0),rho = 0 .. 1));

C
4 (-BesselJZero(s0, n) Bessel^0, BesselJZero(s0, n)) + 2 BesselJd, BesselJZero(s0, BesselJZero(s0, n)³ BesselJ 1, BesselJZero(s0, n ))²

:

n

Yechish uchun yuqorida olingan formuladan foydalanamiz:

> u(t,rho) := Sum(C[n]*exp(-

BesselJZeros(0,n)^A2/r0^A2*a^A2*t)*BesselJ(0,BesselJZeros(0,n)*rho),n = 1 .. 12);

1
f (-0.0625 BesselJZeros0, n )²1 \ ^ . A

8 e BesselJ(0, BesselJZeros( 0, n ) p )

_v BesselJZeros( 0, n )³ BesselJ( 1, BesselJZeros( 0, n )) _y

2

^u(t, p^{) :}= 2

n = 1

Olingan yechimni animasiyali grafik ko’rinishida tasvirlaymiz:

> with(plots):

addcoords(M_cylindrical,[z,rho,theta],[rho*cos(theta),rho*sin(theta),z]); animate(plot3d,[u(t,rho),rho=0..r0,theta=0..2*Pi,coords=M_cylindrical], t=0..10, frames=40,

axes=BOXED,lightmodel=light1,numpoints=1000,orientation=[45,

65],shading=ZHUE,style=PATCHCONTOUR);

Warning, coordinates already exists, system redefined.

b
2 - Misol. > restart;



r


d_

dt


u( t, r ) = a²


^d²


\\


dr²


u( t, r )


+ -


d

¹¹ dr ^{u( t},^r)


r

ir jinsli tenglamani bir jinsli chegaraviy shartlar va quyidagi boshlang’ich shartlar bilan yeching

u(0, r ) = F( r )

bu yerda F(r) funksiya quyidagi ko’rinishda berilgan: > r0:=1; H:=1;a:=0.25;

F(rho):=-(rho * r0)+H;

r0 := 1

H := 1
a := 0.25
^F( P^{) :}= -P + ¹

> addcoords(M_cylindrical,[z,rho,theta],[rho*cos(theta),rho*sin(theta),z]); plot3d(F (rho),rho=0..r0,theta=0..2* Pi,coords=M_cylindrical,axes=BOXED,lig htmodel=light3,numpoints=1000,orientation=[45,65],shading=ZHUE,style=P

A
TCHCONTOUR);

Yoyilmaning koeffisiyentlari:

>

C[n]:=simplify(2/BesselJ(1,BesselJZeros(0,n))^A2*int(rho*BesselJ(0,BesselJZer os(0,n)*rho)*F(rho*r0),rho = 0 .. 1));

C_n := n (-BesselJ(0, BesselJZeros(0, n )) StruveH(1, BesselJZeros(0, n ))

+ BesselJ(1, BesselJZeros(0, n )) StruveH(0, BesselJZeros(0, n )))/(

BesselJZeros(0, n )² BesselJ(1, BesselJZeros(0, n ))²)

Yechish uchun yuqorida olingan formuladan foydalanamiz:

> u(t,rho):=Sum(C[n]*exp(-

BesselJZeros(0,n)^A2/r0^A2*a^A2*t)*BesselJ(0,BesselJZeros(0,n)*rho),n =1..24);

2
9

(-0.0625BesselJZeron)² t)

n StruveH( 0, BesselJZeros( 0, n )) e BesselJ( 0, BesselJZ

BesselJZeros(0, n )² BesselJ( 1, BesselJZeros( 0, n ))

4

^u( t, p^{) :}= X

n = 1

Olingan yechimni animasiyali grafik ko’rinishida tasvirlaymiz:

> with(plots):

addcoords(M_cylindrical,[z,rho,theta],[rho*cos(theta),rho*sin(theta),z]);

animate(plot3d,[u(t,rho),rho=0..r0,theta=0..2*Pi,coords=M_cylindrical],

t=0..10,

frames=20,axes=BOXED,lightmodel=light1,numpoints=1000,orientation=[45,

65],shading=ZHUE,style=PATCHCONTOUR);

W
arning, coordinates already exists, system redefined.

Xulosa

Turmush hayotimizda muhim ahamiyatga ega bo’lgan isssiqlikning to’g’ri chiziq, tekislik va fazoda tarqalish jarayoni, shuningdek diffuziya hodisasi parabolik tipli tenglamalar orqali o’rganiladi. Bu tenglamalar uchun ham to’lqin tenglamasi kabi chegaraviy va Koshi masalari tenglama yechimini bir qiymatli ajratib olishga imkon yaratadi va ular belgilangan rejimga asosan tanlab olinadi.

Biz bu malakaviy bitiruv ishida chekli uzunlikdagi sterjenda qo’yilgan aralash masalalarning limitik holi sifatida aniqlagan chegaralanmagan uzunlikdagi sterjenda issiqlik tarqalish tenglamasiga qo’yilgan Koshi masalasining yechimi xuddi giperbolik tenglamalar uchun chegaraviy masalalrni yechishda qo’llanilgan o’zgaruvchilarni almashtirish yoki Fur’e usuli yordamida topilib, yechim Puasson integrali deb ataluvchi integral shaklda tasvirlanishini o’rgandik.

Bitiruv malakaviy ishida, Maple matematik paketidan foydalanib, yarim chegaralangan sohada issiqlik o'tkazuvchanlik tenglamasini va bir jinsli silindrda issiqlik o’tkazuvchanlik tenglamasini Fur’ye usuli (o’zgaruvchilarni ajratish usuli) yordamida yechish va tenglamani yechishga doir misollar qaralgan, shuningdek, yechimning ikki o’lchovli animasiyali grafigi tasvirlangan.

Foydalanilgan adabiyotlar ro’yxati.

1. 
   Salohiddinov M. S. Matematik fizika tenglamalari, T. “Ozbekiston”, 2002, 448 b.
   
2. 
   Salohiddinov M. S., Islomov B. I. Matematik fizika tenglamalari fanidan masalalar to’plami, T. “Mumtoz so’z” , 2010, 372 b.
   
3. 
   Zokirov O. S. Xususiy hosilali differensial tenglamalar.T. Universitet”2012, 260 b.
   
4. 
   Begmatov A. H. Ochilov Z. H. Matematik fizika tenglamalari O’quv- uslubiy majmua Samarqand - 2013, 552 b.
   
5. 
   Бицадзе А. В. Калинеченко Д. Ф. Сборник задач по уравнениям математической физики. М. 1977.312 С.
   
6. 
   Аладъев В. З. Системы компютерной математики: MAPLE : искусство программирования.// М. Лаборатория базовых знаний, 2006,792 с.
   
7. 
   Аладъев В. З. Бойко В. К, Ровба Е. А. Программирование и разработка приложений в Maple .// Городно, Таллин, 2007, 458 с.
   
8. 
   Г оворухин В. Цибулин В. Компютер в математическом исследовании. Учебный курс. Питер, 2001, 624 с.
   

Internet va ZiyoNet saytlari

1. 
   www. lib. homelinex. org/math
   
2. 
   www. eknigu. com/lib/mathematics/
   
3. 
   www. ekingu. com/info/M mathematics/MC
   
4. 
   www. allmath. ru/highermath/
   

5Koshi masalaning yechimining mavjudligi

Bir jinsli Koshi masalasini qaraymiz:

J(1) u = a2u_xx , - да< x <+да, 0 < t < T;

[ (2) u(x,0) = p(x), -да < x < +да.

2. 
   1-chegaraviy masalani yechimini topayotganimizdek bu yerda ham oldin malum bir almashtirishlarni o’tkazamiz. So’ngra esa hosil bo’lgan funksiya yechim ekanligini ko’rsatamiz.
   

v( x, t) = X (x)T (t).