Differensial tenglamalar kafedrasi

к = 0

eval(subs(x=0,U_x [n](t,x)*k=0)); eval(subs(x=L,U[n](t,x)* k=0));

> simplify((C1[n]*sin(1/2*Pi*(1+2*n))+C2[n]*cos(1/2*Pi*(1+2*n)))*exp(- 1/4*Pi^A2*(1+2*n)^A2/L^A2*a^A2*t)*k = 0) assuming n::integer;

2 2 2,

n
Cl (-1)ⁿ e^v


4 L


к = 0

(1 + 2 n) a t

Bundan:

Cl = 0

n

> U[n](t,x):=C2[n]*cos(lambda[n]*x)*exp(-lambda[n]^A2*a^A2*t);

^f л²(1 + 2 n )² a² t¹
4 L²
U⁽^ ^{x) :}= ^C2„H ^{n (1} + \ ^{n) X 1 e}

^f n²(1 + 2 и )² a² t¹

4 L²

U
U⁽t,^{x) :}= X ^C2_n ^co


n = 1


n (1 + 2 n ) x
2L


e

mumiy yechim quyidagi ko’rinishga ega: > U(t,x):=sum(U[n](t,x), n=1..infinity);

>
X C2 co

n


f n (1 + 2 n ) x


n = 1


t


2 L


= F( x )

eval(subs(t=0,U(t,x)=F(x)));

Shuning uchun, C2_n koeffisiyentlar Fur’ye almashtirishi formulalari orqali topiladi:

>

C2

n


F( ^£ )co


ж (1 + 2 n) £

2L


d£

C2[n]:=(2/L)*int(F(xi)*cos(1/2*Pi*(1+2*n)/L*xi), xi=0..L);

"o

Umumiy yechimni yozamiz.

> h1:=0;

sigma:= -h1*h2*(-T2+T1)/(k*h2+k*h1+h2*L*h1);

kappa:= limit((k*h2*T2+k*h1*T1+h2*L*h1*T1)/(k*h2+k*h1+h2*L*h1), h2=infinity);

F(xi):=f(xi)-kappa-sigma*xi;

U(t,x) := sum(C2[n]*cos(1/2*Pi*(1+2*n)/L*x)*exp(- 1/4*Pi^A2*(1+2*n)^A2/L^A2*a^A2*t),n =0..infinity);

hi := 0 a := 0 к := T2

e

F
U( t, x) :=

r


2 2 2 ^ ж (1 + 2 n) a t


\


TO

I

n = 0


2 co


ж (1 + 2 n ) x
2L


4 L


2


L


^L

(f( £ ) - T2 )co


ж (1 + 2 n ) £
2L


d£


У

( £) := f( £) - T2


u( t, x) :=

	V (		f 2 2 2 , ^ ж (1 + 2 n) a t
	2 coS	'ж (1 + 2 n ) x Л	l ^4 l2 J
TO I n = 0		v ^2^L^J C
	L

V

+ T2

rL

(f( £ ) - T2 )co

ЛЛ

ж (1 + 2 n ) £
2L

d£

JJ

Tenglamani yechishga doir misollar [2,5].

1. 
   - Misol.
   

>
d_

dt


u( t, x )


= a


2


f—

_Vdx²


\

u( t, x )

J

restart;

bir jinsli tenglamani birinchi 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
f(x ) := x ^ piecewise! x <


L

"2’


TO, x < L, 0

:= 12

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

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

f( E, ) := E, piecewisef E, <—, TO, E, < L, 0

Yechish uchun yuqorida olingan formuladan (1- chegaraviy shardan) foydalanamiz:

> C2_0 := 2/L*int(T0,xi = 0 .. L/2);

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

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

C2 0 := TO

\
TO


2 co


u( t, x )


II

n = 0 V


л nx

L


e^V


L


2


sin


Л n


TO


л n



n _--> 2 n sin^ n) = 0

n --> 2 n + 1 sin( л n ) = (-1)ⁿ

Tenglamaning yechimi:

(2 2 2 i %(2 n + l)² a² t i

>
, ч ^T0

u( t, x ) := — +


Z


2 (-l)n co/^n+HX , _e


L


V n = 0 V


L


2


T0


% (2 n + l)


d_

dt

u( t, x )

= a

2

f—

_Vdx²

\

u( t, X )
u(t,x):=C2_0/2+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..infinity);