Differensial tenglamalar kafedrasi

U( t, x) := У C2 cos| | e


L


2


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


У C2 co

n


n = 0


к nx
L


= F( x )


Shuning uchun C2_n koeffisiyentlar Fur’ye almashtirishi formulalari orqali topiladi: > C2[n]:=(2/L)*int(F(xi)*cos(Pi*n/L*xi), xi=0..L);


C2 := y ⁿ L


Umumiy yechimni yozamiz.

> F(xi):=f(xi);

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

F( £) := f( £)


r


2 2 2 ,

к n a t


U( t, x) := У


n = 0


.к nx

2 cog ——— | e L


L


2


^{f( £ )cog} 1^d£

huningdek, umumiy yechimquyidagi ko’rinishga ega: > u(t,x):=subs(F (xi)=f(xi),U(t,x));

( 2 2 2 Л

к
(к nx

2 cos —-— I e


L


2


u( t, x ) := У


n = 0


L


L


^{f(£ )cog} | ^d£

n a t

2. 
   Sterjen oxirlari har xil rejimda bo’lsin.
   

• 
  restart;
  
• 
  lambda[n]:=Pi*n/L;
  

U[n](t,x):=(C1[n]*sin(lambda[n]*x)+C2[n]*cos(lambda[n]*x))*exp(-

lambda[n]^A2*a^A2*t);

к n

^п' L

( 2 2 2 Л

к п a t

^U
2


L

_n⁽t^{x) :}= (ci_n + C2_n | | e

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

C2 e^v

L

2

k = 0

n

Г 2 2 2 Л

к n a t

(C1 sin(к n) + C2 cos(k n)) e^v

L

2

k = 0

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

2 2 2

к n a t

C2 (-1)ⁿ e^v

n ⁴ '

L

2

k = 0

Bundan:

C2 = 0

n

> U[n](t,x):=C1[n]*sin(lambda[n]*x)*exp(-lambda[n]^A2*a^A2*t);

( 2 2 2. j

к n a t

U (t, x) := C1" sin| ^ |e

L

2

Umumiy yechim quyidagi ko’rinishga ega: > U(t,x):=sum(U[n](t,x), n=0..infinity);

to

U( t, x) := X C1

n = 0

к nx

n ^sin|-^ l^e

L

2

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

X C1 si

n

к nx , _N sin I = F(x)

n = 0

ⁿ \ L

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

>
C1

n


F( ^£ )co


ж n £
L


d£


.. ж nx 2 sin e

L

^{f(£ )coS} I^d£


0


Umumiy yechimni yozamiz.


> F(xi):=f(xi);


U(t,x) := sum(C1[n]*sin(Pi*n/L*x)*exp(-Pi^A2*n^A2/L^A2*a^A2*t),n =0.. infinity);


F( £) := f( £)


c


2


ж


U( t, x ) := X

n = 0

Shuningdek, umumiy yechim quyidagi ko’rinishga ega > u(t,x):=U(t,x);


( 2 2 2 Л

ж n a t


u( t, x) := X


n = 0


ж nx

2 sin e


L


2


L


^{f(£ )coS} I ^d£

3. 
   Sterjen oxirlari har xil rejimda bo’lsin.
   

• 
  restart;
  
• 
  lambda[n]:=1/2*Pi*(1+2*n)/L;
  

U[n](t,x):=(C1[n]*sin(lambda[n]*x)+C2[n]*cos(lambda[n]*x))*exp(-

lambda[n]^A2*a^A2*t);

U_x [n] (t,x) :=diff(U [n](t,x),x);

ж (1 + 2 n )

2L


U (t, x) := \C1


n


. (ж (1 + 2 n)x

^sin[ 2L

+ C2 co


n


ж (1 + 2 n ) x

2L


e


n²(1 + 2 n )² a² t ^
4 L²

C1 [n]:=(2/L)*int(F(xi)*cos(Pi*n/L*xi), xi=0..L);

U_x„ ⁽t, х⁾ :=

„, fn (1 + 2 n) x 1 fn (1 + 2 n) x1

2

^f n²(1 + 2 n )² a² t ¹

!
L


2


L

^C1_n ^cosf 2Z 1 ^{n (1 + 2 n )} ! ^C2_n ^sinf Л 1 ^{n (1 + 2 n )}

4 L

e

^f n²(1 + 2 n )² a² t ¹

4 L²

>

1. 
   C1 _n n (1 + 2 n) e
   

2


к


L


= 0


2 2 2,

n (1 + 2 n) a t


C1 sin ⁿ⁽¹ + ²ⁿ> 1 + C2 co/*⁽¹ + ²ⁿ⁾ I Ie


4 L


n


n


2