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-BOB. AMALIY MASALALARNI MATEMATIK PAKETLAR YORDAMIDA SONLI YECHISH Qiziqarli tarixiy masalalar yechimlari



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Oddiy differensial tenglamalarni maple va mathcad matematik pake

3-BOB.


AMALIY MASALALARNI MATEMATIK PAKETLAR YORDAMIDA SONLI YECHISH


    1. Qiziqarli tarixiy masalalar yechimlari





  1. misol (Nyuton misoli) [14, 11-bet]. Nyuton tomonidan o‟rganilgan quyidagi differensial tenglamani y(0)=0 boshlang‟ich shartda darajali qatorlar va Runge- Kutta sonli usuli bilan yeching.

dy 1 3x dx
y x2 xy

Yechimlarning vektor maydonini quring.
Yechish. Avvalo chegaraviy masalani tuzamiz:

    • restart; Order:=10:

de:=diff(y(x),x)=1-3*x+y(x)+x^2+x*y(x); cond:=y(0)=0;

de := d
dx


y( x )


1 3 x


y( x ) x2

y( x )





cond := y( 0 ) 0
Masalaning xususiy yechimini topamiz:

  • dsolve({de,cond},y(x)); y1:=rhs(%);

y( x )


3 e( 1/2 )


e( 1/2 )
erf 2 x
2
erf 2 e
2
2
2
x ( 2 x ) 2
4 e( x
1/2 x2 )
e( x
1/2 x2 ) x 4

Differensial tenglamaning darajali qatorlardagi yechimi:



  • dsolve({de,cond},y(x), series);convert(%,polynom): y2:=rhs(%):




y( x )
x x2 x3 x4 x5 x6 x7
x8 x9
O( x10 )

Ikkala natijani grafiklarda taqqoslaylik (3.1-rasm): p1:=plot(y1,x=-1..2,y=-2..1,thickness=2,color=black): p2:=plot(y2,x=-1..2, y=-2..1,linestyle=3,thickness=2,color=blue):



with(plots): display(p1,p2);


3.1-rasm. Tenglamaning analitik va darajali qator ko‟rinishidagi yechimlarini taqqoslash grafiklari.

Endi tenglamani sonli yechib, yechimlarning vektor maydonini quraylik (3.2- rasm):



  • Eqs:=diff(y(x),x)=1-3*x+y(x)+x^2+x*y(x); icsc:=y(0)=0;

with(DEtools): DEplot(Eqs,y(x),x=-1..2,y=-2..1,{icsc}, linecolor=black,stepsize=0.05,color=black);

Eqs := d
dx
y( x )
1 3 x
y( x ) x2
y( x )



icsc := y( 0 ) 0




3.1-rasm. Tenglamaning sonli yechimi grafigi va yechimlarning vektor maydoni.

Bu misolni har xil chegaraviy shartlarda qaraylik (3.3-rasm):



  • restart;

de:=diff(y(x),x)=1-3*x+y(x)+x^2+x*y(x):
cond:=y(0)=0: dsolve({de,cond},y(x)): y1:=rhs(%):
cond:=y(0)=0.1: dsolve({de,cond},y(x)): y2:=rhs(%):
cond:=y(0)=0.2: dsolve({de,cond},y(x)): y3:=rhs(%):
cond:=y(0)=0.3: dsolve({de,cond},y(x)): y4:=rhs(%): p1:=plot(y1,x=-1..2,y=-2..1,thickness=2,color=black): p2:=plot(y2,x=-1..2,y=-2..1,thickness=2,color=black): p3:=plot(y3,x=-1..2,y=-2..1,thickness=2,color=black): p4:=plot(y4,x=-1..2,y=-2..1,thickness=2,color=black): with(plots): display(p1,p2,p3,p4);

3.3-rasm. Tenglamaning har xil boshlang‟ich shartlardagiyechimlari grafiklari: y(0) = 0; 0.1; 0.2; 0.3.

  1. misol (Kuchsiz maxsuslikka ega bo‟lgan tenglama) [14, 25-bet]. Kuchsiz maxsuslikka ega bo‟lgan quyidagi differensial tenglamanisonli yeching:

dy y ; y(0) = 0
dx
Yechish. x = 0 da yechim maxsuslikka ega, ya‟ni tenglamaning o‟ng tarafidagi f(x,y) funksiya x = 0 da cheksizga intiladi. Tenglamaning q = 2, b = 1 dagi yechimlari vektor maydoni 3.4-rasmda tasvirlangan.

  • de:=diff(y(x),x)=(q+b*x)*y(x)/x; cond:=y(0)=0;

with(DEtools): DEplot(de,y(x),x=-2..2,y=-5..5,{cond}, linecolor=black,stepsize=0.05,color=black);

de := d
dx
y( x )



cond := y( 0 ) 0




a)




b)



3.4-rasm. Tenglamaning a) q = 2, b = 1 va b) q = -1/2, b = 1 dagi yechimlari vektor maydoni.



  1. misol (Eyler tenglama) [14, 44-bet]. x=0 da cheksiz ko‟p yechimga ega bo‟lgan quyidagi differensial tenglamanisonli yeching:

dy y ; y(0) = 0
dx
Yechish. x = 0 da cheksiz ko‟p yechimga ega bo‟lgan tenglamaning berilgan boshlang‟ich shartdagi vector maydoni 3.5-rasmda tasvirlangan:

  • de:=diff(y(x),x)=4*(sign(y(x))*sqrt(abs(y(x)))+max(0,x-

abs(y(x))/x)*cos(Pi*ln(x)/ln(2)));
cond:=y(0)=0; with(DEtools): DEplot(de,y(x),x=0..1,y=-1..1,{cond}, linecolor=black,stepsize=0.05,color=black);

de := d
dx
y( x ) 4
y( x )
4 max
0, x
y( x )
x
cos



cond := y( 0 ) 0


3.4-rasm. Tenglamaning yechimlari vektor maydoni.





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