Annotasiya toshkent axborot texnologiyalar universitei Farg‘ona filiali Kompyuter injiniringi yo‘nalishi 613-18 gurux talabasi Oltmishev Javohirnig Ushbu malakaviy

animate(plot,[u1(t,x),x=-5..5, y=-7..7], t=0..4, frames=30,thickness=3, title="u1(t,x)");
animate(plot,[u2(t,x),x=-5..5, y=-7..7], t=0..4, frames=30,thickness=3, title="u2(t,x)");
animate(plot,[u(t,x),x=-5..5, y=-7..7], t=0..4, frames=30,thickness=3, title="u(t,x)");
animate(plot,[{u1(t,x),u2(t,x),u(t,x)},x=-5..5, y=-2.2..2.2], t=0..4, frames=30,thickness=3, title="u1(t,x), u2(t,x), u(t,x)");

3.2. Furye usuli (o‘zgaruvchilarni bo‘laklash usuli)

Membranani erkin tebranishini ko‘rib chiqamiz. Buning uchun yuir jinsli tenglamani yechamiz.

Boshlang‘ich shartlari
,
.
va chegaraviy shartlari quyidagicha(membranzakreplena po krayu)
,
> restart;
Bir jinsli tenglama va uni o‘zgaruvchilarni bo‘laklash usuli bilan yechimi:
> PDE:=diff(u(t,r),t,t)=a^2*(diff(u(t,r),r,r)+(1/r)*diff(u(t,r),r));
struc:=pdsolve(PDE,HINT=T(t)*R(r));


> dsolve(diff(T(t),`$`(t,2)) = _c[1]*T(t));
dsolve(diff(R(r),`$`(r,2)) = R(r)/a^2*_c[1]-diff(R(r),r)/r);


Doimiyni o‘zgaruvchilarni bo‘laklanishi bilan almashtiramiz:
= = -
> dsolve(diff(T(t),`$`(t,2)) = -lambda^2*a^2*T(t));
dsolve(diff(R(r),`$`(r,2)) = -lambda^2*R(r)-diff(R(r),r)/r);


Ikkinchi tenglamani yechamiz.

Membranani markazida doimiylik shartini kanoatlantirish kerak..
funksiyasi nolda doimiy:
> BesselJ(0,lambda*r)=series(BesselJ(0,lambda*r), r=0,4);

Funksiyasi nolda singulyar:
> BesselY(0,lambda*r)=series(BesselY(0, lambda*r), r=0,4);

Radial tenglamani umumiy yechimiga quyidagini qo‘yamiz:

Shuning uchun radial tenglamani ko‘rinishi quyidagicha bo‘ladi:

(shuningdek doimiy umumiy yechimda xisobga olingan bo‘ladi.).
Membranani (r = r0) chegaraviy shartini xisobga olamiz:

Bu shartni quyidagi funksiyalar qanoatlantiradi.