Dalamber usuli (xarakteristik usul) Tor tebranish tenglamasi. Dalamber formulasi

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Dalamber usuli (xarakteristik usul) Tor tebranish tenglamasi. Dalamber formulasi

Erkin tor tebranish tenglamasini qaraymiz. Buning uchun quyidagi bir jinsli tenglama yechiladi

boshlang'ich shartlar bilan

,

.

> restart;

Bir jinsli tenglama va uning yechimi:

> PDE:=diff(u(t,x),t,t)=a^2*diff(u(t,x),x,x);

pdsolve(PDE);

Yechimni Belgilanishi:

> u(t,x):=U1(x-a*t)+U2(x+a*t);

Boshlang'ich shartlar hisoblanishi:

> u_0(x):=subs(t=0,u(t,x))-F(x)=0;

ut_0(x):=subs(t=0,diff(u(t,x),t))-f(x)=0;

> int(diff(-a*U1(xi),xi)+diff(a*U2(xi),xi)-f(xi),xi=0..x);

> -a*(U1(x)-U2(x))+a*(U1(0)-U2(0))=int(f(xi),xi=0..x);

> -a*(U1(x)-U2(x))=int(f(xi),xi=0..x)+a*C;

Boshlang'ich funksiyalarni ifodalanishi:

> solve({U1(x)+U2(x)-F(x)=0,-a*(U1(x)-U2(x)) = int(f(xi),xi = 0 .. x)+a*C}, {U1(x),U2(x)});

> U1(x):=1/2*(a*F(x)-int(f(xi),xi = 0 .. x)-a*C)/a;