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

Aim.uz

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;

U2(x):=1/2*(a*F(x)+int(f(xi),xi = 0 .. x)+a*C)/a;


Umumiy yechimni ko'rinishi:
> u(t,x):=simplify(subs(x=x-a*t,U1(x))+(subs(x=x+a*t,U2(x))));

> u(t,x):=collect(1/2*(a*F(x-a*t)-int(f(xi),xi = 0 .. x-a*t)+a*F(x+a*t)+int(f(xi),xi = 0 .. x+a*t))/a, a);

Dalamber formulasi:
> u(t,x):=1/2*F(x-a*t)+1/2*F(x+a*t)+1/2*int(f(xi),xi = x-a*t .. x+a*t)/a;

funksiya tenglamani va boshlang'ich shartlarni qanoatlantirishini tekshiramiz.
Buning uchun funksiyani tenglama va boshlang'ich shartlarga qo'yamiz:
> simplify(diff(u(t,x),`$`(t,2)) - a^2*diff(u(t,x),`$`(x,2)));
eval(subs(t=0,u(t,x)));