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

subs(t=0,diff(u(t,x),t));



Tenglama yechimini topishga misollar
1- Misol
> restart;
Bir jinsli tenglamani

quyidagi boshlang'ich shartlar bilan yechish
,
.
bu yerda va funksiyalar quyidagi ko'rinishda berilgan:
> a:=1;L:=1;alpha:=1;
F(x):=x->piecewise(x<-L,0, x<0,alpha*(1+x/L),xL,0);
f(x):=0;





> plot(F(x),-16..16,-0.1..1.1, numpoints=400,color=blue,thickness=3);

Yechish uchun dalamber formulasidan foydalanamiz:
> restart;
> F(x):=x->piecewise(x<-L,0, x<0,alpha*(1+x/L),xL,0);
f(xi):=0;


> u(t,x):=subs(x=x-a*t,1/2*piecewise(x<-L,0, x<0,alpha*(1+x/L),xL,0))+(subs(x=x+a*t,1/2*piecewise(x<-L,0, x<0,alpha*(1+x/L),xL,0)));

Olingan yachimni ikki o'lchamli animirlangan grafik ko'rinishida tasvirlaymiz:
> with(plots):
a:=1;L:=1;alpha:=1;

u1(t,x):=1/2*PIECEWISE([0, x-a*t < -L],[alpha*(1+(x-a*t)/L), x-a*t < 0],[alpha*(1-(x-a*t)/L), x-a*t < L],[0, L < x-a*t]);
u2(t,x):=1/2*PIECEWISE([0, x+a*t < -L],[alpha*(1+(x+a*t)/L), x+a*t < 0],[alpha*(1-(x+a*t)/L), x+a*t < L],[0, L < x+a*t]);
u(t,x):=1/2*PIECEWISE([0, x-a*t < -L],[alpha*(1+(x-a*t)/L), x-a*t < 0],[alpha*(1-(x-a*t)/L), x-a*t < L],[0, L < x-a*t])+1/2*PIECEWISE([0, x+a*t < -L],[alpha*(1+(x+a*t)/L), x+a*t < 0],[alpha*(1-(x+a*t)/L), x+a*t < L],[0, L < x+a*t]);

animate(plot,[u1(t,x),x=-5..5, y=-0.1..1], t=0..4, frames=30,thickness=3, title="u1(t,x)");
animate(plot,[u2(t,x),x=-5..5, y=-0.1..1], t=0..4, frames=30,thickness=3, title="u2(t,x)");
animate(plot,[u(t,x),x=-5..5, y=-0.1..1], t=0..4, frames=30,thickness=3, title="u(t,x)");