Dalamber usuli, Dalamber formulasi

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: u0(x):=subs(t=0,u(t,x))-F(x)=0; ut0(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=0x); -a*(U1(x)-U2(x))+a*(U1(0)-U2(0))=int(f(xi),xi=0x); -a*(U1(x)-U2(x))=int(f(xi),xi=0x)+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):=12*(a*F(x)-int(f(xi),xi = 0 x)-a*C)a; U2(x):=12*(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(12*(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):=12*F(x-a*t)+12*F(x+a*t)+12*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)); 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 restart; F(x):=x-piecewise(x ...