Question

solve the wave equation c^2 uxx = u tt for t ≥ 0 0 ≤ x ≤ π . Take c = 1 , take initial and boundary conditions

u(x , 0) = sin x and ut(x , 0) = 0


u( 0 , t) = u ( L , t ) =0 ?

Answer 1

solve the wave equation c^2 uxx = u tt for t ≥ 00 ≤ x ≤ π . Take c = 1 , take initial and boundary conditions
u(x , 0) = sin x and ut(x , 0) = 0
u( 0 , t) = u ( L , t ) =0 ?
U(x,t)=T(t)*X(x)
T''(t)/T(t)=X''(x)/X(x)=b
X''(x)+bX(x)=0
X(0)=0
X(L=π)=0
X(x)=Csin(nx) , b=1
T''(t)+T(t)=0
T(t)=Asin(kt)+Bcos(kt)
U(x,t)=Summa[n,k,{1, Infinity} (Aksin(kt)+Bkcos(kt))Cnsin(nx)]
U(x , 0) = sin x and Ut(x , 0) = 0
U(x , 0) = sin x = Summa[ (Aksin(0)+Bkcos(0))Cnsin(nx)]
BkCn=1 if n=k=1
Ut(x , 0) = 0= Summa[ (Akcos(0)-Bksin(0))Cnsin(nx)]
AkCn=0

Answer: U(x,t)=sin(x)+sin(t)