Question

Solve the wave equation:

utt = c^{2}u_{xx}, 0<x<pi, t>0

u(0,t)=0, u(pi,t)=0, t>0

u(x,0) = sinx, u_{t}(x,0) = sin2x, 0<x<pi

Answer #1

Solve the following wave equation using Fourier Series
a2uxx = utt, 0 < x < pi,
t > 0, u(0,t) = 0 = u(pi,t), u(x,0) = sin2x - sin3x,
ut(x,0) = 0

Solve the following wave equation using Fourier Series
a2uxx = utt, 0 < x < pi,
t > 0, u(0,t) = 0 = u(pi,t), u(x,0) = sinxcosx,
ut(x,0) = x(pi - x)

Solve the wave equation Utt - C^2 Uxx = 0 with initial condtions
:
1) u(x,0) = log (1+x^2), Ut(x,0) = 4+x
2) U(x,0) = x^3 , Ut(x,0) =sinx
(PDE)

Solve the non homogenous wave equation , Utt - c^2Uxx =1 ,
u(x,0) = sin (x) , Ut(x,0) = 1+x
(PDE)

Solve the following wave equation using Fourier Series
a2uxx = utt, 0 < x < L, t
> 0, u(0,t) = 0 = u(L,t), u(x,0) = x(L - x)2,
ut(x,0) = 0

Solve the below boundary value equation
1. Ut=2uxx o<x<pi 0<t
2. u(0,t) = ux(pi,t) 0<t
3. u(x,0) = 1-2x 0<x<pi

Determine the solution of the following initial boundary-value
problem
Uxx=4Utt 0<x<Pi t>0
U(x,0)=sinx 0<=x<Pi
Ut(x,0)=x 0<=x<Pi
U(0,t)=0 t>=0
U(pi,t)=0 t>=0

We have the Problem:
utt-c2uxx=0,x>=0,t>=0
u(x,0)=g(x),x>=0
ut(x,0)=h(x),x>=0
ut(0,t)=αux(0,t),t>=0
u(x,t)=?

Let U(x,t) be the solution of the IBVP:
Utt=4Uxx, x>0, t>0
ICs: U(x,0) = x, Ut(x,0) = 0, x>0
BCs: Ux(0,t) = 0
Find U(4,1) and U(1,2)

1. Solve fully the heat equation problem: ut = 5uxx u(0, t) =
u(1, t) = 0 u(x, 0) = x − x ^3 (Provide all the details of
separation of variables as well as the needed Fourier
expansions.)

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