Question

Solve the following wave equation using Fourier Series

a^{2}u_{xx} = u_{tt}, 0 < x < L, t
> 0, u(0,t) = 0 = u(L,t), u(x,0) = x(L - x)^{2},
u_{t}(x,0) = 0

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 wave equation:
utt = c2uxx, 0<x<pi, t>0
u(0,t)=0, u(pi,t)=0, t>0
u(x,0) = sinx, ut(x,0) = sin2x, 0<x<pi

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

Use the eigenfunction expansion to solve utt = uxx + e −t
sin(3x), 0 < x < π u(x, 0) = sin(x), ut(x, 0) = 0 u(0, t) =
1, u(π, t) = 0.
Your solution should be in the form of Fourier series. Write
down the formulas that determine the coefficients in the Fourier
series but do not evaluate the integrals

(PDE)
WRITE down the solutions to the ff initial boundary problem for
wave equation in the form of Fourier series :
1. Utt = Uxx ; u( t,0) = u(t,phi) = 0 ; u(0,x)=1 , Ut( (0,x) =
0
2. Utt = 4Uxx ; u( t,0) = u(t,1) = 0 ; u(0,x)=x , Ut( (0,x) =
-x

(PDE
Use the method of separation of variables and Fourier series to
solve where m is a real constant
And boundary value prob. Of Klein Gordon eqtn.
Given :
Utt - C^2 Uxx + m^2 U = 0 ,for 0 less than x less pi , t greater
than 0
U (0,t) = u (pi,t) =0 for t greater than 0
U (x,0) = f (x) , Ut (x,0)= g (x) for 0 less than x less than
pj

Solve the heat equation ut = k uxx, 0 < x < L, t >
0
u(0, t) = u(L, t) = 0, t > 0
u(x, 0) = f(x), 0 < x < L
a) f(x) = 6 sin 9πx L
b) f(x) = 1 if 0 < x ≤ L/2 2 if L/2 < x < L

Find a solution u(x, t) of the following problem utt = 2uxx, 0 ≤
x ≤ 2 u(0, t) = u(2, t) = 0 u(x, 0) = 0, ut(x, 0) = sin πx − 2 sin
3πx.

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