Question

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.

Answer #1

8. Find the solution of the following PDE:
utt − 9uxx = 0
u(0, t) = u(3π, t) = 0
u(x, 0) = sin(x/3)
ut (x, 0) = 4 sin(x/3) − 6 sin(x)
9. Find the solution of the following PDE:
utt − uxx = 0
u(0, t) = u(1, t) = 0
u(x, 0) = 0
ut(x, 0) = x(1 − x)
10. Find the solution of the following PDE:
(1/2t+1)ut − uxx = 0
u(0,t) = u(π,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)

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

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)=?

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

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

find the solution of the initial value-boundry vaule problem
8uxx=ut 0<x<8 t>=0
u(0,t)=0 u(8,t) = 4
u(x,0) = x

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 following initial/boundary value problem:
∂u(t,x)/∂t = ∂^2u(t,x)/∂x^2 for t>0, 0<x<π,
u(t,0)=u(t,π)=0 for t>0,
u(0,x)=sin^2x for 0≤x≤ π.
if you like, you can use/cite the solution of Fourier sine
series of sin^2(x) on [0,pi] = 1/4-(1/4)cos(2x)
please show all steps and work clearly so I can follow your
logic and learn to solve similar ones myself.

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

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