Question

Solve u_{t}=u_{xx}, 0 < x < 3, given the
following initial and boundary conditions:

- u(0,t) = u(3,t) = 1

- u(x,0) = 0

Please write clearly and explain your reasoning.

Answer #1

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here we see that Boundary conditions are not homoegeneous...so first we make these conditions homoegeneous by subtracting 1 from u(x,t)

uxx = ut - u (0<x<1, t>0),
boundary conditions: u(1,t)=cost, u(0,t)= 0
initial conditions: u(x,0)= x
i) solve this problem by using the method of separation of
variables. (Please, share the solution step by step)
ii) graphically present two terms(binomial) solutions for
u(x,1).

Solve the heat equation and find the steady state solution :
uxx=ut 0<x<1, t>0,
u(0,t)=T1, u(1,t)=T2, where T1 and T2 are
distinct constants, and u(x,0)=0

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

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

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 heat equation and find the steady state solution:
uxx = ut, 0 < x < 1, t > 0,
u(0,t) = T1, u(1,t) = T2, where T1
and T2 are distinct
constants, and u(x,0) = 0

Please solve the following:
ut=kuxx+sin3πx, 0<x<1, t>0
u(0,t)=u(1,t)=0, t>0
u(x,0)=sinπx, 0<x<1

(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

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) =...

Find the solution formula for the heat equation ut =
c2 uxx on the half-infinite bar (0 ≤ x <
∞) with Dirichlet boundary condition u(0, t) = a, for some constant
a, and initial condition u(x, 0) = f(x) using the Fourier sine
transform.

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