Question

Solve the heat equation and find the steady state solution:

u_{xx} = u_{t}, 0 < x < 1, t > 0,
u(0,t) = T_{1}, u(1,t) = T_{2}, where T_{1}
and T2 are distinct
constants, and u(x,0) = 0

Answer #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

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.

In each of Problems 1 through 8, find the steady-state solution
of the heat conduction equation α2uxx = ut that satisfies the given
set of boundary conditions.
1. ux (0, t) = 0, u( L, t) = 0
2. u(0, t) = 0, ux ( L, t) = 0

Solve ut=uxx, 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.

Solve heat equation for the following conditions
ut = kuxx t > 0, 0 < x < ∞
u|t=0 = g(x)
ux|x=0 = h(t)
2. g(x) = 1 if x < 1 and 0 if x ≥ 1
h(t) = 0;
for k = 1/2

Use the Fourier sine transform to derive 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).

Once the temperature in an object reaches a steady state, the
heat equation becomes the Laplace equation. Use separation of
variables to derive the steady-state solution to the heat equation
on the rectangle R = [0, 1] × [0, 1] with the following Dirichlet
boundary conditions: u = 0 on the left and right sides; u = f(x) on
the bottom; u = g(x) on the top. That is, solve uxx +
uyy = 0 subject to u(0, y) =...

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)

Using separation of variables to solve the heat equation, ut =
kuxx on the interval 0 < x < 1 with boundary conditions ux
(0, t ) = 0 and ux (1, t ) = 0, yields the general solution,
∞
u(x,t) = A0 + ?Ane−kλnt cos?nπx? (with λn = n2π2)
n=1DeterminethecoefficientsAn(n=0,1,2,...)whenu(x,0)=f(x)= 0,
1/2≤x<1 .

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