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.

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 .

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).

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 12 minutes ago

asked 13 minutes ago

asked 14 minutes ago

asked 15 minutes ago

asked 20 minutes ago

asked 21 minutes ago

asked 28 minutes ago

asked 29 minutes ago

asked 32 minutes ago

asked 32 minutes ago

asked 1 hour ago

asked 1 hour ago