Question

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 u_{xx} +
u_{yy} = 0 subject to u(0, y) = u(1, y) = 0, u(x, 0) =
f(x), and u(x, 1) = g(x). You may assume the separation constant is
negative: F''/F = −k, for k > 0. Extra credit: Plot u(x, y) when
f(x) = sin (πx) and g(x) = sin (2πx).

Answer #1

Doubt in this then comment below...i will explain you..

.

**Please thumbs up for this
solution..thanks..**

**.**

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).

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

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.

To write Laplace’s equation, Uxx + Uyy =
0, in polar coordinates, we begin with
Ux = (∂U/∂r)(∂r/∂x) + (∂U/∂θ)(∂θ/∂x)
where r = √(x2+y2), θ = arctan (y/x), x =
r cos θ, y = r sin θ. We get
Ux = (cos θ) Ur – (1/r)(sin θ)
Uθ , Uxx = [∂(Ux)/∂r] (∂r/∂x) +
[∂(Ux)/∂θ](∂θ/∂x)
Carry out this computation, as well as that for Uyy.
Since Uxx and Uyy are both expressed in polar
coordinates, their sum gives Laplace...

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

Use the seperation of variables to find all the solutions on the
form
F(x)G(y) to the boundary value problems
u(x,0)=0 and u(x,1)=0
Where
uxx=uyy

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

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

Consider the one dimensional heat equation with homogeneous
Dirichlet conditions and initial condition:
PDE : ut = k uxx, BC : u(0, t) = u(L, t) = 0, IC : u(x, 0) =
f(x)
a) Suppose k = 0.2, L = 1, and f(x) = 180x(1−x) 2 . Using the
first 10 terms in the series, plot the solution surface and enough
time snapshots to display the dynamics of the solution.
b) What happens to the solution as t →...

7.4 Solve the Laplace equation u = 0 in the square 0 < x, y
< π, subject to the boundary condition u(x, 0) = u(x, π) = 1,
u(0, y) = u(π, y) = 0.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 12 minutes ago

asked 12 minutes ago

asked 13 minutes ago

asked 16 minutes ago

asked 22 minutes ago

asked 22 minutes ago

asked 25 minutes ago

asked 29 minutes ago

asked 34 minutes ago

asked 34 minutes ago

asked 34 minutes ago

asked 34 minutes ago