Question

Use the Fourier sine transform to derive the solution formula
for the heat equation u_{t} = c^{2} u_{xx}
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).

Answer #1

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.

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 →...

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

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 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

(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

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

(PDE
Use the method of separation of variables and Fourier series to
solve where m is a real constant
And boundary value prob. Of Klein Gordon eqtn.
Given :
Utt - C^2 Uxx + m^2 U = 0 ,for 0 less than x less pi , t greater
than 0
U (0,t) = u (pi,t) =0 for t greater than 0
U (x,0) = f (x) , Ut (x,0)= g (x) for 0 less than x less than
pj

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 .

Let a, c be positive constants and assume that a/ 2πc is a
positive integer. Consider the equation Utt +
aut = c^2Uxx , which represents a damped
version of the wave equation (telegrapher’s equation). Assuming
Dirichlet boundary conditions u(0, t) = u(1, t) = 0, on the
infinite strip 0 ≤ x ≤ 1, t ≥ 0, with initial conditions u(x, 0) =
f(x), ut(x, 0) = 0, complete the following:
(a) Find all separable solutions (of the form...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 11 minutes ago

asked 12 minutes ago

asked 12 minutes ago

asked 12 minutes ago

asked 12 minutes ago

asked 13 minutes ago

asked 16 minutes ago

asked 16 minutes ago

asked 20 minutes ago

asked 38 minutes ago

asked 55 minutes ago

asked 58 minutes ago