Question

Find 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) using the Fourier sine
transform.

Answer #1

ANSWER:

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

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 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 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 Fourier transform to find the solution of the following
initial boundaryvalue Laplace equations
uxx + uyy = 0, −∞ < x < ∞ 0 < y < a,
u(x, 0) = f(x), u(x, a) = 0, −∞ < x < ∞
u(x, y) → 0 uniformlyiny as|x| → ∞.

(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

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 .

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.

Determine the solution of the following initial boundary-value
problem
Uxx=4Utt 0<x<Pi t>0
U(x,0)=sinx 0<=x<Pi
Ut(x,0)=x 0<=x<Pi
U(0,t)=0 t>=0
U(pi,t)=0 t>=0

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 23 minutes ago

asked 30 minutes ago

asked 35 minutes ago

asked 36 minutes ago

asked 46 minutes ago

asked 49 minutes ago

asked 57 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago