Integral Transforms and Partial DEs: Heat Equation of a rod
up vote
0
down vote
favorite
Consider a rod of length $l$ whose lateral surface is insulated. Let its initial temperature be zero, then maintain one of its end faces as at zero temperature and the other at $u_0$.
A. Determine the rod as a function of time.
B. Complete (A) by mode of separation of variables and compare answers.
--
First, I will define some of the equations and the given information.
$frac{d^2u}{dt^2}=frac{1}{alpha}frac{du}{dt}$ where $0 leq x$, $0 <t$, $0leq x leq l$.
$mathscr{L}{frac{d^2u}{dx^2}}=int_{0}^{infty}e^{-st}frac{d^2u}{dx^2}dx$ $=frac{d^2}{du^2}int_{0}^{-infty}e^{-st}u(x,t)dt=frac{d^2 hat{U}(x,s)}{dx^2}$
I believe that this solves A. But how would I approach the solving via seperation of variables?
integration differential-equations fourier-analysis laplace-transform transformation
asked Nov 17 at 22:05
Pascal
15410
add a comment |
up vote
0
down vote
favorite
Consider a rod of length $l$ whose lateral surface is insulated. Let its initial temperature be zero, then maintain one of its end faces as at zero temperature and the other at $u_0$.
A. Determine the rod as a function of time.
B. Complete (A) by mode of separation of variables and compare answers.
--
First, I will define some of the equations and the given information.
$frac{d^2u}{dt^2}=frac{1}{alpha}frac{du}{dt}$ where $0 leq x$, $0 <t$, $0leq x leq l$.
$mathscr{L}{frac{d^2u}{dx^2}}=int_{0}^{infty}e^{-st}frac{d^2u}{dx^2}dx$ $=frac{d^2}{du^2}int_{0}^{-infty}e^{-st}u(x,t)dt=frac{d^2 hat{U}(x,s)}{dx^2}$
I believe that this solves A. But how would I approach the solving via seperation of variables?
integration differential-equations fourier-analysis laplace-transform transformation
asked Nov 17 at 22:05
Pascal
15410
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Consider a rod of length $l$ whose lateral surface is insulated. Let its initial temperature be zero, then maintain one of its end faces as at zero temperature and the other at $u_0$.
A. Determine the rod as a function of time.
B. Complete (A) by mode of separation of variables and compare answers.
--
First, I will define some of the equations and the given information.
$frac{d^2u}{dt^2}=frac{1}{alpha}frac{du}{dt}$ where $0 leq x$, $0 <t$, $0leq x leq l$.
$mathscr{L}{frac{d^2u}{dx^2}}=int_{0}^{infty}e^{-st}frac{d^2u}{dx^2}dx$ $=frac{d^2}{du^2}int_{0}^{-infty}e^{-st}u(x,t)dt=frac{d^2 hat{U}(x,s)}{dx^2}$
I believe that this solves A. But how would I approach the solving via seperation of variables?
integration differential-equations fourier-analysis laplace-transform transformation
asked Nov 17 at 22:05
Pascal
15410
Consider a rod of length $l$ whose lateral surface is insulated. Let its initial temperature be zero, then maintain one of its end faces as at zero temperature and the other at $u_0$.
A. Determine the rod as a function of time.
B. Complete (A) by mode of separation of variables and compare answers.
--
First, I will define some of the equations and the given information.
$frac{d^2u}{dt^2}=frac{1}{alpha}frac{du}{dt}$ where $0 leq x$, $0 <t$, $0leq x leq l$.
$mathscr{L}{frac{d^2u}{dx^2}}=int_{0}^{infty}e^{-st}frac{d^2u}{dx^2}dx$ $=frac{d^2}{du^2}int_{0}^{-infty}e^{-st}u(x,t)dt=frac{d^2 hat{U}(x,s)}{dx^2}$
I believe that this solves A. But how would I approach the solving via seperation of variables?
integration differential-equations fourier-analysis laplace-transform transformation
integration differential-equations fourier-analysis laplace-transform transformation
asked Nov 17 at 22:05
Pascal
15410
asked Nov 17 at 22:05
Pascal
15410
asked Nov 17 at 22:05
Pascal
15410
asked Nov 17 at 22:05
Pascal
15410
asked Nov 17 at 22:05
Pascal
15410
15410
add a comment |
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3002875%2fintegral-transforms-and-partial-des-heat-equation-of-a-rod%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
