Wave equation - domain of dependence
up vote
0
down vote
favorite
Let's say I have the following problem on $ - infty < x < infty {rm{ }},{rm{ }}t > 0$ :
$$left{ begin{array}{l}
{u_{tt}} = {u_{xx}}{rm{ }}\
uleft( {x,0} right) = fleft( x right){rm{ }}\
{u_t}left( {x,0} right) = gleft( x right)
end{array} right.$$
D'alembert's formula states that the solution is given by:
$$uleft( {x,t} right) = frac{{fleft( {x + t} right) + fleft( {x - t} right)}}{2} + frac{1}{2}intlimits_{x - t}^{x + t} {gleft( s right)ds} $$
From this we can conclude that in order to calculate just $uleft( {{x_0},{t_0}} right)$ we don't need to know the values of $fleft( x right),gleft( x right)$ on the entire real line but rather only their values on $left[ {{x_0} - {t_0},{x_0} + {t_0}} right]$
My question concerns the following problem:
$$left{ begin{array}{l}
{u_{tt}} + {u_t} = {u_{xx}}{rm{ }}\
uleft( {x,0} right) = fleft( x right){rm{ }}\
{u_t}left( {x,0} right) = gleft( x right)
end{array} right.$$
Which my professor's notes call "the telegraph equation".
The same claim was made here: that in order to calculate just $uleft( {{x_0},{t_0}} right)$ we don't need to know the values of $fleft( x right),gleft( x right)$ on the entire real line but rather only their values on $left[ {{x_0} - {t_0},{x_0} + {t_0}} right]$ but I fail to see the reasoning behind this.
For the first problem we used D'alemebert's formula but now we're not able to.
So how exactly was the same conclusion made?
I realize the two problems have the same principal part and the same characteristic curves but I haven't been able to find a satisfiable answer.
pde wave-equation
asked Nov 24 at 6:39
zokomoko
132214
add a comment |
up vote
0
down vote
favorite
Let's say I have the following problem on $ - infty < x < infty {rm{ }},{rm{ }}t > 0$ :
$$left{ begin{array}{l}
{u_{tt}} = {u_{xx}}{rm{ }}\
uleft( {x,0} right) = fleft( x right){rm{ }}\
{u_t}left( {x,0} right) = gleft( x right)
end{array} right.$$
D'alembert's formula states that the solution is given by:
$$uleft( {x,t} right) = frac{{fleft( {x + t} right) + fleft( {x - t} right)}}{2} + frac{1}{2}intlimits_{x - t}^{x + t} {gleft( s right)ds} $$
From this we can conclude that in order to calculate just $uleft( {{x_0},{t_0}} right)$ we don't need to know the values of $fleft( x right),gleft( x right)$ on the entire real line but rather only their values on $left[ {{x_0} - {t_0},{x_0} + {t_0}} right]$
My question concerns the following problem:
$$left{ begin{array}{l}
{u_{tt}} + {u_t} = {u_{xx}}{rm{ }}\
uleft( {x,0} right) = fleft( x right){rm{ }}\
{u_t}left( {x,0} right) = gleft( x right)
end{array} right.$$
Which my professor's notes call "the telegraph equation".
The same claim was made here: that in order to calculate just $uleft( {{x_0},{t_0}} right)$ we don't need to know the values of $fleft( x right),gleft( x right)$ on the entire real line but rather only their values on $left[ {{x_0} - {t_0},{x_0} + {t_0}} right]$ but I fail to see the reasoning behind this.
For the first problem we used D'alemebert's formula but now we're not able to.
So how exactly was the same conclusion made?
I realize the two problems have the same principal part and the same characteristic curves but I haven't been able to find a satisfiable answer.
pde wave-equation
asked Nov 24 at 6:39
zokomoko
132214
Nothing to add as an answer, just that this is also known as a damper wave equation.
– DaveNine
Nov 26 at 0:38
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Let's say I have the following problem on $ - infty < x < infty {rm{ }},{rm{ }}t > 0$ :
$$left{ begin{array}{l}
{u_{tt}} = {u_{xx}}{rm{ }}\
uleft( {x,0} right) = fleft( x right){rm{ }}\
{u_t}left( {x,0} right) = gleft( x right)
end{array} right.$$
D'alembert's formula states that the solution is given by:
$$uleft( {x,t} right) = frac{{fleft( {x + t} right) + fleft( {x - t} right)}}{2} + frac{1}{2}intlimits_{x - t}^{x + t} {gleft( s right)ds} $$
From this we can conclude that in order to calculate just $uleft( {{x_0},{t_0}} right)$ we don't need to know the values of $fleft( x right),gleft( x right)$ on the entire real line but rather only their values on $left[ {{x_0} - {t_0},{x_0} + {t_0}} right]$
My question concerns the following problem:
$$left{ begin{array}{l}
{u_{tt}} + {u_t} = {u_{xx}}{rm{ }}\
uleft( {x,0} right) = fleft( x right){rm{ }}\
{u_t}left( {x,0} right) = gleft( x right)
end{array} right.$$
Which my professor's notes call "the telegraph equation".
The same claim was made here: that in order to calculate just $uleft( {{x_0},{t_0}} right)$ we don't need to know the values of $fleft( x right),gleft( x right)$ on the entire real line but rather only their values on $left[ {{x_0} - {t_0},{x_0} + {t_0}} right]$ but I fail to see the reasoning behind this.
For the first problem we used D'alemebert's formula but now we're not able to.
So how exactly was the same conclusion made?
I realize the two problems have the same principal part and the same characteristic curves but I haven't been able to find a satisfiable answer.
pde wave-equation
asked Nov 24 at 6:39
zokomoko
132214
Let's say I have the following problem on $ - infty < x < infty {rm{ }},{rm{ }}t > 0$ :
$$left{ begin{array}{l}
{u_{tt}} = {u_{xx}}{rm{ }}\
uleft( {x,0} right) = fleft( x right){rm{ }}\
{u_t}left( {x,0} right) = gleft( x right)
end{array} right.$$
D'alembert's formula states that the solution is given by:
$$uleft( {x,t} right) = frac{{fleft( {x + t} right) + fleft( {x - t} right)}}{2} + frac{1}{2}intlimits_{x - t}^{x + t} {gleft( s right)ds} $$
From this we can conclude that in order to calculate just $uleft( {{x_0},{t_0}} right)$ we don't need to know the values of $fleft( x right),gleft( x right)$ on the entire real line but rather only their values on $left[ {{x_0} - {t_0},{x_0} + {t_0}} right]$
My question concerns the following problem:
$$left{ begin{array}{l}
{u_{tt}} + {u_t} = {u_{xx}}{rm{ }}\
uleft( {x,0} right) = fleft( x right){rm{ }}\
{u_t}left( {x,0} right) = gleft( x right)
end{array} right.$$
Which my professor's notes call "the telegraph equation".
The same claim was made here: that in order to calculate just $uleft( {{x_0},{t_0}} right)$ we don't need to know the values of $fleft( x right),gleft( x right)$ on the entire real line but rather only their values on $left[ {{x_0} - {t_0},{x_0} + {t_0}} right]$ but I fail to see the reasoning behind this.
For the first problem we used D'alemebert's formula but now we're not able to.
So how exactly was the same conclusion made?
I realize the two problems have the same principal part and the same characteristic curves but I haven't been able to find a satisfiable answer.
pde wave-equation
pde wave-equation
asked Nov 24 at 6:39
zokomoko
132214
asked Nov 24 at 6:39
zokomoko
132214
asked Nov 24 at 6:39
zokomoko
132214
asked Nov 24 at 6:39
zokomoko
132214
asked Nov 24 at 6:39
zokomoko
132214
132214
Nothing to add as an answer, just that this is also known as a damper wave equation.
– DaveNine
Nov 26 at 0:38
add a comment |
Nothing to add as an answer, just that this is also known as a damper wave equation.
– DaveNine
Nov 26 at 0:38
Nothing to add as an answer, just that this is also known as a damper wave equation.
– DaveNine
Nov 26 at 0:38
Nothing to add as an answer, just that this is also known as a damper wave equation.
– DaveNine
Nov 26 at 0:38
add a comment |
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
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%2f3011262%2fwave-equation-domain-of-dependence%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
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%2f3011262%2fwave-equation-domain-of-dependence%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
Nothing to add as an answer, just that this is also known as a damper wave equation.
– DaveNine
Nov 26 at 0:38