Solution to 2D discrete laplacian on a rectangle
$begingroup$
I am attempting to solve the 2D discrete heat equation : Consider a function $f_{i,j}$ with $(i,j)in[0,L+1]^2$. The values of $f_{0,j}$, $f_{i,0}$, $f_{L+1,j}$, $f_{i,L+1}$ are fixed as our boundary conditions. For $(i,j)in[1,L]^2$, $f_{i,j}$ respects begin{equation} 0=f_{i-1,j}-2f_{i,j}+f_{i+1,j}+f_{i,j-1}-2f_{i,j}+f_{i,j+1}end{equation} Is there a simple analytic solution?
boundary-value-problem laplacian discrete-calculus
asked Jan 22 at 19:01
Tony JinTony Jin
183
$endgroup$
add a comment |
$begingroup$
I am attempting to solve the 2D discrete heat equation : Consider a function $f_{i,j}$ with $(i,j)in[0,L+1]^2$. The values of $f_{0,j}$, $f_{i,0}$, $f_{L+1,j}$, $f_{i,L+1}$ are fixed as our boundary conditions. For $(i,j)in[1,L]^2$, $f_{i,j}$ respects begin{equation} 0=f_{i-1,j}-2f_{i,j}+f_{i+1,j}+f_{i,j-1}-2f_{i,j}+f_{i,j+1}end{equation} Is there a simple analytic solution?
boundary-value-problem laplacian discrete-calculus
asked Jan 22 at 19:01
Tony JinTony Jin
183
$endgroup$
1
$begingroup$
How about expanding the solution in terms of the eigenfunctions of the discrete laplacian?
$endgroup$
– Sangchul Lee
Jan 22 at 19:39
add a comment |
$begingroup$
I am attempting to solve the 2D discrete heat equation : Consider a function $f_{i,j}$ with $(i,j)in[0,L+1]^2$. The values of $f_{0,j}$, $f_{i,0}$, $f_{L+1,j}$, $f_{i,L+1}$ are fixed as our boundary conditions. For $(i,j)in[1,L]^2$, $f_{i,j}$ respects begin{equation} 0=f_{i-1,j}-2f_{i,j}+f_{i+1,j}+f_{i,j-1}-2f_{i,j}+f_{i,j+1}end{equation} Is there a simple analytic solution?
boundary-value-problem laplacian discrete-calculus
asked Jan 22 at 19:01
Tony JinTony Jin
183
$endgroup$
I am attempting to solve the 2D discrete heat equation : Consider a function $f_{i,j}$ with $(i,j)in[0,L+1]^2$. The values of $f_{0,j}$, $f_{i,0}$, $f_{L+1,j}$, $f_{i,L+1}$ are fixed as our boundary conditions. For $(i,j)in[1,L]^2$, $f_{i,j}$ respects begin{equation} 0=f_{i-1,j}-2f_{i,j}+f_{i+1,j}+f_{i,j-1}-2f_{i,j}+f_{i,j+1}end{equation} Is there a simple analytic solution?
boundary-value-problem laplacian discrete-calculus
boundary-value-problem laplacian discrete-calculus
asked Jan 22 at 19:01
Tony JinTony Jin
183
asked Jan 22 at 19:01
Tony JinTony Jin
183
asked Jan 22 at 19:01
Tony JinTony Jin
183
asked Jan 22 at 19:01
Tony JinTony Jin
183
asked Jan 22 at 19:01
Tony JinTony Jin
183
183
1
$begingroup$
How about expanding the solution in terms of the eigenfunctions of the discrete laplacian?
$endgroup$
– Sangchul Lee
Jan 22 at 19:39
add a comment |
1
$begingroup$
How about expanding the solution in terms of the eigenfunctions of the discrete laplacian?
$endgroup$
– Sangchul Lee
Jan 22 at 19:39
1
1
$begingroup$
How about expanding the solution in terms of the eigenfunctions of the discrete laplacian?
$endgroup$
– Sangchul Lee
Jan 22 at 19:39
$begingroup$
How about expanding the solution in terms of the eigenfunctions of the discrete laplacian?
$endgroup$
– Sangchul Lee
Jan 22 at 19:39
add a comment |
0
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%2f3083561%2fsolution-to-2d-discrete-laplacian-on-a-rectangle%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
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.
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%2f3083561%2fsolution-to-2d-discrete-laplacian-on-a-rectangle%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
1
$begingroup$
How about expanding the solution in terms of the eigenfunctions of the discrete laplacian?
$endgroup$
– Sangchul Lee
Jan 22 at 19:39