Not sure how to set up the Laplacian/Poisson Equation
$begingroup$
As stated, I am having trouble trying to set up a Laplacian/Poisson Equation. I have boundary conditions with this too, and I've tried using the DirichletCondition function, but I don't know what I'm doing there either. (I have almost zero Mathematica experience, and the Wolfram site's help is just as confusing to me as the program.)
Laplacian[V[x, y], {x, y} == 0;
V[x, 0] == 0;
V[x, 0.05] == 1;
V[0, y] == 0;
V[0.1, y] == 0;]
Plot[{x, -0.25, 0.25}, {y, -0.15, 0.15}]
While I am getting that plot to appear, it's not even close to what I need. What I'm needing is the solution to appear within the region 0 ≤ x ≤ 0.1 and 0 ≤ y ≤ 0.05, as stated by the boundary conditions (rectangular). It's supposed to be a distribution type of plot, kind of like elevation contour graphs and similar.
And for now, the PDE I'm solving is equal to 0, so once I get that done, how do I set up the PDE when it's not equal to 0 (Poisson's Equation)? I would think that since Laplacian is a function, I can't use it anymore since the PDE isn't equal to 0 anymore.
Help is greatly appreciated!
differential-equations
asked 3 hours ago
LtGenSpartanLtGenSpartan
132
New contributor
LtGenSpartan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
As stated, I am having trouble trying to set up a Laplacian/Poisson Equation. I have boundary conditions with this too, and I've tried using the DirichletCondition function, but I don't know what I'm doing there either. (I have almost zero Mathematica experience, and the Wolfram site's help is just as confusing to me as the program.)
Laplacian[V[x, y], {x, y} == 0;
V[x, 0] == 0;
V[x, 0.05] == 1;
V[0, y] == 0;
V[0.1, y] == 0;]
Plot[{x, -0.25, 0.25}, {y, -0.15, 0.15}]
While I am getting that plot to appear, it's not even close to what I need. What I'm needing is the solution to appear within the region 0 ≤ x ≤ 0.1 and 0 ≤ y ≤ 0.05, as stated by the boundary conditions (rectangular). It's supposed to be a distribution type of plot, kind of like elevation contour graphs and similar.
And for now, the PDE I'm solving is equal to 0, so once I get that done, how do I set up the PDE when it's not equal to 0 (Poisson's Equation)? I would think that since Laplacian is a function, I can't use it anymore since the PDE isn't equal to 0 anymore.
Help is greatly appreciated!
differential-equations
asked 3 hours ago
LtGenSpartanLtGenSpartan
132
New contributor
LtGenSpartan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
$begingroup$
As stated, I am having trouble trying to set up a Laplacian/Poisson Equation. I have boundary conditions with this too, and I've tried using the DirichletCondition function, but I don't know what I'm doing there either. (I have almost zero Mathematica experience, and the Wolfram site's help is just as confusing to me as the program.)
Laplacian[V[x, y], {x, y} == 0;
V[x, 0] == 0;
V[x, 0.05] == 1;
V[0, y] == 0;
V[0.1, y] == 0;]
Plot[{x, -0.25, 0.25}, {y, -0.15, 0.15}]
While I am getting that plot to appear, it's not even close to what I need. What I'm needing is the solution to appear within the region 0 ≤ x ≤ 0.1 and 0 ≤ y ≤ 0.05, as stated by the boundary conditions (rectangular). It's supposed to be a distribution type of plot, kind of like elevation contour graphs and similar.
And for now, the PDE I'm solving is equal to 0, so once I get that done, how do I set up the PDE when it's not equal to 0 (Poisson's Equation)? I would think that since Laplacian is a function, I can't use it anymore since the PDE isn't equal to 0 anymore.
Help is greatly appreciated!
differential-equations
asked 3 hours ago
LtGenSpartanLtGenSpartan
132
New contributor
LtGenSpartan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
As stated, I am having trouble trying to set up a Laplacian/Poisson Equation. I have boundary conditions with this too, and I've tried using the DirichletCondition function, but I don't know what I'm doing there either. (I have almost zero Mathematica experience, and the Wolfram site's help is just as confusing to me as the program.)
Laplacian[V[x, y], {x, y} == 0;
V[x, 0] == 0;
V[x, 0.05] == 1;
V[0, y] == 0;
V[0.1, y] == 0;]
Plot[{x, -0.25, 0.25}, {y, -0.15, 0.15}]
While I am getting that plot to appear, it's not even close to what I need. What I'm needing is the solution to appear within the region 0 ≤ x ≤ 0.1 and 0 ≤ y ≤ 0.05, as stated by the boundary conditions (rectangular). It's supposed to be a distribution type of plot, kind of like elevation contour graphs and similar.
And for now, the PDE I'm solving is equal to 0, so once I get that done, how do I set up the PDE when it's not equal to 0 (Poisson's Equation)? I would think that since Laplacian is a function, I can't use it anymore since the PDE isn't equal to 0 anymore.
Help is greatly appreciated!
differential-equations
differential-equations
asked 3 hours ago
LtGenSpartanLtGenSpartan
132
New contributor
LtGenSpartan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 3 hours ago
LtGenSpartanLtGenSpartan
132
New contributor
LtGenSpartan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 3 hours ago
LtGenSpartanLtGenSpartan
132
New contributor
LtGenSpartan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 3 hours ago
LtGenSpartanLtGenSpartan
132
asked 3 hours ago
LtGenSpartanLtGenSpartan
132
132
New contributor
LtGenSpartan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
LtGenSpartan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
LtGenSpartan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
Something like this?
PDE = D[V[x, y], x, x] + D[V[x, y], y, y];
BCs = {DirichletCondition[V[x, y] == 0, y == 0],
DirichletCondition[V[x, y] == 1, y == 0.05],
DirichletCondition[V[x, y] == 0, x == 0],
DirichletCondition[V[x, y] == 0, x == 0.1]};
ufun = NDSolveValue[{PDE == 0, BCs}, V, {x, 0, 0.1}, {y, 0, 0.05}];
ContourPlot[ufun[x, y], {x, 0, 0.1}, {y, 0, 0.05}]
For Poisson equation replace PDE == 0 by PDE == f[x,y], where f[x,y] is an arbitrary function.
answered 3 hours ago
zhkzhk
9,32411433
$endgroup$
$begingroup$
Oh wow, that's exactly what I needed! Also thanks for clearing up the DirichletCondition, the way you did it was much easier than what the Wolfram site had. To make sure I understand the syntax for PDE, i.e does that mean derivative of V with respect to x, and x again (to satisfy a squared partial)?
$endgroup$
– LtGenSpartan
3 hours ago
$begingroup$
@LtGenSpartan Yes!
$endgroup$
– zhk
3 hours ago
$begingroup$
I actually have another question, is there a way to add legends, axes, etc. to this plot?
$endgroup$
– LtGenSpartan
3 hours ago
$begingroup$
@LtGenSpartan Of course you can. Check the documentations onPlot.
$endgroup$
– zhk
3 hours ago
$begingroup$
@LtGenSpartan -FrameLabel -> Automatic, PlotLegends -> Automatic
$endgroup$
– Bob Hanlon
2 hours ago
add a comment |
$begingroup$
DSolve can handle the Lapalace equation if analytic is of any interest. It doesn't do well with Poisson though.
pde = D[V[x, y], x, x] + D[V[x, y], y, y] == 0
bc = {V[x, 0] == 0, V[x, 1/20] == 1, V[0, y] == 0, V[1/10, y] == 0}
DSolve[{pde, bc}, V[x, y], {x, y}] // Flatten;
It gives K[1] as the summation variable. n looks nicer.
% /. K[1] -> n
(*{V[x, y] -> Inactive[Sum][(4 Csch[(n π)/2] Sin[(n π)/2]^2 Sin[10 n π x] Sinh[10 n π y])/(n π), {n, 1, ∞}]}*)
answered 12 mins ago
Bill WattsBill Watts
3,4411620
$endgroup$
add a comment |
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: "387"
};
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: false,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: null,
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
},
onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
LtGenSpartan is a new contributor. Be nice, and check out our Code of Conduct.
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%2fmathematica.stackexchange.com%2fquestions%2f191984%2fnot-sure-how-to-set-up-the-laplacian-poisson-equation%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Something like this?
PDE = D[V[x, y], x, x] + D[V[x, y], y, y];
BCs = {DirichletCondition[V[x, y] == 0, y == 0],
DirichletCondition[V[x, y] == 1, y == 0.05],
DirichletCondition[V[x, y] == 0, x == 0],
DirichletCondition[V[x, y] == 0, x == 0.1]};
ufun = NDSolveValue[{PDE == 0, BCs}, V, {x, 0, 0.1}, {y, 0, 0.05}];
ContourPlot[ufun[x, y], {x, 0, 0.1}, {y, 0, 0.05}]
For Poisson equation replace PDE == 0 by PDE == f[x,y], where f[x,y] is an arbitrary function.
answered 3 hours ago
zhkzhk
9,32411433
$endgroup$
$begingroup$
Oh wow, that's exactly what I needed! Also thanks for clearing up the DirichletCondition, the way you did it was much easier than what the Wolfram site had. To make sure I understand the syntax for PDE, i.e does that mean derivative of V with respect to x, and x again (to satisfy a squared partial)?
$endgroup$
– LtGenSpartan
3 hours ago
$begingroup$
@LtGenSpartan Yes!
$endgroup$
– zhk
3 hours ago
$begingroup$
I actually have another question, is there a way to add legends, axes, etc. to this plot?
$endgroup$
– LtGenSpartan
3 hours ago
$begingroup$
@LtGenSpartan Of course you can. Check the documentations onPlot.
$endgroup$
– zhk
3 hours ago
$begingroup$
@LtGenSpartan -FrameLabel -> Automatic, PlotLegends -> Automatic
$endgroup$
– Bob Hanlon
2 hours ago
add a comment |
$begingroup$
Something like this?
PDE = D[V[x, y], x, x] + D[V[x, y], y, y];
BCs = {DirichletCondition[V[x, y] == 0, y == 0],
DirichletCondition[V[x, y] == 1, y == 0.05],
DirichletCondition[V[x, y] == 0, x == 0],
DirichletCondition[V[x, y] == 0, x == 0.1]};
ufun = NDSolveValue[{PDE == 0, BCs}, V, {x, 0, 0.1}, {y, 0, 0.05}];
ContourPlot[ufun[x, y], {x, 0, 0.1}, {y, 0, 0.05}]
For Poisson equation replace PDE == 0 by PDE == f[x,y], where f[x,y] is an arbitrary function.
answered 3 hours ago
zhkzhk
9,32411433
$endgroup$
$begingroup$
Oh wow, that's exactly what I needed! Also thanks for clearing up the DirichletCondition, the way you did it was much easier than what the Wolfram site had. To make sure I understand the syntax for PDE, i.e does that mean derivative of V with respect to x, and x again (to satisfy a squared partial)?
$endgroup$
– LtGenSpartan
3 hours ago
$begingroup$
@LtGenSpartan Yes!
$endgroup$
– zhk
3 hours ago
$begingroup$
I actually have another question, is there a way to add legends, axes, etc. to this plot?
$endgroup$
– LtGenSpartan
3 hours ago
$begingroup$
@LtGenSpartan Of course you can. Check the documentations onPlot.
$endgroup$
– zhk
3 hours ago
$begingroup$
@LtGenSpartan -FrameLabel -> Automatic, PlotLegends -> Automatic
$endgroup$
– Bob Hanlon
2 hours ago
add a comment |
$begingroup$
Something like this?
PDE = D[V[x, y], x, x] + D[V[x, y], y, y];
BCs = {DirichletCondition[V[x, y] == 0, y == 0],
DirichletCondition[V[x, y] == 1, y == 0.05],
DirichletCondition[V[x, y] == 0, x == 0],
DirichletCondition[V[x, y] == 0, x == 0.1]};
ufun = NDSolveValue[{PDE == 0, BCs}, V, {x, 0, 0.1}, {y, 0, 0.05}];
ContourPlot[ufun[x, y], {x, 0, 0.1}, {y, 0, 0.05}]
For Poisson equation replace PDE == 0 by PDE == f[x,y], where f[x,y] is an arbitrary function.
answered 3 hours ago
zhkzhk
9,32411433
$endgroup$
Something like this?
PDE = D[V[x, y], x, x] + D[V[x, y], y, y];
BCs = {DirichletCondition[V[x, y] == 0, y == 0],
DirichletCondition[V[x, y] == 1, y == 0.05],
DirichletCondition[V[x, y] == 0, x == 0],
DirichletCondition[V[x, y] == 0, x == 0.1]};
ufun = NDSolveValue[{PDE == 0, BCs}, V, {x, 0, 0.1}, {y, 0, 0.05}];
ContourPlot[ufun[x, y], {x, 0, 0.1}, {y, 0, 0.05}]
For Poisson equation replace PDE == 0 by PDE == f[x,y], where f[x,y] is an arbitrary function.
answered 3 hours ago
zhkzhk
9,32411433
edited 3 hours ago
answered 3 hours ago
zhkzhk
9,32411433
answered 3 hours ago
zhkzhk
9,32411433
answered 3 hours ago
zhkzhk
9,32411433
9,32411433
$begingroup$
Oh wow, that's exactly what I needed! Also thanks for clearing up the DirichletCondition, the way you did it was much easier than what the Wolfram site had. To make sure I understand the syntax for PDE, i.e does that mean derivative of V with respect to x, and x again (to satisfy a squared partial)?
$endgroup$
– LtGenSpartan
3 hours ago
$begingroup$
@LtGenSpartan Yes!
$endgroup$
– zhk
3 hours ago
$begingroup$
I actually have another question, is there a way to add legends, axes, etc. to this plot?
$endgroup$
– LtGenSpartan
3 hours ago
$begingroup$
@LtGenSpartan Of course you can. Check the documentations onPlot.
$endgroup$
– zhk
3 hours ago
$begingroup$
@LtGenSpartan -FrameLabel -> Automatic, PlotLegends -> Automatic
$endgroup$
– Bob Hanlon
2 hours ago
add a comment |
$begingroup$
Oh wow, that's exactly what I needed! Also thanks for clearing up the DirichletCondition, the way you did it was much easier than what the Wolfram site had. To make sure I understand the syntax for PDE, i.e does that mean derivative of V with respect to x, and x again (to satisfy a squared partial)?
$endgroup$
– LtGenSpartan
3 hours ago
$begingroup$
@LtGenSpartan Yes!
$endgroup$
– zhk
3 hours ago
$begingroup$
I actually have another question, is there a way to add legends, axes, etc. to this plot?
$endgroup$
– LtGenSpartan
3 hours ago
$begingroup$
@LtGenSpartan Of course you can. Check the documentations onPlot.
$endgroup$
– zhk
3 hours ago
$begingroup$
@LtGenSpartan -FrameLabel -> Automatic, PlotLegends -> Automatic
$endgroup$
– Bob Hanlon
2 hours ago
$begingroup$
Oh wow, that's exactly what I needed! Also thanks for clearing up the DirichletCondition, the way you did it was much easier than what the Wolfram site had. To make sure I understand the syntax for PDE, i.e does that mean derivative of V with respect to x, and x again (to satisfy a squared partial)?
$endgroup$
– LtGenSpartan
3 hours ago
$begingroup$
Oh wow, that's exactly what I needed! Also thanks for clearing up the DirichletCondition, the way you did it was much easier than what the Wolfram site had. To make sure I understand the syntax for PDE, i.e does that mean derivative of V with respect to x, and x again (to satisfy a squared partial)?
$endgroup$
– LtGenSpartan
3 hours ago
$begingroup$
@LtGenSpartan Yes!
$endgroup$
– zhk
3 hours ago
$begingroup$
@LtGenSpartan Yes!
$endgroup$
– zhk
3 hours ago
$begingroup$
I actually have another question, is there a way to add legends, axes, etc. to this plot?
$endgroup$
– LtGenSpartan
3 hours ago
$begingroup$
I actually have another question, is there a way to add legends, axes, etc. to this plot?
$endgroup$
– LtGenSpartan
3 hours ago
$begingroup$
@LtGenSpartan Of course you can. Check the documentations on
Plot.$endgroup$
– zhk
3 hours ago
$begingroup$
@LtGenSpartan Of course you can. Check the documentations on
Plot.$endgroup$
– zhk
3 hours ago
$begingroup$
@LtGenSpartan -
FrameLabel -> Automatic, PlotLegends -> Automatic$endgroup$
– Bob Hanlon
2 hours ago
$begingroup$
@LtGenSpartan -
FrameLabel -> Automatic, PlotLegends -> Automatic$endgroup$
– Bob Hanlon
2 hours ago
add a comment |
$begingroup$
DSolve can handle the Lapalace equation if analytic is of any interest. It doesn't do well with Poisson though.
pde = D[V[x, y], x, x] + D[V[x, y], y, y] == 0
bc = {V[x, 0] == 0, V[x, 1/20] == 1, V[0, y] == 0, V[1/10, y] == 0}
DSolve[{pde, bc}, V[x, y], {x, y}] // Flatten;
It gives K[1] as the summation variable. n looks nicer.
% /. K[1] -> n
(*{V[x, y] -> Inactive[Sum][(4 Csch[(n π)/2] Sin[(n π)/2]^2 Sin[10 n π x] Sinh[10 n π y])/(n π), {n, 1, ∞}]}*)
answered 12 mins ago
Bill WattsBill Watts
3,4411620
$endgroup$
add a comment |
$begingroup$
DSolve can handle the Lapalace equation if analytic is of any interest. It doesn't do well with Poisson though.
pde = D[V[x, y], x, x] + D[V[x, y], y, y] == 0
bc = {V[x, 0] == 0, V[x, 1/20] == 1, V[0, y] == 0, V[1/10, y] == 0}
DSolve[{pde, bc}, V[x, y], {x, y}] // Flatten;
It gives K[1] as the summation variable. n looks nicer.
% /. K[1] -> n
(*{V[x, y] -> Inactive[Sum][(4 Csch[(n π)/2] Sin[(n π)/2]^2 Sin[10 n π x] Sinh[10 n π y])/(n π), {n, 1, ∞}]}*)
answered 12 mins ago
Bill WattsBill Watts
3,4411620
$endgroup$
add a comment |
$begingroup$
DSolve can handle the Lapalace equation if analytic is of any interest. It doesn't do well with Poisson though.
pde = D[V[x, y], x, x] + D[V[x, y], y, y] == 0
bc = {V[x, 0] == 0, V[x, 1/20] == 1, V[0, y] == 0, V[1/10, y] == 0}
DSolve[{pde, bc}, V[x, y], {x, y}] // Flatten;
It gives K[1] as the summation variable. n looks nicer.
% /. K[1] -> n
(*{V[x, y] -> Inactive[Sum][(4 Csch[(n π)/2] Sin[(n π)/2]^2 Sin[10 n π x] Sinh[10 n π y])/(n π), {n, 1, ∞}]}*)
answered 12 mins ago
Bill WattsBill Watts
3,4411620
$endgroup$
DSolve can handle the Lapalace equation if analytic is of any interest. It doesn't do well with Poisson though.
pde = D[V[x, y], x, x] + D[V[x, y], y, y] == 0
bc = {V[x, 0] == 0, V[x, 1/20] == 1, V[0, y] == 0, V[1/10, y] == 0}
DSolve[{pde, bc}, V[x, y], {x, y}] // Flatten;
It gives K[1] as the summation variable. n looks nicer.
% /. K[1] -> n
(*{V[x, y] -> Inactive[Sum][(4 Csch[(n π)/2] Sin[(n π)/2]^2 Sin[10 n π x] Sinh[10 n π y])/(n π), {n, 1, ∞}]}*)
answered 12 mins ago
Bill WattsBill Watts
3,4411620
answered 12 mins ago
Bill WattsBill Watts
3,4411620
answered 12 mins ago
Bill WattsBill Watts
3,4411620
answered 12 mins ago
Bill WattsBill Watts
3,4411620
3,4411620
add a comment |
add a comment |
LtGenSpartan is a new contributor. Be nice, and check out our Code of Conduct.
LtGenSpartan is a new contributor. Be nice, and check out our Code of Conduct.
LtGenSpartan is a new contributor. Be nice, and check out our Code of Conduct.
LtGenSpartan is a new contributor. Be nice, and check out our Code of Conduct.
Thanks for contributing an answer to Mathematica 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%2fmathematica.stackexchange.com%2fquestions%2f191984%2fnot-sure-how-to-set-up-the-laplacian-poisson-equation%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
