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
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LtGenSpartan is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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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.
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Check out our Code of Conduct.
add a comment |
add a comment |
1 Answer
1
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
2 hours ago
$begingroup$
@LtGenSpartan Yes!
$endgroup$
– zhk
2 hours ago
$begingroup$
I actually have another question, is there a way to add legends, axes, etc. to this plot?
$endgroup$
– LtGenSpartan
2 hours ago
$begingroup$
@LtGenSpartan Of course you can. Check the documentations onPlot.
$endgroup$
– zhk
2 hours ago
$begingroup$
@LtGenSpartan -FrameLabel -> Automatic, PlotLegends -> Automatic
$endgroup$
– Bob Hanlon
2 hours ago
add a comment |
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1 Answer
1
active
oldest
votes
1 Answer
1
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
2 hours ago
$begingroup$
@LtGenSpartan Yes!
$endgroup$
– zhk
2 hours ago
$begingroup$
I actually have another question, is there a way to add legends, axes, etc. to this plot?
$endgroup$
– LtGenSpartan
2 hours ago
$begingroup$
@LtGenSpartan Of course you can. Check the documentations onPlot.
$endgroup$
– zhk
2 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
2 hours ago
$begingroup$
@LtGenSpartan Yes!
$endgroup$
– zhk
2 hours ago
$begingroup$
I actually have another question, is there a way to add legends, axes, etc. to this plot?
$endgroup$
– LtGenSpartan
2 hours ago
$begingroup$
@LtGenSpartan Of course you can. Check the documentations onPlot.
$endgroup$
– zhk
2 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
2 hours ago
$begingroup$
@LtGenSpartan Yes!
$endgroup$
– zhk
2 hours ago
$begingroup$
I actually have another question, is there a way to add legends, axes, etc. to this plot?
$endgroup$
– LtGenSpartan
2 hours ago
$begingroup$
@LtGenSpartan Of course you can. Check the documentations onPlot.
$endgroup$
– zhk
2 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
2 hours ago
$begingroup$
@LtGenSpartan Yes!
$endgroup$
– zhk
2 hours ago
$begingroup$
I actually have another question, is there a way to add legends, axes, etc. to this plot?
$endgroup$
– LtGenSpartan
2 hours ago
$begingroup$
@LtGenSpartan Of course you can. Check the documentations onPlot.
$endgroup$
– zhk
2 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
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
2 hours ago
$begingroup$
@LtGenSpartan Yes!
$endgroup$
– zhk
2 hours ago
$begingroup$
@LtGenSpartan Yes!
$endgroup$
– zhk
2 hours ago
$begingroup$
I actually have another question, is there a way to add legends, axes, etc. to this plot?
$endgroup$
– LtGenSpartan
2 hours ago
$begingroup$
I actually have another question, is there a way to add legends, axes, etc. to this plot?
$endgroup$
– LtGenSpartan
2 hours ago
$begingroup$
@LtGenSpartan Of course you can. Check the documentations on
Plot.$endgroup$
– zhk
2 hours ago
$begingroup$
@LtGenSpartan Of course you can. Check the documentations on
Plot.$endgroup$
– zhk
2 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 |
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.
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