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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}]

Plot I'm getting

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!

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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}]

enter image description here

For Poisson equation replace PDE == 0 by PDE == f[x,y], where f[x,y] is an arbitrary function.

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  • $\begingroup$ 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 on Plot. $\endgroup$ – zhk 2 hours ago
  • $\begingroup$ @LtGenSpartan - FrameLabel -> Automatic, PlotLegends -> Automatic $\endgroup$ – Bob Hanlon 1 hour ago

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