Not sure how to set up the Laplacian/Poisson Equation












2












$begingroup$


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}]


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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    2












    $begingroup$


    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}]


    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!










    share|improve this question







    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.







    $endgroup$















      2












      2








      2





      $begingroup$


      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}]


      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!










      share|improve this question







      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}]


      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!







      differential-equations






      share|improve this question







      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.











      share|improve this question







      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.









      share|improve this question




      share|improve this question






      New contributor




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      asked 3 hours ago









      LtGenSpartanLtGenSpartan

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      Check out our Code of Conduct.






















          1 Answer
          1






          active

          oldest

          votes


















          4












          $begingroup$

          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}]


          enter image description here



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






          share|improve this answer











          $endgroup$













          • $begingroup$
            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 on Plot.
            $endgroup$
            – zhk
            2 hours ago










          • $begingroup$
            @LtGenSpartan - FrameLabel -> Automatic, PlotLegends -> Automatic
            $endgroup$
            – Bob Hanlon
            2 hours ago











          Your Answer





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          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          4












          $begingroup$

          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}]


          enter image description here



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






          share|improve this answer











          $endgroup$













          • $begingroup$
            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 on Plot.
            $endgroup$
            – zhk
            2 hours ago










          • $begingroup$
            @LtGenSpartan - FrameLabel -> Automatic, PlotLegends -> Automatic
            $endgroup$
            – Bob Hanlon
            2 hours ago
















          4












          $begingroup$

          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}]


          enter image description here



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






          share|improve this answer











          $endgroup$













          • $begingroup$
            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 on Plot.
            $endgroup$
            – zhk
            2 hours ago










          • $begingroup$
            @LtGenSpartan - FrameLabel -> Automatic, PlotLegends -> Automatic
            $endgroup$
            – Bob Hanlon
            2 hours ago














          4












          4








          4





          $begingroup$

          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}]


          enter image description here



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






          share|improve this answer











          $endgroup$



          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}]


          enter image description here



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







          share|improve this answer














          share|improve this answer



          share|improve this answer








          edited 2 hours ago

























          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 on Plot.
            $endgroup$
            – 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












          • $begingroup$
            @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
            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










          LtGenSpartan is a new contributor. Be nice, and check out our Code of Conduct.










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          LtGenSpartan is a new contributor. Be nice, and check out our Code of Conduct.
















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