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I want to define a function $f_c(x,y)$ as the solution of the PDE $\Delta f_c(x,y) = g_c(x,y)$, where $g_c$ is a given function depending on a constant $c$. I tried the following:

domain = Rectangle[{0, 0}, {1, 1}];
solution[c_] := NDSolve[{Laplacian[h[x, y], {x, y}] == x + c*y, DirichletCondition[h[x, y] == 0, True]}, h, {x, y} \[Element] domain];
mem : f[x_?NumericQ, y_?NumericQ, c_?NumericQ] := mem = Part[Evaluate[h[x, y] /. solution[c]], 1];
f[0.1, 0.2, 3]
Plot3D[f[x, y, 3], {x, y} \[Element] domain]

The fourth line returns the approximate value of $f_3(0.1,0.2)$, which to me suggests that everything works. But the plot fails saying NDSolve (...) is neither a list of replacement rules nor a valid dispatch table, and so cannot be used for replacing and a bunch of other similar errors.

I understand the problem probably lies in the way I define $f$, but I don't see how to correct it.

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This is what ParametricNDSolve is for

domain = Rectangle[{0, 0}, {1, 1}];

Create a parametric function in the parameter c:

pfun = ParametricNDSolveValue[{Laplacian[h[x, y], {x, y}] == x + c*y, 
    DirichletCondition[h[x, y] == 0, True]}, 
   h, {x, y} \[Element] domain, c];

Use a specific value for c to create the interpolating function:

fun = pfun[3];
Plot3D[fun[x, y], {x, y} \[Element] domain]

enter image description here

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  • $\begingroup$ Thanks, but I get problems when I want to take g[x_, y_, c_] := Sum[Part[coefficient, i]*g[x, y, i], {i, 1, c}, with c an integer.. Then I get the error The expression i cannot be used as a part specification when trying to evaluate ParametricNDSolveValue. $\endgroup$ Nov 18 '21 at 8:26
  • $\begingroup$ How is you comment related to the question? $\endgroup$
    – user21
    Nov 19 '21 at 6:22
  • $\begingroup$ The function $g_c$ I want to use, uses the parameter $c$ as a summation limit ($x + cy$ was perhaps a wrong MWE). ParametricNDSolveValue does not appear to be able to handle that, hence my question. Is it worthy of a separate post? $\endgroup$ Nov 19 '21 at 10:51
  • $\begingroup$ @BrazilianCérebro, yes, make new post about that. $\endgroup$
    – user21
    Nov 19 '21 at 14:30
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You are making things way too complicated. I could not understand it. I like to keep things simple myself. Why not just do this?

ClearAll[c, x, y, h, sol];

domain = Rectangle[{0, 0}, {1, 1}];

sol[c_?NumericQ] := NDSolve[{Laplacian[h[x, y], {x, y}] == x + c*y, 
    DirichletCondition[h[x, y] == 0, True]}, h, Element[{x, y}, domain]];

Plot3D[Evaluate[h[x, y] /. sol[3]], Element[{x, y}, domain]]

Mathematica graphics

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