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I am not sure if Mathematica is capable of solving integral equations in 2D/3D. I found this page in the documentation, but this is just for 1D.

The following is what I would like to solve, it can considered an electromagnetics problem but that is besides the point. Let the incident field $u_{x_0}^{in}(x)$ by given by a point source at $x_0$:$$ u_{x_0}^{in}(x) = \frac{e^{ik|x-x_0|}}{4\pi|x-x_0|}, $$ Then, I need to find $\varphi \in L^2(\Gamma) $ such that$$ S_\Gamma^k[\varphi](x) = u_{x_0}^{in}(x), \quad \quad \forall x \in \Gamma, $$ where $$ S_\Gamma^k[\varphi](x) := \int_\Gamma \frac{e^{ik|x-y|}}{4\pi|x-y|}\varphi(y) \, \mathrm{dS}(y), $$ and $\Gamma \in \mathbb{R}^3$ is the triangle defined by its vertices as $$\Gamma := \{v_1,v_2,v_3\}, $$ with \begin{align} v_1 & = (4,0,0), \\ v_2 & = (8,0,0), \\ v_1 & = (6,2,0). \end{align}

The problem is in 3D, but the domain of integration is a 2D triangle on the $x$-$y$ plane, with a singular integrand when $x=y$.

Is it possible to solve this problemwith Mathematica?

ManUtdBloke
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    It is possible, but there is no built-in BEM-method, yet. So one has to implement everything by hand. – Henrik Schumacher Mar 06 '19 at 08:19
  • @HenrikSchumacher That's a pity, hopefully they'll implement some BEM routines in the near future! – ManUtdBloke Mar 06 '19 at 09:56
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    @MK Well, integral equations of such type actually do appear in BEM. The problem is to implement the matrix-vector multiplication in an efficient way (e.g., by fast multipole method or hierarchical matrices). Another issue is to find a good iterative solver. I know that multigrid-preconditioned CG-method works exceptionally well here... – Henrik Schumacher Mar 06 '19 at 10:31
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    I actually work at something in that direction but my code is really, really experimental (and so far only for the 2D-case (so integration goes along a curve)). Unfortunately, I cannot find the time at the moment to finish this project. As for the linear solver, this post might be of interest. I also work at a tool for mesh-decimation for producing a multigrid hierarchy; this one is somewhat more mature; maybe I manage to post it on the upcoming weekend (but I would not bet on that). – Henrik Schumacher Mar 06 '19 at 10:32
  • Although it might be related, the posted equation is not BEM. 2. Could you be more precise with your notation: x and y are once used as different points in 3D space and then as Cartesian coordinates of one of them? 3. Your write that \phi is in \Gamma, which does not make sense: \varphi is a function that takes a complex value and elements of \Gamma are points in 3D space. Maybe you mean that the function is defined on this 2D domain? 4. If your source point x_0 is in \Gamma, then you can sole it analytically, namely Dirac delta of y-x_0 is a solution.
    • – MK. Mar 06 '19 at 10:34
  • Could you please elaborate on the physical interpretation of your problem. Due to my geometric understanding of it, it seems that there is no solution for the case if x_0 does not belong to \Gamma.
    • – MK. Mar 06 '19 at 10:35
  • @MK I guess $\mathrm{d} y$ is meant to denote the area element of the surface $\Gamma$; at least is not uncommon to use this notation. Simply read it as $\mathrm{d} \mathscr{H}^2(y)$ where $\mathscr{H}^2$ is the two-dimensional Hausdorff measure. – Henrik Schumacher Mar 06 '19 at 10:42
  • @Henrik Schumacher: I agree that the integral appearing on the LHS of the posted equation is "of BEM type", however the Boundary Element Method is just one way of solving equations involving already some discretization, while the posted equation is a continuous problem. This is what I meant by "they might be somehow related". – MK. Mar 06 '19 at 10:44
  • Let us continue this discussion in chat. – Henrik Schumacher Mar 06 '19 at 10:45
  • @MK 3. I've fixed the $\varphi$ typo, its just and $L^2$ function on the triangle $T$. This question could be classified as BEM for a screen problem. The source is some point $x \in \mathbb{R}^3\setminus \Gamma$. – ManUtdBloke Mar 06 '19 at 13:09
  • @MK The physical interpretation could be acoustic scattering for a screen with Dirichlet boundary conditions. The scattered field is then $u^s(x) = S_\Gamma[\varphi]$. To determine this scattered field we need to solve the given integral equation for $\varphi$. – ManUtdBloke Mar 06 '19 at 13:12
  • I was hoping Mathematica could do evaluate simple some BEM problems on canonical geometry so I can check some my own implementation against it. – ManUtdBloke Mar 06 '19 at 13:13