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I gain some introductory knowledge from the materials I read. I feel Ok with the numerical implementation part of boundary element method when the integral equation has been formulated. But the concept of fundamental solution confuses me. They appear in the integral equation during the deduction. I wonder what does fundamental solution stand for? I tried to read some materials on bem, but they just don't help. I'm a beginner in BEM, thus I think I need some intuitive explanation, which I could not find in books.

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In the framework of mathematical physics, the fundamental solution is the response of an infinite domain to a point source.

E.g. in electrostatics the electric potential field $\varphi$ satisfies the Poisson equation \begin{equation} \nabla^2 \varphi = -\frac{\rho_f}{\epsilon_0} \end{equation} where $\epsilon_0$ is permittivity and $\rho_f$ is the free charge density (i.e. charge per unit of volume). What happens if we try to obtain the electric potential of a point charge $Q$ at $y$? $\rho_f(x)$ is everywhere zero, except at point $y$, where it becomes infinite—i.e. a Dirac-delta function in the generalized functions sense. The solution to this problem is the well known Coulomb potential \begin{equation} \varphi(r) = \frac{1}{4\pi\epsilon_0}\frac{Q}{r} \end{equation} where $r=\| x-y \|$ is the euclidean distance. And indeed, if we take $Q/\epsilon_0=1$, the Coulomb potential coincides with the fundamental solution of the Laplace/Poisson equation.

The same holds for elasto-statics: the fundamental solution (Kelvin) is the response of the elastic space to a point force; and so on for other interesting equations.

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The fundamental solution is the Green's function of the PDE you are trying to solve.

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In the BIE field usually one distinguishes the fundamental solution (defined by $L \, I(x-y) = \delta(x-y)$ with radiation conditions at infinity) from the Green's function which is the same but with homogeneous conditions at a finite boundary $\partial\Omega$. –  Stefano M Nov 6 '12 at 18:16
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