Computational Science Stack Exchange is a question and answer site for scientists using computers to solve scientific problems. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|improve this question
up vote 2 down vote accepted

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.

share|improve this answer

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

share|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.