Applications
As we have seen, the boundary element method initially emerged as a numerical technique for solving various boundary value problems governed by linear partial differential equations. Earlier applications included scalar potential problems that arise in many areas of physical sciences (e.g. electrostatics, steady state heat conduction, etc.) and vector elasticity problems. After these initial successes, the method was quickly applied to model problems of wave propagation, elastodynamics, fracture mechanics, diffusion, fluid flow, transient heat transfer, plates and shells, thermoelasticity, contact mechanics, etc. Recent advances extended the applicability of the method to problems that include material and geometric nonlinearities and heterogeneities.
The method is only 50 years old, but we no longer need to ask the questions that were so important only 35 yeas ago when it was written
"Today's applied mathematician is much better off in this regard. Because of the electronic computer, he can now look at integralequation formulations of his problems with hope, and increasingly he has found them useful. Here we shall try to answer the questions, "Why are integral equations applicable?" and "In which directions may one reasonably expect to find applications?"
To learn more about the applications of the boundary element method see the following references and the bibliography therein.
References

Cruse, T.A. 1996. BIE fracture mechanics: 25 years of developments. Computational Mechanics 18:111

Lonseth, A. T. 1977. Sources and applications of integral equations. SIAM Review 19:241278
Online sources

http://perso.univrennes1.fr/martin.costabel/publis/Co_PrinciplesBEM.pdf

http://www.infam.tubs.de/infam2/include/Studium/vorlesungen/bemshort.pdf