To derive the BEM, one must replace the partial differential equation that governs the solution in a domain by an equation that governs the solution on the boundary alone. This is particulary difficult tast because there are only certain class of partial differential equations that are feasible. For example the Laplace equation

governing the interior to a domain D bounded by a surface S can be replaced by an integral equation of the form

In operator notation the integral equation (1) can be written in the alternative shorthand notation

In a typical boundary-value problem we may be given f(q), ¶f(q)/ ¶n_q or a combination of such data on S: equation (1) is a means of determining the unknown boundary function(s) from given boundary data.

The integral operators can be viewed as matrices, the boundary functions as vectors.

The application of such a technique transforms the equation (3) to an equation of the form

where the components of the vectors f and v represent the values of the function f(p) and ¶f/ ¶n_q(p) at a set of points on the boundary and the ^s represent their approximations that may arise through measurement of the physical quantities or through numerical solution.

There are a variety of techniques for deriving the system of linear equations from a given integral equation. In general, a method can be derived by replacing the integrals in an integral equation by a quadrature formula or by a weighted residual method such as the Galerkin method.

Domain Solution

On solution of the integral equation the unknown boundary function(s) will be known on S. In most cases a solution in the domain D is required and this can be found using the following equation

The equation (6) is also an outcome of Green's second theorem. The domain solution f(p) for any point p in the domain can be obtained through simply evaluating the integrals in (6).

Direct and Indirect Boundary Integral Equations

There are two fundamental approaches to the derivation of an integral equation formulation of a partial differential equation. The first is often termed the direct method and the integral equations are derived through the application of Green's second theorem. The other method is called the indirect method. This is based on the assumption that the solution can be expressed in terms of a source density function defined on the boundary. For example it is assumed that the solution of the Laplace equation can be written in the form

In operator notation the above integral equations are written

Now we have f(p)  v(p)  in terms of matrices, hence one can able to dicretize the PDE.

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