Interval boundary element method

Interval boundary element method is classical boundary element method with the interval parameters.
Boundary element method is based on the following integral equation

c ⋅ u = ∫ ∂ Ω ( G ∂ u ∂ n − ∂ G ∂ n u ) d S

The exact interval solution on the boundary can be defined in the following way:

u ~ ( x ) = { u ( x , p ) : c ( p ) ⋅ u ( p ) = ∫ ∂ Ω ( G ( p ) ∂ u ( p ) ∂ n − ∂ G ( p ) ∂ n u ( p ) ) d S , p ∈ p ^ }

In practice we are interested in the smallest interval which contain the exact solution set

u ^ ( x ) = h u l l   u ~ ( x ) = h u l l { u ( x , p ) : c ( p ) ⋅ u ( p ) = ∫ ∂ Ω ( G ( p ) ∂ u ( p ) ∂ n − ∂ G ( p ) ∂ n u ( p ) ) d S , p ∈ p ^ }

In similar way it is possible to calculate the interval solution inside the boundary Ω .