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 {\displaystyle c\cdot u=\int \limits _{\partial \Omega }\left(G{\frac {\partial u}{\partial n}}-{\frac {\partial G}{\partial n}}u\right)dS}
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 ^ } {\displaystyle {\tilde {u}}(x)=\{u(x,p):c(p)\cdot u(p)=\int \limits _{\partial \Omega }\left(G(p){\frac {\partial u(p)}{\partial n}}-{\frac {\partial G(p)}{\partial n}}u(p)\right)dS,p\in {\hat {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 ^ } {\displaystyle {\hat {u}}(x)=hull\ {\tilde {u}}(x)=hull\{u(x,p):c(p)\cdot u(p)=\int \limits _{\partial \Omega }\left(G(p){\frac {\partial u(p)}{\partial n}}-{\frac {\partial G(p)}{\partial n}}u(p)\right)dS,p\in {\hat {p}}\}}
In similar way it is possible to calculate the interval solution inside the boundary Ω {\displaystyle \Omega } .